history of induction and recursion b

Alexandre BorovikMath40000

History of Mathematical Induction & Recursion [B].

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Introduction:

On the 3rd March 1845, Georg Ferdinand Ludwig Philipp Cantor was born in St Petersburg,

Russia. Cantor would soon move to Germany where he would spend most of the remainder of his

life. Having studied to become an engineer at his fathers’ request, Cantor would turn to

mathematics. Cantor was said to have shown exceptional skill in mathematics whilst at school,

particularly in trigonometry. However, whilst at university in Berlin, Cantor came under the influence

of three great mathematicians – Karl Weierstrass, Leopard Kronecker and Ernst Kummer – and it was

under their influence that Cantor showed an interest in number theory.

Having finished university, Cantor would eventually find a position at the University of Halle

where he worked alongside Heinrich Halle, who challenged Cantor to prove the problem of the

uniqueness of representation of a function as a trigonometric series. With this Cantor turned his

attention from number theory to analysis. Cantor solved the problem within a year of taking up the

challenge. Trigonometric series representation would draw Cantor’s interest away from analysis and

back towards number theory.

In 1873, Cantor produced three ground-breaking results. The first two showed that the

rational numbers and the algebraic numbers were both countable, i.e. could be put into a one-to-

one correspondence with the natural numbers. The third result was to show that the real numbers

could not be put into a one-to-one correspondence with the natural numbers, thus making them

uncountable. The result was very controversial at the time and would lead to a lot of tough years for

Cantor as he tried to justify his result to those who doubted it.

One reason for the ridicule which Cantor’s work received was because of its introduction of

new concepts such as his transfinite numbers into mathematics. Another reason was its level of

abstraction and use of philosophical arguments. With Cantor introduction of new concepts he often

argued extensively from a philosophical point of view, distancing his reasoning from mathematics.

However the main cause of confrontation towards Cantor’s work came from his use of the infinite.

We know that both the natural numbers and the real numbers have no greatest number.

Both are infinite totalities. Cantor’s result that the real numbers were not denumerable essentially

said that the real numbers had a different infinite entity than that of the natural numbers. What

Cantor implied is that they are more than one type of infinity. In order to justify this, Cantor

embarked on a long, enduring journey to develop his theory of transfinite numbers and solve his

Continuum Hypothesis.

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The concept of infinity has always been a difficult one. From the time of the early Greek

mathematicians up until the time 18th Century, infinity was always seen as the potential infinite;

everywhere finite but still without end. This was due to Aristotle who had banned the actual infinite

in the 4th Century B.C. due to its contradictive characteristics. However, in the 19th Century,

philosophers such as Bolzano began to work on the infinite and investigated the paradoxes which it

led to within mathematics. The main digress from the use of infinity in mathematics came from its

paradoxes. Dedekind turned this negative into a positive and was one of the first mathematicians to

challenge the Aristotle stance on the infinite.

The paradoxes that were produced as a result of infinity often stemmed from their

contradictions of laws and definitions that had been constructed only to incorporate the finite.

Dedekind took one such contradiction and used it to define an infinite set. From this he then defined

the natural numbers as one such infinite set. However, before the time of Cantor and Dedekind,

mathematicians had constructed their own procedure of dealing with infinite quantities.

Within mathematics there are a varied collection of mathematical procedures and concepts

which we frequently use. Some we use to create and develop new results, others we use to verify

and prove ones in which we already have. As mathematics has developed with time, it has seen the

introduction of new concepts which have served in helping with such development. These new

concepts are not always perfect in their beginning but have been moulded into more desirable

forms. With regards to handling infinite quantities, the principles of induction and recursion made it

possible to manage the infinite with respect to finite terms.

In this piece of work I wish to investigate the origins of the principles of induction and

recursion and how the two confine the natural numbers to a couple of finite steps. I shall give

account for how mathematical recursion was naturally introduced to mathematics and how it was

used to produce one of the most important results of the philosophy of mathematics by Gödel.

I shall then work my way back, from Pascal who gave structure to the concept of induction,

through those who were the first to realize the benefits and uses of such a principle and showed

traces of the concept throughout their work, using their own similar versions to obtain results, and

mostly proof to results, in short, finite, structured manners. Then I shall address the name of

mathematical induction and how the principle became popular in the 18th and 19th Century’s, paying

particular attention to how beneficial the concept was to Peano and Dedekind.

The principles of induction and recursion are of a major benefit in tackling the totality of the

natural numbers, however, they cannot be applied to Cantor’s transfinite numbers. Thus, I shall give

an account of the origins Cantor’s theory of transfinite numbers, the reasons he felt they were

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needed, the struggle he had for their acceptance, the different paths he took in their development

and the resulting theory he left behind. After Cantor’s last major contribution to the theory in which

he produced, it was taken up by others who tried to solve any problems which Cantor left behind.

The subject would come under the influence of Ernst Zermelo who would add axioms to Cantor’s

naïve set theory. Throughout the development of the theory of transfinite numbers we shall

produce the concepts of transfinite induction and recursion and give an account of them towards

the end.

I expect the reader to be familiar with a basic understanding of set theory and have a

university level of mathematical understanding.

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Contents:Introduction:......................................................................................................................................2

[1] Induction & Recursion:.....................................................................................................................6

[1.1] The Natural Numbers:...............................................................................................................6

[1.2] Mathematical recursion:...........................................................................................................8

[1.3] Gödel’s Theorems:..................................................................................................................12

[1.4] Mathematical Induction:.........................................................................................................16

[1.5] The Origin of Mathematical Induction:...................................................................................20

[1.6] Maurolycus & Gersonides:......................................................................................................25

[1.7] The Name – Mathematical Induction:.....................................................................................30

[1.8] Bibliography:...........................................................................................................................33

[2] Set Theory and Transfinite Numbers:.............................................................................................34

[2.1] Giuseppe Peano:.....................................................................................................................35

[2.2] Richard Dedekind:...................................................................................................................38

[2.3] Arithmetices principia and Was Sollen:...................................................................................43

[2.4] Dedekind’s Interest in Infinity:................................................................................................46

[2.5] Georg Cantor:..........................................................................................................................49

[2.6] Cantor’s Quest for the Transfinite Numbers:..........................................................................54

[2.7] Cantor’s Struggle:....................................................................................................................61

[2.8] Post-Grundlagen:....................................................................................................................65

[2.9] Contributions to the Founding Theory of Transfinite Numbers:.............................................72

[2.2.1] Part I:................................................................................................................................72

[2.2.2] Part II:...............................................................................................................................82

[2.10] Aleph-One and Transfinite Induction:...................................................................................88

[2.11] Post-Beiträge:........................................................................................................................92

[2.12]: Axiomatic Set Theory and Transfinite Recursion:.................................................................94

[2.13] Summary:..............................................................................................................................99

[2.14] Bibliography:.......................................................................................................................103

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[1] Induction & Recursion:

[1.1] The Natural Numbers:

In mathematics, there are numerous things that we take for granted. As mathematics has

progressed through the ages, it has grown and become more abstract. When we say ‘5’ what do we

actually mean by ‘5’; 5 apples, 5 minutes, 5 metres, 5 grains of sand? When we say that ‘ten equals

two fives’ we know that when we have 2 sets/groups/clusters, each of size/quantity/length 5, and

then our total is 10, regardless of any 10 possible concrete examples. Back in the ages of the ancient

Greeks, where mathematics had essentially grown from, philosophers like Plato, Socrates, the

Pythagoreans et al, began to ignore concrete numbers of ‘things’ and took to counting ‘things’

philosophically, with no consideration of what actually counting anything, and that lives on today.

Building upon this, there is much more we take for granted with regards our number

systems e.g. the definitions of odd, even, prime, complex, irrational, integer. And it doesn’t stop at

definitions, we can continue to include basic, essential propositions, (1+1=2 ;4 is an even number;

Pythagoras’ Theorem). There are formulas and summations that we expect any other pier, with

similar levels of mathematical knowledge as ourselves, to know and to believe to be true.

One thing we always use, without any consideration as to why it is true, is the notion that, as

an example, 2<3. Of course this does seem obvious; if one has 2 apples but 3 oranges then of

course one has more oranges than apples, which is true syntactically. But why do we consider the

number 2 to be strictly less than 3? And why in turn is 3 strictly less than 4, 4 less than 5? We

overlook the fact that the natural numbers are in a natural orderingi. Starting from 1, 1 is succeeded

by 2, which is succeeded by 3, and that by 4 and that by 5, then 6, 7, 8…

We obtain any natural number nfrom 1. The Greeks thought of 1 not as a number, but as

‘unity’ or as the ‘One’. Thus any number is a collection of units. We obtain one number by taking its

predecessor and adding 1 to that. But how do we obtain said predecessor? Well, from its

predecessor; and that predecessor from its own predecessor; this in turn has its own predecessor.

Eventually we arrive at 1, which, thinking as the Greeks do, is where we can go no further with the

natural numbers.

The natural numbers are a simple concept, beginning from 1, we add 1 to it n−1 times to

obtain the natural number n. Then adding 1 to n we obtain n’s successor, n+1. Thus every natural

number has a successor; each depends on the one defined before it. This latter notion of defining

something in terms of how we define another brings us to mathematical recursion.

i In fact they are ‘a well ordering’ and we shall come to that later.7


According to the Oxford dictionary, recursion is the process of ‘repeated application of a

rule, definition, or procedure to successive results’. We see resemblances of recursion in everyday

life, e.g. the Russian Matryoshka Dolls, The Droste Effect; however we mainly use recursion in

mathematics in order to define such things as sequences, series, relations and functions. We can use

recursion to define sets, or to define unions and intersections of sets. And there are many more

uses of recursion in different areas of mathematics - logic, statistics, graph theory…

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[1.2] Mathematical recursion:

Mathematical recursion is the process of defining a mathematical process by repetition; a

function or procedure defined in terms of itself. We define the natural numbers by recursive

definition: starting from 1, add 1 to obtain the next natural number. From this new number, namely

2, add 1 to it to obtain the next natural number, 3. From 3, we add 1 to obtain 4 . Etc.

Before we start to look further into mathematical

recursion I would like to look at a well-known and common

example, the Fibonacci Numbers, attributed to Leonardo Pisano

Bigollo (Leonardo Fibonacci) [ca.1170- ca.1250] of Pisa. The son of

a wealthy business man, Fibonacci had an extended interest in the

mathematics of the East and the Arabs. In 1202, after returning to

Italy having visited Egypt, Sicily, Greece and Syria, Fibonacci

published Liber abaci, a text mainly focusing on the base 10

arithmetic of al-Khwārizmī and Abū Kāmil. In Liber abaci appears

the Fibonacci Numbers.

The Fibonacci Numbers are defined as:

Define F1=F2=1;

For the natural number n>2,define Fn=Fn−1+F (n−2 ).

Thus the 1st few terms of this sequence are 1 ,1 ,2 ,3 ,5 ,8 ,13 ,21 ,34 ,55 ,… .

This process for defining a structure through an arbitrary number is what is known as

Recursive Definition in mathematics. Notice that we have essentially 2 parts to the definition of the

Fibonacci Numbers:

(1) A starting point, indexed by the natural numbers 1 & 2;

(2) A rule for the formulation of greater Fibonacci Numbers, indexed by the naturals greater

than 2. The rule corresponds to the 2 previously defined Fibonacci numbers.

Generally, to define a mathematical procedure recursively, we:

(1) assign a Base Case;

(2) set up a Recursive Step.

The Base Case serves as our starting point. 1 being the base case of the natural umbers; F1

and F2 serve as the base case of the Fibonacci Numbers. The base case is our reference point from

where we can continue, to formulate the remainder of mathematical procedure. And we do this

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Figure 1: Leonardo Fibonacci.


from our Recursive Step. The recursive step allows us to continue to

formulate more examples of a procedure; it extends our definition

towards a possible infinite number of terms for our procedure. The

recursive step allows us to describe an infinite number of instances

in a finite quantity and the natural number n holds the infinite

factor.

We can think of a recursive definition as a sequence,

u1 ,u2 , u3 , u4 ,…,un ,… indexed by the natural numbers. This allows

us to visualize the sequence in an order parallel to the naturals and

see the sequence as being a countable set of terms. We can draw up

results, formulas, ratios etc. between the natural number n and the

term un of our sequence or even define the our un in terms of the

natural number n. An example of the latter is the factorial function,

which we know to be n !=∏1

n

k.

However we can define the factorial function recursively as follows:

Base Case: let 1 !=1 ;

Recursive Step: define, for ¿1 , n!=n ∙ (n−1 )! .

Now if we were to rename our factorial function in the structure of a sequence, then we

could look at the previous definition as:

Base Case: let u1=1 ,

Recursive Step: define, for n>1 , un=n∙un−1

Thus the 1st few terms of the sequence are: u1=1 , u2=2 , u3=6 ,u4=24 ,u5=120...

The Degree of recursion is said to be the number of predecessors that are used in defining

any term by the recursive step i.e. it is the number of terms defined in the base case.

Looking back at examples previously defined, the Fibonacci numbers are of degree 2

whereas the factorial function is of degree 1. Beginning with the 1st 3 Fibonacci numbers as a new

base case, we can define the Tribonacci Numbers by taking the sum of the previous 3 defined

numbers as opposed to 2. Thus the Tribonacci Numbers are of degree 3. Similarly we can define the

Fibonacci n-step Number Sequence which will be of degree n.

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Figure 2: The Sierpinski Triangle – showing iterations of the Recursive Step.


Now that we have looked at the degree of recursion I give a standard procedure for

definition of recursive definition:

1. Base Case: let u1=n1 , u2=n2 ,…,um=nm where m is the degree of recursion for this

definition and n1 , n2 ,… ,nm are m pre-defined terms.

2. Recursive Step: let um+1= f (u1 , u2 ,…um ) where f is an m-dimensional function or relation.

Sets are another important entity in mathematics that can be defined by recursion; we can

define N , the set of natural numbers, recursively in the following manner:

Base case: 1∈N ;

Recursive Step: if n∈N , then n+1∈N ;

Extremal Clause: ℕ is the smallest set satisfying these conditions.

There are other properties that accompany our set N of the natural numbers, namely the

Peano-Dedekind Axioms; however if we look solely at N as a set of elements, without any regards to

the ordering, then we can define N recursively as above. Other examples of sets that we can define

by recursion are the positive even integers, the integers, the set of triangular numbers, and the set

of squared integers. In set theory all of these examples would be accompanied with extra criteria

with regards their ordering, but for now I choose to overlook this as we are still just looking at sets

with specific elements.

Notice that I have added another piece of criteria to the definition of ℕ, the Extremal

Clause. We do this to distinguish between 2 different sets which may satisfy both the Base Case and

the Recursive Step but yet may still be 2 different sets. For example, the set

{1 ,1.5 ,2 ,2.5 ,3 ,3.5 ,4 ,…} satisfies the Base Case and Recursive Step afore mentioned yet it is not

the set of natural numbers as it includes some rational numbers which are not ‘whole’ numbers.

Taking a (possibly infinite) collection of sets, we can define recursively their unions,

intersections and Cartesian cross products. This is possible because of the associativity of these

actions. For example, suppose we have the collection of sets S1 , S2 , S3 ,…Sn ,… where the indices

are randomly assigned to the sets of the collection, then

a) The union can be defined as:

¿ i=1 ¿1Si=S i;

¿ i=1 ¿n+1S i=(¿i=1¿nS i)Sn+1.

b) The intersection can be defined as:

¿ i=1 ¿1Si=S1;

¿ i=1 ¿n+1S❑i=(¿ i=1¿nSi)Sn+1.

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c) The Cartesian cross product can be defined as:

∏i=1

1

S i=S1;

∏i=1

n+1

S i=¿¿ .

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[1.3] Gödel’s Theorems:

Mathematical recursion is a tool used throughout mathematics; a few examples we have

seen. It can be used in every area of mathematics, from logic to geometry, from statistics to graph

theory. Recursion seems to be a natural process of mathematics, one without any origin. It appears

to be a tool for building structures such that we can produce theorems, propositions, or algorithms

based upon these structures.

Recursive definition seems to follow from the natural numbers since we use the naturals to

structure this form of definition. As with any definition in mathematics, we can just make any

assumption or assign a certain criteria or a specific procedure; the truth of which we just assume to

hold. We can do this because we are more interested in the results that we can produce from these

assumptions than the assumptions themselves. We use the definition as our point of reference from

which we aim to build upon, to explore and investigate. There is no requirement to verify that our

definition is true, the truth value follows from the natural numbers and the axioms exhibited there.

The origin of recursion therefore cannot be pinned

down to one specific point in time. The Fibonacci numbers

were developed by Fibonacci in the 12th century but they were

said to be known to the Indians before that. Fibonacci was also

known to have read and studied a lot of Indian and Arabic text

in his time, traces of which can be found in Liber abaci. Each

row of Pascal’s Triangle, used to define binomial coefficients

(which can themselves be recursively defined), can be

recursively defined by the row above it. However, although it

is named after Blaise Pascal who lived during the 17th century,

‘his’ triangle was investigated by the Geeks, Chinese, Hindu

and Arabic mathematicians before him.

The origin of mathematical recursion can be hard to

trace but its popularity and importance is well known. One

such important result to come from recursion is that of Kurt

Gödel.

Born 1906 in Brno, now of the Czech Republic, Gödel was a philosopher as well as a

mathematician who focused mainly on the logical aspects of mathematics. With the Czechoslovak

Republic declaring its independence from the Austro-Hungarian Empire in 1918, Gödel moved to

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Figure 3 - A young Kurt Gödel ca. 1922.


Vienna in 1924 to study at university, having always considered himself to be an Austrian living in a

Czechoslovak majority.

In 1931, Gödel had published the paper ‘Über formal unentscheidbare Sätze der Principia

Mathematica und verwandter Systeme I’ which can be translated as ‘On Formally Undecidable

Propositions of Principa Mathematica and Related Systems I’. The paper, originally appearing in

‘Monatshefte der Mathematik und Physik’ Vol.38, was of significant importance to mathematical

logic and philosophy as it contained Gödel’s highly important Incompleteness Theorems.

The theorems were used to answer the 2nd problem of David Hilbert’s “bootstrapping”

Program which asked “to prove that they (the axioms of arithmetic) are not contradictory, that is,

that a finite number of logical steps based upon them can never lead to contradictory results .”

The axioms of arithmetic that Hilbert is referring to are the axioms of the Peano Arithmetic

for the set of natural numbers, N , which appeared in Peano’s 1889 book, The Principles of

Arithmetic by a New Method:i

1. 1∈N ;

2. a∈N ,∃a'∈N s. t . a'=a+1;

3. ∄a∈N s. t . a+1=1;

4. a ,b∈N ,a=b⇔a+1=b+1;

5. For M⊂N , if

1∈N ;

a∈N⇒ a+1∈N ;

thenM=N .

In effect, the problem is asking – is there any proof that this arithmetic is consistent i.e. yield

no contradictions? Gödel used his theorems to prove that the answer to this question was in fact,

no. Hilbert’s Program wanted to secure the foundations of mathematics and then develop its

foundations further. Hilbert wanted to find out what lay in store for the future generations of

mathematicians, what possible new techniques they would use, and most importantly, what results

would they yield. But some thought Gödel’s answer sought to destroy such a possibility, whereas

Gödel himself saw it as a route to develop further Hilbert’s work.

i I only give 5 of the original 9 axioms as they are sufficient to describe the Peano Arithmetic. The remaining 4 axioms deal with the transitive, reflexive and symmetric properties of equality in the natural numbers.

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The Incompleteness Theorems:

1. Any effectively generated theory capable of expressing elementary arithmetic cannot be

both consistent and complete. In particular, for any consistent, effectively generated

formal theory that proves certain basic arithmetic truths, there is an arithmetical statement

that is true, but not provable in the theory.

2. Any formal effectively generated theory that includes basic arithmetical truths and also,

certain truths about formal provability, if the theory includes a statement of its own

consistency, then the theory is inconsistent.

Gödel's paper consisted of 11 propositions. Propositions VI and XI are now what are called

the Incompleteness Theorems and it is proposition XI, the 2nd of the Incompleteness theorems, that

answers Hilbert’s problem. Once Gödel has established the 1st of the 2 Incompleteness Theorems he

proceeds to build upon it to produce the 2nd.

First I state Proposition V:

“Every recursive relation is definable in the system P (interpreted as to content),

regardless of what interpretation is given to the formulae of P.”

After proving Proposition V, Gödel states Proposition VI. He does so in this order as he

requires the recursive element of Proposition V to prove VI. The 1962 English translation, by B.

Meltzer, of Gödel’s paper is introduced by Richard B Braithwaite, who highlights the importance of

recursion in mathematics and the importance it had for Gödel in his piece of work:

“Recursive definition enables every number in a recursively defined infinite sequence to

be constructed according to a rule, so that a remark about the infinite sequence can be

constructed as a remark about the rule of construction and not as a remark about a

given infinite totality.”

“For the proof of Gödel’s ‘Unprovability’ theorem the importance of recursiveness lies in

the fact (Proposition V) that every statement of a recursive relationship holding between

given numbers x1, x2 ,…, xn is expressible by a formula f of the formal system P which

is ‘provable’ within P if the statement is true and ‘disprovable’ within P ...if the

statement is false.”

Recursion remains an essential and valuable asset throughout mathematics. The ability to

represent the infinite in the terms of the finite allows for an efficient and more abstract practice of

mathematics. Its quick and effective method has been known throughout the history of

mathematics, from the time of the ancient Greeks, through to the Renaissance and onwards to

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today, where it remains in common use, from mathematics used in primary school, to the highest

and most intellectual of levels of mathematics. The main purpose of mathematical recursion is for

stating definitions or defining functions, sequences, series, etc. However, as mathematics became

more abstract and ever closer to logic, the need for mathematical proof became greater. Through

this, a close relative of mathematical recursion developed; that of mathematical induction.

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[1.4] Mathematical Induction:

“Even in mathematical sciences, our principal instruments to discover the truth are

induction and analogy.”

– Pierre-Simon Laplace, Essai Philosophique sur les Probabilités.

Mathematical induction, often referred to as The Principle of Induction, has close ties to

mathematical recursion; if recursion is the process of building in mathematics, then induction is the

process of checking that build. Essentially induction is a form of proof for mathematical procedures

that are defined recursively in mathematics. Alternatively, once we have defined a procedure

recursively, we can check its validity by induction. Mathematical induction is not a method of

discovery, but a method of proving that which has already been discovered. In arithmetic induction

proves that some property holds for all positive integers. In logic it proves that that a property holds

for a language based upon the length of a sentence. We can define sets recursively, and prove

properties about these recursively defined sets by induction.

Mathematical induction bears a close resemblance to the induction used in the other

sciences. The inductive methods of other sciences look at generalities from specific examples in

order to formulate a general and common conjecture that can be put forward and used in other

cases to test its strength. Similarly in mathematics we can usually see that if property holds or does

not hold for arbitrary cases. But this is not proof that that property does in fact hold for every case

but it is an indicator that it might just do so.

Where mathematical induction differs is in the fact that one case depends upon another due

to the fact that we have recursively defined our procedure, and that is where we form the ‘proof’

aspect. If we find a certain property associated to a certain case then we look to the next case that

follows to see if the same property holds in that case as well. However we usually need a starting

point, so we prove that a property holds for the most trivial of cases first. From that we look at the

next case after it. If the property holds for that case as well, then we look at the succeeding case and

see if the property still holds. And so on. But can we test that a property holds for a structure of a

vast size, or a set of infinite cardinality? It would be very time consuming to check if it is true that

1+2+3+4+…+n=n (n+1 )2

for n=12 orn=12222.

In most cases we use the truth which we install in the axioms for the natural numbers. This

allows us to prove that a property holds for all the natural numbersn without having to inspect each

17


case one by one. But in other cases, as with sets or logic, we use our recursive definitions. These

recursive definitions do have a starting point, namely their base cases, which we have previously

seen. And we know the natural numbers have the base case 1 i. Mathematical induction has the

same structure; a base case and the step that passes from one case to another, although we refer to

this as the Inductive Step as it is slightly different from our recursive step. It is this Inductive Step that

proves that a property holds for every case in the structure.

We can state The Principle of Induction for the natural numbers in the following axiom:

Suppose that P (n ) is a statement involving a general natural numbern. Then P(n) is

true for all for the natural numbers if;

1. P (1 )is true, and

2. P (k )⇒P (k+1 )for all natural numbers k .

Notice the comparison of this axiom and the 5th Peano axiom stated in section [1.3]. They

are, albeit slightly worded differently, the same. The Principle of Induction is an extension of the

algebraic and orderii axioms that we have for the natural numbers. Induction is merely an extension

of something we often overlook about the natural numbers; starting from 1, we can reach any other

natural number by simply adding 1.

The statement P (1 )is the Base Case of The principle of induction, whereas our second

statement is the Inductive Step. Notice that there is an implication in the inductive step. So, if P (k )is

true, then P (k+1 )is also true. We may think that this means that we have to prove that P (k )is true

and thus it follows that P (k+1 )is true also, but what we have to do is show that – if it was the case

that P (k )was true, then from this P (k+1 )is also true. This is where we get our Inductive Hypothesis;

we assume, as a hypothesis, that P (k )is true and under that assumption we prove that P (k+1 ) is

true.

The concept of mathematical induction is – for a general propertyP (n ) of a natural number

n, whatever it may be, the base case checks that P (1 ) is true and then, by the inductive hypothesis,

P (1 )⇒P(2), and again by the inductive hypothesis P (2 )⇒P (3 ) and thus

P (3 )⇒ P ( 4 )⇒P (5 )⇒…⇒P (n )⇒P (n+1 )⇒…

Now I would like to give an example of proof by induction to illustrate how we use the

inductive hypothesis to prove the inductive step. I shall prove that the following statement holds:

i In some cases when using mathematical induction, we can assume that 0∈N as a property may hold for the case when n=0. ii I have not mentioned the ordering axioms at this point but they will appear in a later section.

18


For all natural numbers n, the number n2+nis even.

So P (n )is the statement ‘n2+n is even’. Thus we are looking for a natural number m such

that 2m=n2+n. We proceed now with the structure laid out in the induction axiom before.

Base case: When n=1 ,n2+n=1+1=2thus we have verified that P (1 )is true.

Inductive Step: Assume now for an Inductive Hypothesis that, for an arbitrary k , ‘k 2+k is

even’, then we seek to show that (k+1 )2+( k+1 )=2 p for some natural number p.

From our inductive hypothesis we can assume further that there exists q belonging to

the natural numbers such that, 2q=k2+k . However

(k+1 )2+( k+1 )=k 2+2k+1+k+1=(k2+k )+2k+2=2q+2k+2=2(q+k+1). So if we

take p=q+k+1 then we have a natural number p such that 2 p=(k+1 )2+( k+1 ) as

required. ∎

Notice that, from the assumption that P(k) is true for an arbitrary k , we were able to prove

that P(k+1) was also true by manipulating what we knew about P (k+1 ) to reach a trivial

conclusion based on P (k )’s assumption and general arithmetic.

We do not necessarily have to start with n=1 for the base case, there are certain functions

or theorems that only satisfy ‘for n>n0’ for some certain natural number n0i.e. P (n ) is true for all

natural numbers n≥n0 . If this is the case we just need to make 2 slight adjustments to our usual

procedure – the base case is changed to P (n0 ) and the inductive step no longer applies to all natural

numbers k , but only for k ≥n0 . Such an example of this can be seen when comparing n2 and 2n.

When n=1,2,3 we have n2>2n. However, for n≥4, we have n2≤2n. So in proving that the latter

holds, we use induction with the base case for n=4. Then for he inductive step we prove

k 2≤2k⇒ (k+1 )2≤2k+ 1 for all k ≥4.

Another variation of mathematical induction is the method of Strong induction. We require

this method when we have the case that P (k ) alone is not enough to imply that P (k+1 ) is holds but

we actually require some, or maybe even all, of P (1 ) , P (2 ) ,…, P (k−1 ) also.

The axiom of Strong Induction is given as:

Suppose that P (n ) is a statement for some natural number n. Then P(n) is true for all

natural numbers n if:

1. P (1 ) is true, and

2. P(n) holds for all n≤k⇒P(k+1) holds for all such k .

19


If anything this is just a stronger version of the axiom of induction, as all statements proven

by our standard induction can also be proven by strong induction. We may want to use strong

induction to prove that n !=n ∙ (n−1 ) ! .

20


In Peter Eccles An Introduction to Mathematical Reasoning, he gives the following analogy of

The Principle of Induction which really sums up its strength:

Suppose that we think of the integers lined up like dominos. The inductive step tells us

that they are close enough for each domino to knock over the next one, the base case

tells us that the first domino falls over, the conclusion is that they all fall over.

Once it is known that the dominos are so close together in order to knock another over, then

there is no need to push each one over, and we just need to knock over the first. Induction with the

natural numbers is the same, once we have seen that P (n ) shows that P (n+1 ) holds, then all that is

required is to show that the base case is true.

Mathematical induction developed around the natural numbers and their ordering. Then,

induction was applied to other areas of mathematics where objects can be put into an ordering, to

prove properties and their theorems, and to tackle the problem of representing the infinite in these

areas – we have seen that we can define an infinite collection of sets, so we may use induction to

show the property of inclusion amongst these sets, based on their cardinality. In logic we can prove

a theorem for a language by proof by induction on the length the sentences belonging to this

language.

Mathematical induction today is an essential tool within every area of mathematics.

Induction allows us to represent the infinite with the finite. It removes all the complexity and

longevity associated with proofs concerning the infinite. It is a mathematical tool that has grown in

importance until today. In proving theorems, propositions, lemmas, whatever it may be, we have a

certain group of techniques in mathematics that we can use, and induction is one of these; its

importance is unrivalled.

21


[1.5] The Origin of Mathematical Induction:

As I have said, mathematical induction is of great importance within mathematics, but

where did the principle originate from? Who was the 1st person to develop the principle? Who was

the first to realize its importance? Who even, was the first to name the principle mathematical

induction? What we already know at this point is that induction was developed from the natural

numbers as that is the basis of its structure; the ordering in which the natural numbers exhibits. The

exact origin of the principle of mathematical induction is unclear and it could be debated to be

attributed to many different great mathematicians or to various different periods within the history

of mathematics.

We know that the method bears a close resemblance to the inductive processes of other

sciences which tend to be based on observations and to a degree, that is one way in which

mathematical induction was developed. Induction in mathematics is generally a method of proof; we

do not just develop any old procedure and set out to prove its validity. Instead we look at patterns

from examples and try to develop a natural property. Once we have this property, we use induction

to see if it is true. This can be seen in Mathematics and Plausible Reasoning, Vol. 1 – Induction and

Analogy in Mathematics (1954) by George Pólyai:

Wanting to develop a formula for the sum of the squares of the 1st nnatural

numbers, Pólya compares the ratio of ∑1

n

i and ∑1

n

i2. Here I wish to define ∑1

n

i=Sn

and ∑1

n

i2=Sn

2.

In doing this, Pólya states, without proof, that Sn=1+2+3+4+…+n=n (n+1 )2

, which

is true and can be proven, by induction of all things.

Looking at the 1st 6 cases for possible n, Pólya observes that the ratios are given by:

n 1 2 3 4 5

Sn

Sn2

33

53

73

93

113

From this Pólya claims that Sn/(Sn2) ¿(2n+1)/3 which is certainly true for n=1,2 ,…,6

. Taking this into consideration and multiplying both sides thru by Sn=n (n+1 )

2, it is

obtained that

i George Pólya was an Hungarian mathematician [1887-1985].

22


Sn2=1

6n (n+1 ) (n+2 ) .

This remains true for n=1 ,2 ,…,6.

What Pólya has done is establish a ‘conjecture’ as he calls it, all from taking at a few

examples and looking for a pattern in order to present his equation for Sn2. Now that he has seen

that it works from the 1st six cases, he goes on to prove that it holds for n=7 by simply plugging in

the numbers of his equation. Not wanting to prove that that his equation is true for n=8 ,9 ,10 ,…

Pólya proceeds to the inductive step; assuming his equation holds for an arbitrary natural number n,

he proceeds in showing that it does also hold for n+1. Pólya concludes the proof of his equation for

the sum of the squares of the first n natural numbers by saying:

“If our conjecture is true for a certain integer n, it remains necessarily true for the next

integer n+1. Yet we know that the conjecture is true for n=1 ,2 ,3 ,4 ,5 ,6 ,7. Being

true for 7, it must be true also for the next integer 8; being true for 8, it must be true for

9; since true for 9 , also true for 10 , and so also for 11, and so on. The conjecture is true

for all integers; we succeeded in proving it in full generality.”

The procedure in which he developed his equation is similar to that of induction in other

sciences, by making observations. However in proving that his equation does in fact hold for all the

natural numbers Pólya requires to show the inductive step. Making observations and showing that

they are true for certain cases, does not provide a full proof; that is why Pólya requires the inductive

step and that is where mathematical induction differs from the induction of other sciences. This

induction obviously influenced mathematicians to look for recurring patterns in order to formulate

‘conjectures’ like Pólya and then try to prove their validity. Like Pólya says:

“Has mathematical induction anything to do with induction (in science)? Yes, it has.”

However this would only be the inspiration for the base case; the inductive step would not

have followed from the observation of examples.

There are traces of recurrent and inductive processes in the works of some Greek and Hindu

mathematicians. However, rather than trying to generalize a particular mathematical method, they

sought to find one particular solution from another. Here they were showing signs of an inductive

step, although from one specific case to the next, as opposed to an abstract case to the following

abstract case. Examples of such are found in the work of both Theon of Smyra (fl. 100 A.D.) and

Proclus (412-485 A.D.) on their work on finding numbers representing sides and diagonals of

squares. Bhāskara’s (1114-1185) cyclic method for solving indeterminate equations of the form

23


a x2+bx+c= yalso shows traces. Further traces can be found in Euclid’s Elements IX in Proposition

XX where Euclid shows that the number of primes is infinite. However any trace of mathematical

induction that did appear in the work of the Greeks and Hindus would not have been in the same

modern form in which we see it today.

During the 17th century, the French mathematician Pierre de

Fermat was known to have focused a lot of his work on

infinitesimal calculus and it was in a letter to Christiaan Huygens

in which Fermat claimed to use a method called la descente

infinite ou indefinite. The letter was titled ‘Relation des

découvertes en la science des nombres’ and was not discovered

until 1879, amongst a group of papers that had belonged to

Huygens. In the paper, Fermat claims he had found that his new

method was applicable in proving the impossibility of certain

mathematical statements, before realizing that it was also

applicable in proving the assertion of statements too. Fermat’s

method of infinite descent did contain recurrent modes of

inference, however, it was not mathematical induction in its

purity as Fermat would often jump from a certain case n to n−i ,

for some i<n, skipping over several cases at a time. Instead of looking to prove a specific base case

and an inductive step, Fermat chooses to prove a statement for one certain case and then make a

connection to another in order to prove this second case. Here Fermat does show a trace of an

inductive step.

Fermat was known for publishing only his results and excluding his methods; it was not until

1995 that Fermat's famous Last Theorem was proveni; Fermat having died in 1665. This could be the

cause for, by pure chance, not seeing Fermat’s method of infinite descent until 1879. The only

occasions on which Fermat made his methods known were in response to critiques of his work, who

did not believe his results to be true or who sought a clearer explanation of his results. One of these

reasons could be why Huygens possessed a copy of Fermat's method of infinite descent, and the

reason that we now know that a recurrent mode of proof was known to Fermat. There is further

evidence for this in the fact that Adrien-Marie Legendre, Gustav Lejeune Dirichlet and Leonhard

Euler all used similar methods of Fermat's method of infinite descent to prove a lot of Fermat’s

i Andrews Wiles proof of Fermat’s Last Theorem was published in 1995. The famous theorem had been written in 1637 in the margin of Fermat's copy of Diophantus’ Arithmetica to which Fermat said the margin was not big enough to contain the proof as well.

24

Figure 4 - Pierre de Fermat.


propositions in the space of time after his death and before the discovery of Fermat’s letter to

Huygens.

There are also traces of mathematical induction that can be found in the work of Blaise

Pascal. Also a French mathematician of the 17th century, Pascal was worked closely with Fermat on

the foundations of modern probability theory. Pascal is famous today for Pascal’s Triangle which was

mentioned earlier, however it was known to many in the days of Pascal and Fermat as the

arithmetical triangle. In 1665, Pascal's Traité du triangle arithmétiquei was published. Pascal used the

treatise to show the applications of the arithmetical triangle in the theory of combinations, the

theory of probabilities and to the calculations of the powers of the binomial coefficients. Pascal

showed that the binomial coefficient C kn can be found by the equation

n ∙ (n−1 ) ∙ (n−2 ) ∙…∙ (n−k+1 )k !

which, if we multiply thru by (n−k )!(n−k )!

we obtain the form that we

use today n!

(k !) (n−k )! .

Pascal used the arithmetical triangle to calculate the share

of a total stake in a game of dice between 2 players which has been

stopped prematurely. Let the first of the 2 players be person A and

the second person B. With person A needing n points to win and B

needing m points, Pascal used the arithmetical triangle to calculate

that the ratio of A to B should be given as the sum of the 1st n

numbers of the jth row of the arithmetical triangle to the sum of

the remaining m numbers of the same row, where j=m+n.

Alternatively, let numbers of the jth row of the arithmetical triangle

be given by the sequence a1 , a2 ,…,an , b1 ,b2 ,…,bm. Then the

ratio of A to B is given by ∑a i:∑b j .

Here j=m+n is the number of throws of the dice remaining in the game when it is stopped

early. Pascal proved in his treatise, through a lemma, that his ratio is correct for j=1 and then

proved in the next lemma that, if it is also correct for some natural number, then it is also correct for

the next natural number greater than it. Then Pascal concluded that his ratio is correct for every

natural number that j can be. This is precisely mathematical induction of today. Pascal did not refer

to it as mathematical induction or use a 2 step procedure; instead he had proven a base case in one

i Translated into English as - A treatise on the Arithmetical Triangle; it appeared first in 1654.

25

Figure 5 - Blaise Pascal.


lemma and then proven an inductive step in the next lemma before he made a separate conclusion –

that his ratio is true for every natural number.

Due to the popularity of Pascal's treatise on the arithmetical triangle, he could be deemed as

one of the reasons as to how the principle of mathematical induction became known at the time of

and in the time after the 17th Century. Pascal could have also brought the principle of mathematical

induction to the attention of Fermat through their work together on probability theory. Pascal was

certainly one of the first mathematicians to use the principle of induction similar to its current form,

in a systematic way and to realise the implication of the inductive step. However, to Pascal was

known the work of an Italian mathematician called Maurolycus. In a letter from Pascal to Pierre de

Carcavii, Pascal refers to Maurolycus for the proof that twice the nth triangular number minus n

equals n2 . Although Pascal does not mention Maurolycus in his Traité du triangle arithmétique, he

was well aware of Maurolycus whose work could have been the inspiration of Pascal's method of

induction.

i In the letter Lettre de Dettonville á Carcavi.

26


[1.6] Maurolycus & Gersonides:Franciscus Maurolycus (1494-1575) was an Italian

mathematician who worked on translations from some of the most

famous Greek mathematicians – Euclid, Archimedes, Theodosius. His

work was of great importance for the transit of Greek work to Europe.

In 1575, Maurolycus published a treatise on arithmetic titled

Arithmeticorum libri duo found in his book D. Francisci Maurolyci

Opuscula Mathematica. Here, Maurolycus uses a mode of inference in

a systematic way, building up from the first case to the next, to

demonstrate simple propositions before moving on to prove harder,

more complicated ones in a similar fashion. Proposition XI of this

treatise is the proposition to which Pascal credited Maurolycus in the

letter to Carcavi. Maurolycus proved this proposition through 2 previously stated propositions and

various definitions. Therefore Pascal did not get his idea for his inductive proof from the Maurolycus

proposition that he had referenced to Carcavi. However in the same treatise, 2 other propositions

may have caught the eye of Pascal.

Propositions XIII and XV of the treatise as follows:

(13)Every square number plus the following odd number equals the following square number.

(15)The sum of the first n odd integers is equal to the nthsquare number.

In modern notation this would precede as follows:

(13)(n+1 )2=n2+On+1 for On=2n−1.

(15) O1+O2+…+On=n2 .

In the modern notation it is clear to see that there is a connection between the 2 propositions.

Maurolycus states his proof of Proposition XV as follows

“By a previous a previous proposition (namely Proposition XIII) the first square

number (unity) added to the following odd number (3) makes the following square

number (4); and this second square number (4) added to the 3rd odd number (5) makes

the 3rd square number (9); and likewise the 3rd square number (9) added to the 4th odd

number (7) makes the 4th square number (16); and so successively to infinity the

proposition is demonstrated by the repeated application of Proposition XIII.” i

i This is the translation given by W.H. Bussey in American Mathematical Monthly, No.5, Vol.14, May 1917.

27

Figure 6 - Franciscus Maurolycus.


Proposition XV is achieved by repeated use of proposition XIII which acts as an inductive step.

Clearly Maurolycus proof is an example of mathematical induction. Again it is not in the current

structure as we use today but it is in a systematic, step-by-step progression from the first case to the

next, and then to the next, and to the next, and so on towards infinity. One difference between

Pascal's mode of inference and that of Maurolycus, is Maurolycus does not make an inductive

assumption/hypothesis whereas Pascal does, making his method of proof more abstract.

Now I come to Levi Ben Gershon, (Gersonides in

Latin). A Rabbi born in 1288 in Languedoc in what would

now be the southern coast of modern day France,

Gersonides had a broader background to his

mathematical career than Maurolycus. Although

Gersonides did too work on Euclid’s Elements, he also

worked on a lot of the old Arabic and Hindu texts, such as

Bhāskara, who I mentioned before, who is known to have

used his cyclic method to solve indeterminate equations

of the form a x2+bx+c= y .

Gersonides’ 1321 work – Maasei Hoshevi, could be

called a piece of work that was ahead of its time;

Gersonides uses letters to represent arbitrary numbers,

only Jordanus Nemorarius was known to have also have

done this at that time. Another reason that Maasei Hoshev

could be considered as ahead of its time is because of Gersonides use of the method of what he

called rising step-by-step which has similarities to Maurolycus’ work on representing the infinite. This

step-by-step method can be found in Propositions 9 to 12 inclusive, which I now give (in modern

notation):

(9) a (bc )=b (ac )=c (ab).

(10)a (bcd )=b (acd )=c (abd )=d (abc ) .

Gersonides used Proposition 9 to prove Proposition 10 before making the following statement:

In this manner of rising step-by-step, it is proved to infinity. Thus, … the result of

multiplying one number by a product of other numbers contains any one of these

numbers as many times as the product of all the others.”

i Title is taken from Exodus 26:1 to roughly mean The Work of the Calculator.

28

Figure 7 - A stamp of Isreal showing Gersonides' invention, Jacob's Staff, which was used for measuring nautical and astronomical measurements.


Which I interpret as a❑ j (a1a2…ai…an )=ai (a1a2…a j…an ).

(11) a (bcd )=(ab ) (cd )= (ac ) (bd )=…

Again Gersonides wrote towards an extension towards infinity:

Similarly, it is shown to infinity by the same kind of demonstration. Therefore, any

number contains the product of any two of its factor as many times as the product of the

remaining factors.

(12) In modern language, Proposition 12 states that multiplication is both associative and

commutative. Gersonides proves this using the previous 3 propositions to show how factors

can be grouped into different strings of different lengths, which is not really a full proof.

Although none of these propositions are proved by induction, they do show what Gersonides

meant by rising step-by-step and it does bare some resemblance to an inductive step. Gersonides

continued to state and prove a long series of propositions; I skip to Propositions 63 – 65 on

permutations:

(63) Pn+ 1=(n+1 ) Pn.

(64)P2n=n (n−1 ).

(65)P j+1n =(n− j ) P j

n.

Gersonides proved each one of these propositions in turn, using Proposition 63 to prove

Proposition 65. Once this is done, Gersonided concludes by saying:

Thus it has been proven that the permutations of order a given number from a second

given number of elements equals the numbers whose factors are as many as the first

given number and they are the integers in their natural order, the last being the second

given number.

Thus Gersonides had proven that P jn= ∏

n− j+1

n

i. One can see that, when we take j=2 we get

Proposition 64, which had been proven previously by Gersonides. Gersonides claimed that these 3

propositions are enough to prove this general result, and they are. From Proposition 64, we use

Proposition 65 to prove the case for j=3 and from that, we apply it again to prove that the result

holds for j=4 and apply it again for j=5, and so on. Proposition 65 acts as an inductive step, from

any arbitrary j to j+1.

Thus Gersonides had used induction to prove his result on the number of permutations of

order j of n elements. Again we cannot say that this is how we use induction today, but Gersonides

29


method does contain the essence of modern induction having proven a particular case as well as a

recursive step. The proof of proposition 42 in this work was also proven in a similar fashion.

However, there was a lack of an assumption in order to make the inductive step, similar to that of

Maurolycus and again, Gersonides work was constructive, building up to the result, whereas with

induction today, we seek to prove what we have already stated.

The source as to Gersonides influence to investigate and use a recursive mode of inference

in order to prove mathematical procedures lies in the Hebrew community of Gersonides time where

the subject was investigated at an early stage. As well as Bhāskara, Gersonides could have been

influenced by the works of Sefer Yetsirah and his Book of Creation, which involves a recursive mode

of looking at the permutations of the twenty-two letters of the Hebrew alphabet and is believed to

be from the second century. Another possible influence is Rabbi Shabbetai Ben Abraham Donnolo,

(913-970), who proved that n letters can be arranged n ! ways in a similar fashion to a recursive

method.

In Massei Hoshev, Gersonides mentioned that the reader of his text should be aware, and

capable of understanding, the 7th, 8th and 9th books of Euclid’s Elements where we have already seen

a small trace of a recursive mode of inference. However, it would be possible to assume that

Gersonides found modes of recursive definition and induction, throughout his mathematical career,

in the works of others and we could even assume further that Gersonides had set out in Massei

Hoshev to investigate the application of recursively defined structures and the application of

recursion in mathematical proof in a structured, systematic way.

And it seems to me that Gersonides was the first to do this. I feel that Gersonides was the

first to realize the significance of the ordering of the natural numbers and how he could apply it to

proofs in order to represent the a property of the infinite in terms of the finite.

There is only one further mathematician that I feel we could consider for the invention of

mathematical induction and that is Campanus. An Italian mathematician of the 13 th century,

Campanus of Novara worked on translating

Euclid’s Elements into Latin. In doing this he

included his own version of the proof that the

golden ratioi is irrational. The method used by

Campanus in his proof was similar to that of

Fermat. Campanus, like Fermat, used a

i (1+√5 )

2 is the Golden Ratio, which is said to be found to occur naturally throughout all of life.

30

Figure 8 - The Golden Ratio.


descending method of progression, jumping sporadically over certain cases, to prove other certain

cases.

However, I do not feel that this method is fully representative of the mathematical induction

of today. Modern induction represents every possible case whereas the methods of Campanus and

Fermat appear disjoint and full of gaps. Although their methods do show traces of induction, they

lack that continuous, connected, one after another sequence. I feel that the methods of Gersonides,

Maurolycus and Pascal are stronger and closer to the method in which we now use today, not

because they precede from a certain, finite, base case towards infinity, but for the reason that these

methods would not skip over any particular case; they represent the continuous progression from

one discrete case to another.

In addition, I believe that the same three men, Gersonides, Maurolycus and Pascal were

more aware of the significance in their method; Fermat could also be included in this regard but

since he never released his arguments, we will never know if he was aware of such significance. As

for Campanus, I would have to regard him as one of the men who influence the former four. I regard

him in the same group as Bhāskara, Theon, Proclus, Euclid and others, who showed traces of the

mode of inference and recursion, who used a method similar to induction as a one-off proof,

significant only at that one particular time. Moreover I would consider Gersonides to be the first

mathematician to understand the meaning and significance of a recursive mode of inference and to

give it a step by step structure. Furthermore I regard Maurolycus as the first man to see the

significance of applying this recursive mode of inference to proofs whereas Pascal brought it to the

attention of many others.

31


[1.7] The Name – Mathematical Induction:

For the origin of the term Mathematical Induction, a group of other mathematicians who

were aware of the method are to be credited. Fermat referred to his new method as la descente

infinite ou indefinite, Gersonides referred to his method of rising step-by-step. However Maurolycus

and Pascal did not assign any particular name to their mode of inference. It is evident that the term

mathematical induction is derived initially from the observational induction of sciences as many seen

it as an adaptation of that concept.

In John Wallis’ 1656 work, Arithmetica Infinitorum,

Wallisi used the method of induction used in science and

simply refers to this method as induction. In Proposition 16 of

this work, Wallis looked to find the ratio of the 1st nsquared

numbers to the product (n+1 )n2. Wallis proceeds to observe

that, in the 1st 6 cases, the ratio turns out to be 13+x ii where

x<1 decreases as the size of n increases. Wallis then

concluded that limn→∞

x=0. This method was referred to by

Wallis as per modum inductionis.iii As Wallis proceeded

through the remainder of this piece of work, he relied freely

on this method of induction similar to that of natural science.

Wallis did feel strongly about the scientific induction and that it could easily be applied to

mathematics. In his 1685 treatise on Algebra, Wallis stated:

“Those Propositions... demonstrated by way of Induction: which is plain, obvious, and

easy; and where things proceed in a clear regular order (as here they do), very

satisfactory.”

“I look upon Induction as a very good method of investigation; as that which doth very

lead us to the easy discovery of a General Rule.”

However, in 1686, Jacob Bernoulliiv recommended in his Acta Eruditorum that Wallis could

improve his method of induction by introducing the argument from an arbitrary n to n+1. This

appears to be the 1st appearance of the Inductive Hypothesis and the beginning of the modern form

i John Wallis was an English mathematician, (1616 – 1703).ii Just like Polya noted in section [1.5].iiiLatin for - by way of induction.iv Jacob Bernoulli (1655 – 1705) was a Swiss mathematician and part of a large mathematical family.

32

Figure 9 - John Wallis.


of Mathematical Induction. Bernoulli used this new n to n+1 argument to prove the binomial

theorem in his Ars Conjectandii.

Florian Cajoriii, a historian in mathematics, refers to the method used by Wallis as incomplete

and refers to is as Incomplete Induction, which gives rise to the Complete Induction that Cajori

defined as the method by Bernoulli. For more than a century after Bernoulli’s recommendation to

Wallis, Induction was being used as the name for both the methods of Wallis and Bernoulli. The two

methods were seemingly unpopular in this time and it was in fact Bernoulli's method that was less

known at the time Most who used either method actually used the method without assigning a

specific name.

However in the 1830s, this changed. George

Peacock (1791 – 1858) was an English mathematician who

published his Treatise on Algebra in 1830. In this Treatise,

Peacock talked of a “law of formation extended by

induction to any number”. In explaining the argument

from n to n+1, Peacock referred to his method as

Demonstrative Induction.

Augustus De Morgan (1806 – 1871) was a British

mathematician whose name is mainly associated to the

laws of negation on the conjunction and disjunction of

sets. In 1838, De Morgan published in the Penny

Cyclopaedia his Induction (Mathematics) in which he described clearly mathematical induction and

its similarities/differences to induction in physics. De Morgan showed how mathematical induction

should be applied through two clear, well-described examples where a proposition is stated and

then proven via an inductive step (using an inductive hypothesis only in the 1st example), before

referring back to a base case. De Morgan referred to induction as successive induction at the

beginning of this piece of work, however he later refers to the method as Mathematical Induction;

the first published occasion on which the term had been used.

Both the terms Demonstrative Induction and Mathematical Induction became popular in the

time after but the former term fell into disuse as most mathematicians began to adopt the latter.

The term Vollständige Induktion was used by German mathematicians in the 19th century, most

notably by Richard Dedekind in his 1887 Was Sind und Was Sollen die Zahlen. It was this usage by

Dedekind that popularized the method in Germany, although the method was slightly different to

i Published in 1713, after Jacob Bernoulli’s death. ii Origin of the Name Mathematical Induction, The American Mathematical Monthly, vol. 25, number 5, 1918.

33

Figure 10 - Augustus De Morgan.


that of Peacock and De Morgan. In 1863, Isaac Todhunter (1820 – 1884), an English mathematician,

popularized De Morgan’s mathematical induction in his Algebra for Beginners using the method to

prove various examples. One such example that Todhunter spoke of was:

“The sum of (the first) n terms of the series 1 ,3 ,5 ,7 ,… is n2. This assertion we can see

to be true in some cases... we wish to however to prove this theorem universally”.

Using induction, in the same manner as De Morgan, Todhunter proved the above universally,

for all possible cases of n. Realising the full benefit of mathematical induction, Todhunter then

stated:

“The method of mathematical induction may be thus described: we prove that if a

theorem is true in one case, whatever that case may be, it is true in another case which

may be the next case; hence it is true in the next case, and hence in the next to that, and

so on; hence it must be true in every case after that which it began..... The method of

mathematical induction is as rigid as any other process in mathematics.”

Todhunter referred to the method directly as mathematical

induction – the title that De Morgan assigned to it and the title

which we use today. In the century that followed Todhunter’s

Algebra, mathematical induction has become even more abstract

and has been accepted across the mathematical world as an

essential tool for mathematical proof. Today its structure has

become more like that of a procedure that is followed in a step by

step manner as we have seen earlier. However the name and the

basic concept have remained intact; from Gersonides, to Pascal,

through to De Morgan and onwards until its modern form today.

34

Figure 11 - Isaac Todhunter


[1.8] Bibliography:

1. N L Briggs, Discrete Mathematics, Revised Edition, 1989, Oxford University Press, P8-10.2. J L Hein, Discrete Mathematics, 2nd Edition, 2003, Jones and Bartlett Publishers, P145-146.3. H Eves, An Introduction to the History of Mathematics, 4th Edition, 1976, Holt, Rinehart &

Winston, P209-212.4. R C Penner, Discrete Mathematics: Proof Techniques and Mathematical Structures, 1999,

World Scientific Pub. Co. Inc., P141.5. K Gödel, on Formally Undecidable Propositions of Principa Mathematica and Related

Systems, English Translation by B.Meltzer , 1962, Oliver & Boyd LTD. Introduction by RB Braithwaite FBA.

6. S C Kleene, Mathematical Logic, 1967, John Wiley & Sons, Inc., P250.7. J W Dawson Jr., Logical Dilemmas, the Life and Work of Kurt Gödel, 1997, A. K. Peters, P3-21,

P53-79.8. David Hilbert, Mathematical Problems, Bulletin of the Mathematical Society, 1902, Vol. 8,

Number 10, P437-479, translated by M Winston Newson.9. I Grattan-Guinness, Search for Mathematical Roots 1870-1940, 2000, Princeton University

Press, P227. 10. I Grattan-Guinness, Search for Mathematical Roots 1870-1940, 2000, Princeton University

Press, P227. 11. P Eccles, An Introduction to Mathematical Reasoning, 2007, Cambridge University Press,

P39-51.12. G Polya, Mathematics and Plausable Reasoning, Vol. 1: Induction and Analogy in

Mathematics, 1954, Oxford University Press, P108-11113. F Cajori, Über das Wesen der Mathematik, Bulletin of American Mathematical Society, 1909,

Vol. 15, Number 8, P407.14. F Cajori, History of Mathematics, 5th Edition, 1991, Vol. 2, Chelsea Publishing Company, P142.

Also, whole text used for birth/death dates of Mathematicians and their work.15. G Vacca, Maurolycus, The First Discoverer of the Principle of Mathematical Induction, Bulletin

of the American Mathematical Society, 1909, Vol. 16, Number 2, P70-73.16. N L Rabinovitch, Rabbi Levi Ben Gershon and the Origins of Mathematical Induction, Archive

for History of Exact Sciences, 1970, Vol. 6, Issue 3, communicated by C Truesdell.17. F Cajori, Origion of the Name ‘Mathematical Induction’, The American Mathematical

Monthly, 1918, Vol. 25, Number 5, P197-201.18. A De Morgan, Induction (Mathematics), Penny Encyclopaedia, 1838, Vol. 12, London.19. I Todhunter, Algebra for Beginners, 4th Edition, 1866, Macmillan & Co., P281-284.

35


[2] Set Theory and Transfinite Numbers:

The history of how mathematical induction and recursion came to be is an important one.

The journey that both have taken to progress through the ages to their modern forms has been a

labouring one. Both mathematical applications have shown traces of their importance as far back as

the time of Pythagoras and Plato, partly due to the results that they were able to produce, partly

due to the struggle with the infinite at the time. Most of this has been addressed in the first part of

this piece of work. One thing I did not investigate fully was the significance that induction had on

mathematics in the late 19th and early 20th Century’s and in particular, its effect on number theory.

During this time, number theory became an important interest to a lot of mathematicians,

the most important being Giuseppe Peano of Italy, Richard Dedekind and Georg Cantor, both of

Germany. The importance number theory had at the time was not just from a mathematical point of

view but also from a philosophical one. The main aim was to establish foundations for the theory of

numbers and its arithmetic such that it should be sound and free of contradiction. The work of

Peano and Dedekind had the most significance in this area.

Mathematical induction was an essential tool for Giuseppe Peano in establishing his now

famous, and universally accepted, axioms which I have previously mentioned. The use of induction

by Peano helped him convert mathematics into a symbolic, logical form; in fact the simplicity of

induction shines through in Peano’s symbolic notations. Therefore I would like to start by looking at

the work of Peano, most notably his Arithmetices principia, nova method exposita, as it continues on

from the work done in the previous section. This will then lead me to Richard Dedekind.

In his day, Dedekind published two famous pieces of work – Stetigkeit und irrationalle Zahlen

and Was Sind und Was Sollen die Zahlen? The latter of which has a real significance on this piece of

work, although I shall mention the former briefly in various areas beyond this point. Although

Dedekind’s Was Sollen was published before Peano’s Arithmetices principia, I wish to look at both in

this order as the work of Dedekind, baring huge similarity and importance to the work of Peano, also

has an overlapping connection to the work of Cantor.

The work of Georg Cantor is the main interest of this section. The development of the

transfinite ordinals and cardinals by Cantor in the late 18th century as a new subject in mathematics

was not an easy one but it does remain to this day and it became one of the most important areas of

mathematics of the 20th century, bringing mathematics into a logical form and bringing mathematics

closer to its philosophical roots. Towards the end of this work, having introduced Cantor’s naïve set

36


theory, we shall have come across transfinite induction and recursion, which are slightly different

forms of the induction and recursion already investigated.

37


[2.1] Giuseppe Peano:Giuseppe Peano was an Italian mathematician who was born in Cuneo in north-west Italy,

near the border of Austria, on 27th August 1858. Having moved to Turin in 1871 with his uncle to

further his studies, Peano enrolled in a graduate program in mathematics at the University of Turin

in 1876. Upon graduating in 1880, Peano was offered a position of work at the university which he

accepted and where he remained until his death in 1932. Peano soon became a professor of the

infinitesimal calculus at the university, although he did have other interests outside of this area,

most notably, the foundations of mathematics and, to fill the time, Peano had a keen interest in

linguistic studies.

Peano published in excess of 200 papers during his life and the first of these came in 1884,

entitled Calcolo differenziale e principii di calcolo integrale. Peano dedicated this work to a former

teacher, Angelo Genocchi, by publishing it under Genocchi’s name and assigning his own name as a

subtitle of the work. It was in this piece of work, and his work as a lecturer, that the need for a

higher standard of rigour in mathematics became clear to Peano. When writing towards a

publication, Peano liked to keep to his own high standard of rigour while at the same time making

his work simple and easy to understand and follow, so it was usual for Peano, just like in ancient

Greek scripts, to show a demonstrative form of writing.

All of this is evident in Peano’s 1889 Arithmetices principia, nova method exposita, which can

be translated as The Principles of Arithmetic, Presented by a New Method. The need for such work

was apparent to Peano, to work towards that more rigorous and simple foundation of mathematics

38

Figure 12 - Giuseppe Peano (1858-1932)


that he thought was necessary, and to achieve this, Peano introduced symbols to represent full

mathematical structures and also used letters to represent whole propositions and propositional

functions; he was the first to do so, there was no need for Peano to write full labouring sentences

explaining what he wanted to achieve. When published, his symbolic work probably appeared as a

code to its readers, but the simplicity and easy flow of reading would have been easy to pick up.

Other symbols that were also introduced which we still use today were ∈ for the inclusion of an

element in a set, ⊂ for the containment of a subset in a larger parent set, ∪ and ∩ for the union

and intersection, respectively, of two sets. Peano paid special attention to the distinction between ∈

and ⊂ as to lead to no ambiguity.

For Peano to establish his new foundations he had to start at the most essential and basic

area of mathematics, the natural numbers. Peano was looking to establish an axiomatic system for

looking at the natural numbers and their properties, presented through his symbols. With the

natural numbers axiomatized, any proposition or theorem in mathematics could be verified by the

truth held in the axioms. To quote Peano:

“with these notations, every proposition assumes the form and the precision

that equations have in algebra”.

Arithmetices principia was published as a 36 page pamphlet. The opening introduction was

16 pages devoted to defining and explaining, rigorously of course, the new symbols that Peano

wished to introduce. The remainder began with four definitions and these were followed by the

axioms of which they were nine in total. Thus, Arithmetices principia began as follows:

“The sign N means number (positive integer).

The sign 1 means unity.

The sign a+1 means the successor of a, or a plus 1.

The sign ¿ means is equal to.

1. 1∈N .

2. a∈N ,a=a .

3. a ,b∈N , if a=b then b=a .

4. a ,b , c∈N , if a=b∧b=c , thena=c .

5. if a=b ,∧b∈N , thena∈N .

6. a∈N ,a+1∈N .

7. a ,b∈N , if a=b , then a+1=b+1.

8. a∈N ,a+1≠1.

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9. K⊂N ,if (a )1∈K∧(b )if , for a∈N , thena+1∈N ;thenK=N .”

Axioms 2, 3, 4 & 5, are the axioms that we have not seen before. Today these can be considered

as trivial or as part of the underlying logic of equality. Notice that axiom 9, the axiom of induction, is

stated slightly different from that of before.i With the induction axiom being stated last, Peano

follows his axioms with defining the natural numbers, by induction:

2=1+1 ;3=2+1; 4=3+1; and so forth.

With this, Peano introduces addition, subtraction, maximum and minimum numbers,

multiplication, powers, division and then proceeds to move on to theorems on number theory,

rational and irrational numbers, theorems on open and closed intervals. Most of the work in

Arithmetices principia is proven by induction or constructed from N and the successor functionii

under a mode of recursion. The details of these however I do not wish to investigate, as we shall see

all of these from a set-theoretic point of view when looking at the work of Dedekind.

From the axiom of induction, Peano does not have to deduce the validity of the properties of his

natural numbers. This is contrasting with Dedekind as we shall see. We know the axiom carries

properties over the whole of the natural numbers when the property holds for the base case and,

under the inductive hypothesis, it holds for the inductive step. This axiom, along with the, what we

may call logical, preceding axioms, is all Peano requires to define structures of, and properties on,

the natural numbers.

Today, Peano’s axioms are considered a reference point for the natural numbers. However, they

are not always referred to as exclusively belonging to Peano; some refer to the axioms as the Peano-

Dedekind Axioms.

i This is due to wanting to keep to the original translation of Peano’s own statement of the axiom, found in Jean van Heijenhoort’s Frege to Gödel.ii f (a )=a+1. This is simply the third of the four terms defined by Peano at the beginning and how I shall refer to it from here.

40


[2.2] Richard Dedekind:

Figure 13 - Richard Dedekind (1831-1916)

“One of the wholly great in the history of mathematics, now and in the past”

– Edmund Landau on Dedekind.

Julius Wilhelm Richard Dedekind was born the 6th of October 1831 in Brunswicki, Lower

Saxony, Germany. When he was younger, Dedekind found himself more interested in physics and

chemistry than mathematics, however, at the age of 17 and discontented by the lack of reasoning he

found in physics, Dedekind turned his attentions towards mathematics in search for more logically

sound reasoning. It was in 1850 that Dedekind began to attend the University of Göttingen. Here,

having been born in the same town and attended the same college, Dedekind would study under the

influence of Carl Friedrich Gauss.

After two years, Dedekind graduated with his doctorate before taking up a lecturing role at

the university in 1854. It is believed that Dedekind was the first, in 1857, to give a course on Galois

Theory at a university level; he gave the course only to two students. In the same year, Dedekind

moved on to take up a position at a polytechnic in Zurich. 1862 saw Dedekind return to Brunswick to

take up a position as a professor at a technical high school. This seems like two steps backwards for

Dedekind, but it appears that Dedekind lived an obscure, secluded life, away from the demand and

i This is an Anglicization of Braunschweig.

41


attention that other, less capable, mathematicians were receiving. In 1904, it appeared in Teubner’s

Calendar for Mathematicians that Dedekind had died on the 4th of September, 1899, although

Dedekind did inform the editor that he was in fact “in perfect health” on that day having spent it

with his “honoured friend Georg Cantor of Halle”.

In 1888, a year before his fictional death, Dedekind published his famous Was Sind und Was

Sollen Die Zahlen? The second edition of this was published in 1893 and this was translated in 1901

by Wooster W. Beman into English and giving the title the Nature and Meaning of Numbers,

although a direct translation would be What Are Numbers and What Should They Be? From this

work, like that of Peano, Dedekind was looking to establish a higher rigour of mathematics, but in

doing so, he tackles the age old elephant in the room – infinity. Infinity plays a major role in what is

to come in later parts of this work and we have touched on the infinite before with mathematical

induction and recursion. I shall say no more on the complexities of the infinite at present as I would

like to give a brief summary of Was Sollen, highlighting the key points of interest.

Was Sollen differs in approach from Peano’s Arithmetices principia as Dedekind does not take

the axiomatic approach of Peano, instead insisting on a set-theoretic approach to defining the

natural numbers. In doing so, Dedekind begins with the infinite and defines a finite set i as a set

contrasting from that of an infinite one; this part of Dedekind’s work is what he shares with Cantor.

Dedekind uses this method of working back from the infinite to define the natural numbers.

Was Sollen consists of 172 paragraphs; each assigned a number in unison by Dedekind. These

paragraphs are then grouped together in 14 different sections. The second edition has a new preface

attached where Dedekind wanted to addresses some of his critiques; however he could find no

justification for any criticism that the first edition met and the new preface is merely an

acknowledgement that Dedekind had tried to address any issues.

In the preface to the first edition of Was Sollen, Dedekind addresses his desire for a more

rigorous foundation to mathematics:

“In science nothing capable of proof ought to be accepted without proof. Though

this demand seems so reasonable yet I cannot regard it as having been met even in the

most recent methods of laying the foundations of the simplest science ii… numbers are

free creations of the human mind… It is only the purely logical process of building up the

science of numbers and by thus acquiring the continuous number-domain that we are

i Dedekind referred to a set as a system.ii Here Dedekind singles out, in a footnote, the work at the time of Kronecker, Schröder, and von Helmholtz.

42


prepared accurately to investigate our notions of space and time by bringing them into

relation with this number-domain created in our mind.”

In Was Sollen Dedekind referred to sets as systems consisting of things (Dinge) that could be

“considered from a common point of view, associated in the mind”. Although, in later sections of

Was Sollen, Dedekind referred to these things simply as elements and took the same concept of

Peano by using lowercase letters to represent these elements. Dedekind also referred to functions as

transformations.

As already noted, Dedekind and Peano tackle their work from different angles. Peano’s

axiomatic approach allows Peano to give his axioms, along with his 4 new definitions, and expand

from there to produce desired results. Thus Peano establishes the arithmetic of the natural numbers

from his axioms. In contrast, Dedekind’s construction of the natural numbers has to be more

progressive. The foundations of the work by Dedekind consist of a series of definitions and

corresponding theorems. Thus the early sections of Dedekind’s Was Sollen address these

requirements.

In the third section, Similarity of Transformations. Similar Systems, Dedekind says that two

sets are similar if they can be put in to a one-to-one correspondence under a similar function, one

that is bijective. This is followed by the definition of a class, by which all sets that are similar to one

another can be grouped together and any one of the sets in the class can be the representative of

the class as a means of identifying the class from any other class of similar sets. The concept of

similarity between sets is how Dedekind defines infinite sets and from this, finite sets.

¶64. “Definition. A system S is said to be infinite when it is similar to a proper part of

itself; in the contrary case, S is said to be finite.”

Thus, a set S is an infinite set if it has a one-to-one correspondence to a proper i subset of

itself. On the other hand, if this is not the case, then S is said to be a finite set. Dedekind has now

defined what it means for a set to be infinite and highlighted its difference from a finite set.

¶66. “Theorem. There exists infinite systems.”

Although in 1872’s Stetigkeit und Irrationalle Zahlen, Dedekind had already used infinite

collections when creating his idealsii, the need to justify the existence of infinite sets in Was Sollen

was because his definition of the natural numbers depended upon their existence. Thus, according

to Dedekind, if infinite sets exist, then the natural numbers exist; a somewhat pressure to accept

i U is a proper subset of Sif it is not equal to S, i.e. U is strictly contained in S ;U⊂S.ii Dedekind defined an ideal as an infinite collection of algebraic real numbers; numbers that are the root of a non-zero polynomial in 1 variable, with rational coefficients.

43


their existence. The proof of this theorem is completely philosophical and non-mathematical and

shall be returned to in due course. Dedekind continued by distinguishing between infinite and finite

sets, however before Dedekind could define the natural numbers there was the need to define a

simply infinite set.

A set S is simply infinite when there is a similar function ϕ :S→S,such that Scan be

obtained from the union of repeated application of ϕ on an element of Sthat is not in ϕ (S )i, i.e. S is

generated by an element that is not in the image of ϕ.

This definition is then referred to a particular set, N , and an element that generates this set;

the Base-element “which we shall denote by the symbol 1”. The definition of a simply infinite set, N ,

was simplified by Dedekind to the existence of a similar function ϕ of N and an element 1 such that

the following conditions are satisfied:

α : Define N '=ϕ (N ), then N '⊂N .

β :N=ϕ (1) ,that is, 1is the base-element of N .

γ :1∉N ' .

δ : ϕis similar, i.e. a one-to-one function.

It would be possible to take the four requirements α ,β , γ , δ ,and assume their existence as

axioms for a set N , however Dedekind did not feel there was a need for axioms and defined the

natural numbers as follows:

¶73. “Definition. If in the consideration of a simply infinite system N set in order by a

transformation ϕ we entirely neglect the special character of the elements; simply

retaining their distinguishability and taken into account only the relations to one

another in which they are placed by the order-setting transformation ϕ, then are these

elements called natural numbers or ordinal numbers or simply numbers, and the base-

element 1 is called the base-number of the number-series N . With reference to this

freeing the elements from every other content (abstraction) we are justified in calling

numbers a free creation of the human mind. The relations or laws which are derived

entirely from the conditions α ,β , γ , δ ,in (71) and therefore are always the same in all

ordered simply infinite systems, whatever names may happen to be given to the

individual elements, from the first object of the science of numbers or arithmetic ii.”

i Thus S=∪ϕn ( s ) , s∈S, where ϕn is the nth iterate of ϕ .ii From here the elements of the set N are referred to as numbers.

44


The remainder of this definition focused on how the elements and subsets of N are closed

under ϕ. It was also noted by Dedekind that “the transform n ' of a number n is also called the

number followingn”.

This was how Dedekind defined the natural numbers; it was sufficient to have the simply

infinite set N , a whole, single entity, ordered by the repeated use of the function ϕ such that, for

n∈N ,ϕ (n )∈N .

The definition of the natural number was followed by a series of theorems that coincide with

this new definition of the natural numbers. These theorems can be seen as Peano’s axioms. For

example, Peano’s 8th axiom, a∈N ,a+1≠1, can be compared to paragraph 79 of Was Sollen which

says “every number other than the base-number 1 is an element of N '.”

The following paragraph, paragraph 80, is when Dedekind defined induction upon the

natural numbers. Dedekind had previously defined induction on the notions of chains in an early

section but stated that that formation, the “theorem of induction”, served only as a basis for

induction upon the natural numbers. As was stated in the previous section of this work, Dedekind

referred to induction as Völlstandige Induktioni and from this point in Was Sollen, it was used to

prove the majority of the theorems that are to follow. In paragraph 126, the “theorem of the

definition by induction” is introduced; this was then used to define addition, multiplication and

exponentiation with their subsequent theorems and properties proven by induction.

i Beman translates this as complete induction.

45


[2.3] Arithmetices principia and Was Sollen:

The authors of Arithmetices principia and Was Sollen both had the same objective in writing

their publications; to establish that more rigorous foundation to mathematics that both Dedekind

and Peano thought mathematics required at the time. Although Dedekind and Peano attacked their

problem from different angles and taking different approaches, both were successful in obtaining

their objective and their ideas were so strong that they are still being used as a basis for

mathematics today.

Dedekind’s opposite approach from Peano was in how the natural numbers were defined;

having first defined what it means to be an infinite set and saying that the natural numbers are such

a set under a particular one-to-one function, whereas Peano built up from unity, using his axioms. As

we have seen Peano’s use of induction enabled him to work with the infinite from a finite

perspective; it allowed Peano to represent the potential infinity, being finite at each point and never

actually reaching an infinite totality. In contrast, Dedekind’s definition of the natural numbers is not

free of the existence of the actual infinite. At the time this was a bold and unusual approach and

Peano’s axioms would have been more readily accepted by the mathematical community.

For Dedekind, the infinite was easier than its counterpart; the existence of an infinite set and the

definition of the natural numbers, paragraphs 66 and 72 respectively, were addressed before the

specific finite case of theorem 81; n is not equal to its successor n ' . To accept the existence of an

actual infinite set was what every other mathematician at the time was avoiding or whose existence

was absurd to them but it led to a new way of looking at the natural numbers.

This new outlook on the most simplest of all mathematical areas had a huge bearing on the

works of others (those who didn’t reject it) and the proof of the existence of an infinite set helped in

mathematicians accepting the work of Georg Cantor who at this time had already begun to publish

his works on transfinite numbers.

We have seen that Dedekind’s definition of the natural numbers, paragraph 73, was very

wordy and it came from a philosophical viewpoint, just like the ‘proof’ of the existence of simple

infinite sets. Dedekind could have defined the natural numbers in a more mathematical structure

but this approach allowed Dedekind to abstract number from any need of justification of thought,

much like in the same vein as Peano’s use of the induction axiom. This method of abstraction was

fundamental to both Peano and Dedekind to create their foundations; it allowed both men to

distance their ideas from any concrete thing (Dinge) or from any justification in nature and their

abstraction lead them to generality. Their rigorous foundations were laid out and today they still

exist, together in fact, as the Dedekind-Peano Axioms.

46


Dedekind’s acceptance of an actual infinite set is worth consideration. Before the theorem of

paragraph 66, there exist infinite sets, any theorem that had been stated was always followed by a

proof that stuck to mathematical reasoning. However with this theorem, Dedekind began his proof

with “My own realm of thoughts” which immediately suggests that there were to be no

mathematical reasoning involved with this proof, and indeed that was the case. Dedekind instead

argues his case from a philosophical point of view, but why?

This could be because of the status that the infinite had within mathematics at the time –

banned by Aristotle in the age of the great Greek mathematicians, no one dared to accept its

existence from then on, whether it be potential or actual. An example of a famous mathematician

who would never accept the infinite is Galileo i who believed in the axioms of Euclid’s Elements and in

particular – the whole is greater than the part. Therefore, to Galileo, to have a bijective

correspondence between a set and a proper subset of that set would mean denying the truth of

Euclid’s axiom. Dedekind tried addressing this by taking this paradox and using it as a definition for

the infinite, essentially saying, that the infinite is that which is not infinite. Thus, the infinite would

have different properties to its complement and vice-versa.

However, any consideration of the infinite at the time would have been met with stern objection

by many. Most of the ‘justifications’ that were used to do this usually came from a philosophical

point of view. Thus was this why Dedekind found it necessary to argue his point in such a manner?

Another, simpler reason for this could be that Dedekind simply believed it to be true, but could not

formulize a mathematical proof to justify it.

However the question arises, should theorem 66 be a mathematical theorem or should it

come under the postulate/axiom title and just be accepted as true, in order for Dedekind to

continue his work maintaining a continuous mathematical truth?; for every theorem that is based

upon that of 66 would be based on a philosophical justification.

The structure of Dedekind’s Was Sollen is different from that of Peano’s. Dedekind did not

feel the need to define the usual arithmetic definitions of addition and multiplication immediately

after defining the natural numbers, instead defining the structure of the natural numbers and the

ordering of its elements. One reason for this could have been because Dedekind’s definition of the

natural numbers depended on the existence of infinite sets, so a need to investigate the simple

infinite set N further may have stood in the way. Another reason could be because Dedekind did not

feel the need to do so as the need to define the structure of N and the order of its elements where

more important. This allowed Dedekind to distinguish further between finite and infinite sets, use

i Galileo Galilei (1564-1642), Italy.

47


definition by induction, and talk of the classes of infinite sets; all of this before addition of the

natural numbers. Maybe there was a hidden agenda of tackling the actual infinite that Dedekind

wanted to disguise behind the natural numbers.

Whatever reason there was for the order of Dedekind’s work, it remains that Dedekind had

more material that he wanted to publish and show to others than simply defining the natural

numbers. This extra material was mainly focused on the actual infinite and it was parallel to the

work being done by Georg Cantor at the time.

48


[2.4] Dedekind’s Interest in Infinity:

We have seen that Dedekind found significance in the infinite; from the infinite ideals of

Stetigkeit to the infinite systems of Was Sollen. In his work, Dedekind tried to justify the use of

infinity; by using infinite sets as a basis for his definition of the natural numbers, Dedekind tried to

make it difficult and contradictory to deny the existence of such sets. We know that Dedekind was

seeking to establish sounder foundations for mathematics, so to include infinite sets in his

foundations, Dedekind must have also thought that infinite sets were an important part of

mathematics.

Once the natural numbers were defined and those theorems similar to the axioms of Peano

stated, Dedekind proceeded to define the structure of the natural numbers under the greater than

and less than relations. After this Dedekind stated an important result, one which Cantor would find

important in his own work on well-ordered sets:

¶96. “Theorem. In every part Tof N there exists one and only one least number k ,i.e. a

number kwhich is less than every other number contained in T .”

Thus, in any subset of the natural numbers, there is a minimal element, one smaller than

every other element in the subset, according to the order on the natural numbers. This can be

extended to any set that is similar to the natural numbers or a subset of the natural numbers; so it

applies to all simply infinite sets. This coincides with the theorem in paragraph 72, which says that

every simply infinite set has a subset that is similar to the set of natural numbers. Eventually

Dedekind proved that all simply infinite sets are similar to the set of natural numbers and these

simply infinite sets can be grouped into different classes corresponding to which subset of the

natural numbers they are similar to.

Today, we would say that this means that any simply infinite set is one that is countable and

therefore a model of the natural numbers. In the time of Dedekind, the natural numbers were

known to be countable (they count themselves essentially), but other number systems were a bit of

a grey area at the time, especially the real numbers. Cantor believed that the real numbers were of a

different size than the natural numbers, but this faced stiff opposition from Kronecker, who said that

the idea of two different (potentially) infinite sizes was ridiculous. However, to say that all simply

infinite sets can be classed together, one could work out that that would mean, since they do not

have a least element, the set of real numbers could not be put in to the same class as the natural

numbers. Elaborating on this, not every subset of the real numbers would have a least member, take

the open interval (0 ,1) and compare it with [0 ,1].

49


The need for Dedekind to distinguish between the finite and the infinite is an essential part of

Was Sollen. The differences between the two are laid out in simplistic form so that they can be easily

followed and more readily accepted. What Dedekind tried to establish, was to stop mathematicians

from tackling the infinite from a finite perspective. As mathematicians were applying properties and

theorems defined on the finite, to the infinite, they were not getting consistency or producing the

results that were expected. This is evident in the case of Galileo and Euclid’s axiom above and this

treatment of the infinite still happened in the 18th century. Thus, there were some who were

unwilling to investigate and work with the infinite. But the work of Dedekind tried to curb this way of

thinking and attract more mathematicians to accept the concept of the actual infinite.

Another important aspect of Was Sollen is Dedekind’s final section. It is entitled The Number

of Elements in a Finite System and it addresses the size of a finite set. The most notable paragraph in

this section is 161 which I shall give in its entirety:

¶161. “Definition. If a set Σis a finite system, then there exists one and only one single

number n to which Zn Σ, for a system Zn . This number n is called the number (Anzahl)

of elements in Σ; it shows how many elements are contained in Σ. If numbers are used

to express accurately this determinate property of finite systems they are called cardinal

numbers. If there is a similar transformation ψ :Zn→Σ ,then we can say that the

elements of Σare counted and set in order by ψ in determinate manner, and call am the

mth element of Σ; if m<n ,then amis called the element following am, and anis called the

last element. In this counting of the elements therefore the numbers m appear again as

ordinal numbers.”

Thus, the elements of the finite set can be indexed by the natural numbers and if the order

of the elements of the finite set are arranged such that they follow the order of their index, then we

can say that they follow the order of the natural numbers and can thus be called ordinal numbers, as

stated in Dedekind’s definition of the natural numbers. Conversely, the definition says that, if we

abstract from the order of the elements in a set, then we obtain the cardinal number of the set.

Dedekind does not state when a number is an ordinal number, but the implication is that this is

when the number follows the ordering of the numbers that precede it i. Both cardinal and ordinal

numbers were the key interests of cantor at the time, who was trying to introduce both to his

transfinite set theory. The backing of Dedekind would have been a major boost to their acceptance

as the opposition against Cantor and his theory was mounting.

i I think this should have said an is the number following am.

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Cantor began to work on infinite sets when the need to investigate further the cardinality of

infinite sets and the number of points in the continuum ii became clear. It appears that Dedekind saw

cardinality as an important aspect of every set; from the remark at the end of Was Sollen, Dedekind

states that the cardinality of a set does not change under similar functions, thus there can be many

sets that share the same cardinality and be collected together and treated as one and the same.

However, Dedekind does not investigate this as it “does not lie in the line of this memoir to go further

into their discussion”, nor is there any consideration of the cardinality of infinite sets, surely they just

have infinite cardinality anyway.

The new foundations of the infinite, and the finite, would have served as a basis for the work

of Georg Cantor, not for Cantor to build his work upon it as he had already begun to publish work on

his transfinite numbers, but for other mathematicians to reference any doubts in the work of Cantor.

ii Thankfully Cantor gives a better definition of ordinal number.

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[2.5] Georg Cantor:

Figure 14 -A mid-aged Georg Cantor.

Georg Ferdinand Ludwig Philipp Cantor was born, 1845, in St. Petersburg, Russia. Cantor’s

father was a broker who originally came from Denmark and upon retirement, moved the family to

Wiesbaden and then to Frankfurt in 1856 in search for a warmer climate and escape from the cruel

St. Petersburg winter climate. In 1862, Cantor moved to Zurich to study engineering in the

Polytechnic of Zurich at his father’s request. After much persuasion, Cantor was permitted to study

mathematics instead. When Cantor’s father died a year later, Cantor moved on to Berlin to study at

the university there.

In Berlin, Cantor came under the influence of the triumvirate that dominated mathematics in

Berlin and Germany at the time – Leopold Kronecker, Karl Weierstrass and Ernst Kummer. Two of

these mentors would play an important role in the remainder of Cantor’s life; one for the better, one

for the worst. The three had all come to the University of Berlin around the same time, both

Kronecker and then Kummer in 1855, followed by Weierstrass a year later (much to the effort of

Kummer). In fact, Kronecker was a former student of Kummer, so there was a close connection

between them and the three remained good friends for 20 years. Under the reign of these three

influential men, the university became one of the lading mathematical centres in the world; a great

attraction for a young mathematician like Cantor.

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In 1869 Cantor gave a seminar under Weierstrass where he had shown that the rational

numbers were denumerable, despite the fact that it was dense i, using a one-to-one correspondence.

At the age of 22, Cantor had completed his PhD at Berlin; having briefly taught at a girl’s school in

Berlin, Cantor was on the move again, this time to Halle to take up a Privatdozent position where he

worked and studied under Eduard Heine, a former student of Weierstrass, and in 1879 became a

professor of mathematics at Halle.

In the late 18th and early 19th Century’s, Joseph Fourier ii had been working on the problems

related to the conduction of heat. This sparked wide interest into the continuity of the convergence

of an infinite series and it led to much investigation on to the theory of functions. In turn, the theory

of functions branched into two different directions and they gradually took their own paths:

1. Dirichlet’siii foundations of Fourier’s results led to the investigation of real-valued functions

and the trigonometric development of functions.

2. Cauchy’siv and Weierstrass’ independent treatment of functions of a complex variable.

i The real numbers, or equivalently, the open interval (0 ,1) of the real numbers.ii A set is dense if, for a ,b in the set, there is always another element in the set, x, such that a< x<b with respect to the ordering on the set. iii Jean Baptiste Joseph Fourier (1768-1830) was a French mathematician and physicist known for his Fourier series.iv Johann Peter Gustav Lejeune Dirichlet (1805-1859) was a German mathematician who, when at Berlin, helped get Kummer elected there too.

53

Figure 15 – (L-R): Ernst Kummer (1816-1893), Karl Weierstrass (1815-1897), Leopard Kronecker (1823-1891).


Both directions influenced Riemanni to take up their investigation and it was he who

influenced both Hankelii to investigate the complex variable and Heine to look at the trigonometric

functions.

Under the influence of Heine, Cantor turned his attention to analysis; Heine had challenged

Cantor to tackle the uniqueness problem of representing a function as a trigonometric series which

had been unsuccessfully attempted by Heine himself, as well as Weierstrass and Riemann before.

Cantor developed a basic theorem that – if a trigonometric series converges to zero everywhere, then

all of its coefficients are zero. In the following two years, Cantor would release further papers on

trigonometric series.

In 1872 Cantor married a friend of his sister. Whilst on honeymoon in Switzerland, Cantor

became acquainted with Dedekind and that is where their friendship began. This was the year that

Dedekind published Stetigkeit und irrationale Zahlen which contained the first appearance of

Dedekind’s cuts which Dedekind used to define irrational numbers – a cut partitions the rational

numbers into two nonempty subsets S and T such that all the elements of S are less than those of

T and S has no greatest element. Then, if T does not have a smallest element then the cut defines

an irrational number that fills the ‘void’ left between S and T .

Although the real numbers were a common part of mathematics at the time, there was

strangely no proper definition for them. Others at this time were also inventing ways of defining the

real numbers in terms of rationals. The formulation of the real numbers allowed mathematics to

consider infinite collections and to consider them as unitary objects. Indeed Dedekind did this with

his cuts and ideals.

Cantor described his theorem above on trigonometric series in terms of point sets, or point

aggregates as Cantor referred to them at the time. For a collection P of real numbers define P(1 ) to

be the collection of limit points of P, and further define P(n ) to be the nth iteration of this operation,

i.e. P(n ) is the set of limit points of P(n−1 ). Cantor was able to summarize his 1870 theorem as if a

trigonometric series converges to zero everywhere, except in some P, where P(n ) is empty for some n,

then all the coefficients of the series are zero.

It was with this formulation that Cantor shifted his attention to the continuum in order to

further his work on the trigonometric series. Cantor took to the task of formulating the real numbers

in terms of sequences of rational numbers in 1872. The specific purpose of this was to formulate a

proof for the theorem above and thus Cantor was able to show that, if there existed n such that P(n )

i Augustin-Louis Cauchy (1789-1857), French mathematician.ii Georg Bernhard Riemann (1826-1866), German mathematician.

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was empty, then P was denumerable. Cantor wanted to then apply this method to the real

numbers.

In a letter to Dedekind on 2nd December 1873, Cantor expressed that it may be easier to

assume that there was no one-to-one correspondence between the real numbers and the natural

numbers, and thus the real numbers were not denumerable. Cantor’s main conclusion in this letter

was that, if the real numbers were not denumerable, then it would serve as a similar proof of

Liouville’si theorem that there exist transcendental numbersii. Five days later, Cantor wrote to

Dedekind again saying that he had discovered that it was impossible to produce a bijective

correspondence between the real and natural numbers.

In his Stetigkeit, Dedekind had stated, in his own terminology, that the cardinality of the real

numbers is greater than that of the rational numbers. What Dedekind overlooked was, how much

richer in cardinality the real numbers are than the rationals; Cantor, on the other hand, did not. The

1873 proof was sent to Weierstrass who was so impressed that he urged Cantor to publish it; it

appeared in Crelle’s Journal in 1874 under the title Über eine Eigenschaft des Inbegriffes aller reellen

algebraischen Zahlen, translated as On a Property of the Class of All Real Algebraic Numbers. The

title of the paper is misleading as it focused more on the real numbers than the algebraic numbers,

and the most important result of the paper was the undenumerable characteristic of the real

numbers.

The denumerability of the real numbers and existence of transcendental numbers rested

upon the proof that for any countable sequence of reals, every interval contains a real number that is

not in the sequence. Cantor showed this in the following argument:

For an interval I and a sequence of real numbers s, let a<b be the first two

numbers of s, if any, found in I . Then let a '<b ' be the first two numbers of s, if any, in

the open interval (a ,b); and then let a ' '<b' ' the first two numbers of s, if any, in the

interval (a ' , b' ); and so on. However long this process continues, the non-empty

intersection of these intervals cannot contain any member of s.

Although Cantor did not give any examples of transcendental numbers, he had proven that

they do exist. Later in the same year, it was proven by Charles Hermite that e is transcendental and

Ferdinand von Lindemann proved that π is also transcendental in 1882. Both would help in aiding

others to accept Cantor’s proof that the real numbers were not denumerable.

i Hermann Hankel (1839-1873), German mathematician.ii Joseph Liouville (1809-1882), French mathematician who proved that there exist transcendental numbers, in 1844, by constructing an infinite class of such numbers using continued fractions.

55


During 1877 and 1878, Cantor’s focus began to shift away from trigonometric series and to

bijective correspondences between two sets, specifically sets that are said to be well-ordered. No

more was Cantor worried about the trigonometric series. Cantor no longer cared if his investigations

into the continuum were significant to the trigonometric series; the existence of transcendental

numbers and the richness of the real numbers sparked a major interest into the continuum itself and

it would overwhelm and engulf the remainder of Cantor’s mathematical life. In this two year period,

Cantor released some important new concepts that would establish the foundations of his

transfinite numbers.

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[2.6] Cantor’s Quest for the Transfinite Numbers:

It is obvious from an early age that, given two whole numbers, one is bigger than the other,

and furthermore, given a collection of distinct numbers, then there is only one that can be the

smallest of them all. This concept is essentially well-orderedness, i.e. the natural numbers are well-

ordered.

A set S is well-ordered if:

1. There is a total-ordering between any pair of elements of the set, i.e. ∀a ,b∈S ,

a<b ,a=b , or a>b, where ¿ is the ordering of the elements in S.

2. For any nonempty T⊂ S, there is a unique least element, with respect to ¿, belonging to

T , i.e. ∃t∈T , such that ∀ t'∈T ,t<t ' .

In his memoirs of 1877 and 1878, Cantor developed the concept of one-to-one

correspondence and developed a few important results along with new definitions that would

correspond to his set theory of transfinite numbers. If two well-ordered sets can be put in to a one-

to-one correspondence then Cantor said they have the same power (Mächtigkeit) and for finite sets,

this would be when two sets have the same number (Anzahl) of elements. Cantor showed that any

proper subset (Bestandteil) of a finite set will always have a power that is less than that of its parent

set.

Cantor stressed that this may not necessarily be the case for an infinite parent set, for

example, the set of even integers and its parent set, the integers. Consequently, for infinite T and S

such that T⊂ S, then the power of T is less than the power of S if their two respective powers are

unequal. As for the natural numbers, Cantor said that they have the smallest, and first, power which

is infinite and that every infinite subset of the natural numbers (or any infinite set that is equivalent

to any subset of the natural numbers) has the same power as that of the natural numbers. The union

of such aggregates would also have that same power.

This bears a resemblance to Dedekind's definition of an infinite set; a set which is equivalent

to a proper subset of itself. In the same way, Cantor defined the power of an infinite set. All of this

helped Cantor to develop his idea that the real numbers were not denumerable i.e. the continuum

would have a different and greater power than that of the natural numbers and thus they cannot be

in a bijective correspondence to each other.

In 1877, Cantor wrote to Dedekind to show him a proof of a bijective correspondence

between the interval [0 ,1] and an n-dimensional space. This came as a shock even to Cantor who

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wrote “I see it, but I don’t believe it!” in the letter. Cantor published a proof of this in Crelle’s Journal

after much delay due to Kronecker’s refusal, as editor, to publish it.

Besides all this work on developing powers, Cantor could only find two possibilities for the

powers of infinite sets:

“Every infinite set of reals is either countable or has the power of the

continuum.”

Essentially, the set of real numbers is the next largest after the natural numbers. Cantor

referred to this as The Continuum Hypothesis and it became his main focus of investigation over the

next few years as he sought for a proof. After 1878, Cantor began to publish his work in in

Mathematische Annalen in which he desired to provide a foundation and introduction to his set

theory under the editorship of Felix Kleini. One such publication provided a simpler proof that the

continuum is not of the first power. Another, in 1882, introduced enumeral which was used to

describe a set that can be put into a one-to-one correspondence with the natural numbers, thus

having the same order as the natural numbers. The work done by Cantor in this year led to the

definition of a series of ascending infinite powers.

Having published four pieces of work in Klein’s Mathematische Annalen, the criticism and

opposition to Cantor’s set theory began to gather. The need to bring together the progress of the

transfinite numbers and present them clearly in order to win over its critiques had become clear to

Cantor. The fifth paper that Cantor published in Mathematische Annalen was 1883’s Grundlagen

einer allgemeinen Manningfaltigkeitslehre (Foundations of a General Theory of Aggregates) and it

was with this paper that Cantor hoped to sway his doubters. The Grundlagen contained as much

philosophy as it did mathematics and iterated, in the introduction to the original German text, that

the two were in fact related. As we have seen with Dedekind, it was not new to join mathematics

with philosophy, but Cantor was one of the first to give philosophy an equal measure to

mathematics since the latter had regressed from philosophy under the influence of the ancient

Greeks.

The Grundlagen was free of any opposition against its publication as Cantor had found an

ally in Klein and thus, Cantor was free to publish the first defence of the actual infinite, before its use

by Dedekind in 1888’s Was Sind. Grundlagen was a way to show that the old traditional customary

objections of the actual infinity should be dropped to allow it to be investigated properly. Cantor

wanted to present it in such a way that not even Aristotle could even rejects is claims. Cantor

believed that simply arguing the justification of his new ideas were no longer enough and that he

i These are real numbers that are not the root of a polynomial equation with integer coefficients.

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had to defend his work from any external attack, so Cantor tackled any area that he thought possible

of rejection with a philosophical mind and laid them out in the Grundlagen.

Cantor credits Leibnizi and Bolzanoii in the Grundlagen for the effect their work on the actual

infinity had on his approach and idea of the subject. According to Cantor, Leibniz believed that

nature makes frequent use of the actual infinite everywhere and that “the last particle ought to be

thought of as a world full of an infinity of different creatures.” Cantor accepted this idea and

developed upon it. Bolzano’s work on the actual infinite considered the paradoxes that surrounded

it in mathematics, trying to explain their creation and, if any, justification. Bolzano introduced

completed infinities into mathematics that were without contradiction and worked on separating the

differences between the potential and actual infinity, much like Cantor does in the Grundlagen.

The natural numbers are defined successively, adding one to a preceding number that has

already been defined. In the Grundlagen, Cantor defines this process as the first principle of

generation (Erzeugungsprincip).

“The infinite numbers, if they are to be considered in any form at all, must (in

their contrast to the finite numbers) constitute an entirely new kind of number, whose

nature is entirely dependent upon the nature of things and is an object of research, but

not of our arbitrariness or prejudices.”

With this, Cantor dropped the potential infinity of ∞ and introduced a new number to

represent the actual infinite, ω, the last letter of the Greek alphabet, omega; no more was the Greek

concept of number that had abolished the actual infinity. This new number was not the greatest of

the setiii {1 ,2 ,3 ,4 ,…}, but a number that expresses that this whole set is given by its natural law

of succession. There is no greatest number in the finite numbers, as to think of one, say k , then we

can always have a k+1, thus obtaining a potential infinite. But there is no contradiction in assuming

that there is a new, non-finite number that follows all the finite numbers and is the first number to

do so. Thus ω was given this property, the first non-finite number to follow the succession of finite

numbers, the first transfinite number.

The infinite was avoided due to its contradiction of finite properties, hence Aristotle’s

abolishment, as we have seen. Aristotle assumed that there only existed finite numbers, for each

number that was reached became finite, adopting a potential infinite perception. Consequently, it

was believed that only enumeration of finite numbers was possible; eluding all consideration of

infinite numbers from an early stage. Aristotle believed that the sum of any finite number and an

i Felix Christian Klein (1849-1925), German mathematician. ii Gottfried Wilhelm Leibniz (1646-1716), German mathematician and philosopher.iii Bernhard Bolzano, (1781-1848), Bohemian mathematician, philosopher and Catholic priest.

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infinite one would annihilate/engulf the finite number i.e. ∞+a=∞. However, in Grundlagen,

Cantor showed the distinction between ω and ω+1.i Cantor showed justification in the infinite and

that it is improper to think that infinite numbers should be constrained to the properties of the finite

numbers.

With this, Cantor extended ω to ω+1,ω+2 ,…,ω+n ,… which also has no greatest

number. This is referred to as set theory because we allow ω to represent the set of natural

numbers or as a collection of points, which has one end point and is not dense. Then ω+1

represents the set of natural numbers along with an additional point, represented by a number,

assigned to the end to which there is no point. The natural numbers of ω increase towards this

number but they never reach it; there will always be a chasm of numbers in between this number

and any specific number in the natural numbers. Thus, ω+n represents the set of natural numbers

with n new numbers, in their ordering, succeeding it. The numbers in the set of natural numbers will

never reach the n new numbers assigned to the end, i.e. the collection of natural numbers are

always regarded as being less than the n adjoined numbers to the end, with respect to the

successive ordering of the naturals.

A simpler way of thinking of this is to represent the natural numbers as the von Neumann ii

natural numbersiii where each natural number simply represents a specific set, where the elements

are the natural numbers which precede.

0 : {∅ } ,1 : {0 } ,2: {0 ,1 } ,…,n+1: {0 ,1 ,2 ,…,n },…

Thus, k represents the set with k elements in it, arranged from 0 to k−1. Alternatively we

can define the von Neumann numbers inductively by

n+1 : {0 ,1 ,2 ,…, (n−1 ) }∪ {n }=n∪{n }.

We can define new transfinite numbers using this inductive method too:

ω :{0 ,1 ,2 ,…,n ,…}

ω+1:ω∪ {ω}={0 ,1,2 ,…,n ,…,ω}

ω+2=(ω+1 )+1:(ω+1)∪ {ω+1 }={0 ,1,2 ,…,n ,…,ω ,ω+1 }

… … …

ω+ (n+1 ) : (ω+n )∪ {(ω+n ) }={0 ,1,2 ,…,n ,…,ω ,ω+1 ,…,ω+n } .

i Cantor now uses set (Menge) as he shifted away from the point-manifold and point aggregate of his trigonometric interests.ii Although 1+ω=ω – think of Hilbert’s paradox of the Grand Hotel. iii John von Neumann (1903-1957), Hungarian mathematician.

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Cantor, in the same way that it is possible to imagine ω, deems it possible to imagine

ω+ω=2ω, and again we obtain the sequence 2ω+1 ,…,2ω+n ,… .This process can be extended

to higher multiples of ω.

The sequence ω ,2ω,3ω,…,nω,… Cantor defines as the second principle of generation.

Combining the first and second principles together we obtain the sequence:

1 ,2 ,…,n ,…,ω,ω+1 ,ω+2 ,…,2ω,2ω+1 ,…,3ω,…,aω,aω+1 ,…. .

This new sequence has no greatest number aω+n, so again Cantor defines a new number

that is greater than all the numbers of the form aω+n and denotes it as ω2. In the same way that

the natural numbers never reach ω, the sequence of finite and transfinite numbers above never

reaches ω2. In a similar fashion as previously shown, ω2 can be extended to transfinite numbers of

the form bω2+aω+n. Building upon this to higher powers of ω, Cantor produced sequences of

numbers of the form v0ωμ+v1ω

μ−1+…+vμ−1ω+vμ where μ , v0 , v1 ,…,vμ are all finite, natural

numbers, but not all 0.

With μ increasing ‘towards infinity’ it will never reach infinity, so by Cantor’s reckoning it

would be possible to introduce ωω such that the second principle of generation tends towards this

number but never reaches it. Thus any number that precedes ωω is of the form

v0ωμ+v1ω

μ−1+…+vμ−1ω+vμ as defined above.

Figure 16 - Each rotation represents one multiple of ω.

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The series of transfinite numbers was shown by Cantor to fall in to different number classes

by one-to-one relations so that one and only one power belonged to each number class. The first

number class is 1 ,2 ,3 ,…,n ,… whose aggregate, the natural numbers, has the first power. The ω’s

– ω ,ω+1 ,…, (v¿¿0ωμ+v1ωμ−1+…+vμ−1ω+vμ),…,ωω ,…,α ,…¿ – were deemed by Cantor

as the second number class whose aggregate has a different power from that of the first power and

this power is the next higher power, such that there is no power in between it and the first.

Following on from the second class of new numbers, the second principle demanded the

creation of a new number which was greater than all of the numbers in the second number class.

And Cantor gave it one, Ω, which is the first number of the third number class. This can be extended

further to create further number classes. There was a third principle given by Cantor in the

Grundlagen which was called the limiting principle and it restricted each number class to its own

numbers.

Cantor extends his earlier definition of an enumeral of a set to represent the power as well

as the order of the elements set upon by the set. Thus, enumeral is not exclusively restricted to

those sets that are isomorphic to the natural numbers. For finite sets, the enumeral remained the

same, no matter what order was placed upon its elements. However, Cantor stated that this is not

the same with infinite sets; an infinite set will have a different enumeral when its order is altered.

Cantor understood a well-ordered set to be one that has a definite succession upon its

elements, such that there exist a first, or a least, element from which every other element succeeds

it. In the Grundlagen, Cantor showed that two well-ordered sets, S andT , have the same enumeral

if they can be put in to a one-to-one correspondence such that, for a ,b∈S, and their corresponding

a ' , b'∈T , if a<b, then a '<b ' and vice-versa.

A connection between enumeral and power does exist. Cantor explains this connection as

follows:

“Every aggregate of the power of the first class is enumerable by numbers of the

second class and only by these, and the aggregate can always be so ordered that it is

enumerated by any prescribed number of the second class; and analogously for the

higher classes.”

Thus, the enumeral of any well-ordered set that has the first power is a number belonging to

the second number class. Furthermore, any set of the first power can be put into an ordering such

that its enumeral is any one of the second number class.

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Addition and multiplication of the new transfinite numbers are defined by Cantor.

Associativity does hold for addition and multiplication, but commutation does not; a characteristic

that distinguishes the arithmetic of the transfinite numbers from the finite numbers.

The key concept of the Grundlagen was well-ordering. The 1883 piece contained Cantor’s

Well-Ordering Principle – it is always possible to put a well-defined set into the form of a well-

ordered set. Cantor saw this merely as a “law of thought”, yet “fundamental, rich in consequences,

and particularly remarkable for its general validity”. Although a simple concept, it was an essential

one for the development of Cantor’s set theory. It was consistent with Cantor’s view that the finite

and transfinite make up one entity; that they coincide with one another in mathematics and one

should not take precedence over the other.

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[2.7] Cantor’s Struggle:

As we have seen, Cantor put a lot of philosophy in to the Grundlagen in order to formulate

and justify the introduction of the transfinite numbers. Striving for the acceptance of this new area

of mathematics, Cantor reminded his readers how mathematics was initially formed and how it was

allowed to grow and develop; reminding his readers how other new concepts had been introduced

to mathematics even though they are as abstract and removed from the metaphysics as the

transfinite numbers. Mathematics is open to the introduction of new concepts:

“Mathematics is, in its development, quite free, and only subject to the self-

evident condition that its conceptions are both free from contradiction in themselves

and stand in fixed relations, arranged by definitions, to previously formed and tested

conceptions. In particular, in the introduction of new numbers, it is only obligatory to

give such definitions of them as will afford them such a definiteness, and, under certain

circumstances, such a relation to the older numbers, as permits them to be distinguished

from one another in given cases. As soon as a number satisfies all these conditions, it

can and must be considered as existent and real in mathematics. In this I see the

grounds on which we must regard the rational, irrational, and complex numbers as just

as existent as the positive integers.”

Cantor believed that we can think of ω as the limit to which all finite numbers tend towards,

in the same way that √2is the limit of a certain increasing sequence of rational numbers. However

there is a slight difference between thinking of ω and √2 as limits of sequences. The difference

between these rational numbers and √2 becomes infinitely smaller as the sequence continues,

whereas ω−n=ω. This does not mean that ω should be thought of as less definite or complete

than √2, as there are as many numbers tending towards ω as there are towards √2.

Thus, the transfinite numbers can be considered as the new irrationals. Cantor argued that

the same principle applies to defining the transfinite numbers as does to defining the irrational

numbers which represent an infinite series of decimal places or continued fractions. Therefore if the

transfinite numbers are to be rejected, then so should the irrational numbers. But Kronecker would

have probably rejected both anyway. I feel that a similar argument could have been to argue the

introduction of the complex numbers which depends on a number that is known not to exist.

Cantor argued on behalf of his transfinite numbers using the irrational numbers in his 1883

Freiberg lecture and again in lectures in 1884 and 1886. The introduction of Kummer’s ideals was

used as an example of how mathematics is open to the introduction of new concepts in the

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Grundlagen. Using such an example could have been to try to win over Kronecker who, although

having fallen out with Weierstrass at this stage over the work of Cantor, remained close friends with

Kummeri. This was a typical trait of Kronecker, to attack others or to fall out with them if they

disagreed with him mathematically.

Leopard Kronecker was a popular mathematician in his day, yet he did have his fair share of

foes. The main interest of Kronecker’s work was in the theory of equations and number theory, with

his main contributions being afforded to elliptic functions and the theory of algebraic numbers.

Never known for having a lot of students in his lectures, Kronecker restricted himself to the finite

within mathematics, believing that all of mathematics can be reduced to the integers and a finite

number of steps. Kronecker is famously quoted as to saying “God created the integers, all else is the

work of man.” This is how he limited himself, and tried to limit others, to essentially the potential

infinite.

In 1882, Lindemann proved that π is a transcendental number and having presented his

proof in front of Kronecker that year, Kronecker was full of praise and complements for Lindemann,

but only for the quality of the proof. Kronecker regarded the proof as redundant as it proved nothing

since transcendental numbers do not exist. Kronecker was a highly influential man, mostly because

of the positions he held within mathematics, most notably his position in Berlin, being co-director of

the Berlin seminar founded by Kummer and Weierstrass, and his position as editor of Crelle’s

Journal, but also because he made it his aim to establish various foreign contacts and offer them the

hospitality of his own home in Berlin.

It is obvious that Kronecker was a difficult man, especially to work with. His views were often

different from those of others and it was his personality that often forced him to try and supress

those views of others with his own. Weierstrass even believed that Kronecker was going as far as

trying to demerit the value of his work, by telling his students that the work of Weierstrass was of no

value to them or the future of mathematics. With this, Weierstrass was ready to leave Berlin and

move to Switzerland but he soon changed his mind as he knew that Kronecker’s position in the

mathematical world would simply increase with his departure from Berlin, probably making him

more unbearable.

The views of Kronecker were well known to his fellow mathematicians even though he had

never voiced them publically. That changed in 1886 when he argued against the use of irrational

numbers by Heine, Dedekind and Cantor:

i I include 0 now as Cantor does not omit it from any of his work.

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“the introduction of various concepts by the help of which it has frequently been

attempted in recent times (but first by Heine) to conceive and establish the “irrationals”

in general. Even the concept of an infinite series, for example one which increases

according to definite powers of variables …certain assumptions must be shown to hold

which are applicable to the series like finite expressions, and which thus make the

extension beyond the concept of a finite series really unnecessary.”

Singling Heine out to the public in 1886 was not the only time that Kronecker had attempted

to diminish his work. Previously Kronecker tried to prevent Heine from publishing his Über

trigonometrische Reihen but Heine prevailed, much to his own satisfaction, in having it published:

“My little work “On trigonometric series”, the publisher’s proof of which I have at

this moment in my hands, and which, moreover, had been greatly debated with

Kronecker who wanted to persuade me to withdraw it, has appeared in the current (71st)

volume of the journal, and has made me very happy. I sent it to Borchardt in February.

Kronecker saw it, and kept it without my knowing until I came to Berlin.”

Heine’s remark really summarizes the type of man that Kronecker was; he would have been

a pain to work with or to study under. However, whilst at the University of Berlin, Cantor was shown

great support by Kronecker who was an admirer of his early work on number theory, even offering

advice to Cantor. There had been a sense of admiration towards Kronecker from Cantor when he

was a student. This was to change when Cantor sought to further the theory of numbers for the

benefit of his work on the trigonometric series.

Cantor would have known, more so than some others, the stance Kronecker had towards

mathematics and would have been aware of the previous situation between Heine and Kronecker.

This would have been in the back of Cantor’s mind when he went to publish On a Property of the

Collection of the Real Algebraic Numbers in 1874. Cantor’s first major publication was more focused

on the real numbers than the algebraic numbers. From correspondence with Dedekind, it was

known that Cantor was not even working on algebraic numbers, but it was an area of mathematics

in which Kronecker was deeply involved.

The reason most believed for Cantor giving a misleading title to his work was so it could

easily get past the eye of Kronecker and so it did. The main result of the 1872 paper said that the

real numbers were not denumerable. This was contradictory of Kronecker’s mathematical beliefs

and he certainly would not have knowingly allowed Cantor to publish any such result in Crelle’s

Journal under his editorship. In a letter to Dedekind in 1873, Cantor said he gave the paper the title

due to “conditions in Berlin”, although he did not elaborate any further.

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This act of ‘betrayal’ to Kronecker triggered the beginning of a rift between the two men. In

the time after the incident, Kronecker failed to agree with much of Cantor’s work and the transfinite

numbers, although there was never any chance that Kronecker would accept them, came under

heavy fire from Kronecker who made it his objective to highlight any possible flaws. Being in a highly

influential position, Kronecker’s opinions were often picked up by others.

Cantor’s 1877 result, which we can at this point state as ‘R and Rn have the same power’ was

the next major result on the continuum that Cantor wanted to publish in Crelle’s Journal. This time

Cantor did not try and disguise the result behind a misleading title; its publication was initially

refused by Kronecker. Cantor wrote to Dedekind to vent his frustration at Kronecker’s spineless act

and in turn Dedekind wrote to Kronecker to persuade him to change his mind on the publication.

The paper appeared in the journal in the following year with Kronecker making sorry excuses for its

delay. It was the last time that Cantor submitted anything to be published in Crelle’s Journal.

Shortly after this, Cantor was to end his correspondence with Dedekind. With the death of

Heine in 1881, Cantor recommended Dedekind for the vacant position left behind at Halle. Dedekind

turned down the opportunity in favour of remaining where he was. We have seen that Dedekind

lived a slightly secluded life, so it does not come as much of a surprise that he did refuse the

opportunity presented to him. The decision did not sit well with Cantor and they began to

correspond less from 1882 onwards.

At the same time Cantor began to communicate with Mittag-Leffler i and began to publish in

the Frenchmen’s Acta Mathematica which coincided with publishing in Mathematische Annalen.

This relationship was a short-lived one. They fell out when Cantor decided to no longer publish with

Acta Mathematica after Mittag-Leffler persuaded Cantor to withdraw a submission as a proof was

“about one hundred years too soon”.

Opposition towards Cantor’s set theory did come from other mathematicians than

Kronecker. Old university friend Schwarzii ended his long correspondence with Cantor in 1880 due to

the levels of opposition that was gathering against Cantor’s work and he no longer supported the

direction of Cantor. Poincaréiii had translated Cantor’s work submitted with Mathematische Annalen.

Initially a supporter of Cantor, Poincaré began to doubt his work on the transfinite numbers when

paradoxes were found by Russell and Burali-Forti. Poincaré went as far as describing Cantor’s work

as “a perverse mathematical illness that would one day be cured.”

i The friendship may have been a tenuous one at the time as Kummer tried to remain friends with both Kronecker and Weierstrass after their falling out in 1875, up until his retirement in 1883.ii Magnus Gösta Mittag-Leffler (1846-1927), Swedish mathematician.iii Hermann Amandus Schwarz, (1843-1921), German mathematician.

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[2.8] Post-Grundlagen:

Later in September 1883, Cantor gave a lecture in Freiberg on his transfinite numbers in

which he made some significant changes that have remained in the subject until now. Cantor

emphasized a key aspect to the introduction of his transfinite numbers was – when a set has a

proper subset that has the same power as the set itself, then both the set and that subset are infinite .

Then, for the first time, Cantor begins to refer to power as cardinal number and introduces ordinal

type for sets, with ordinal number representing the order type of a well-ordered set and replacing

the old enumeral of the Grundlagen. The root of the transfinite ordinal numbersi lies in the words of

Cantor:

“Two ordered aggregates have one and the same ordinal type if they stand to

one another in the relation of ‘similarity’, which relation will be exactly defined.”

At the end of the lecture, Cantor promised to publish all of his results, complete with their

new definitions, in the near future, although the first contribution to a complete version would not

come until 1895.

The ordinal numbers were to provide the framework for Cantor to tackle his Continuum

Hypothesis. Cantor began to advance his work on order types as the Grundlagen mainly focused on

the well-ordered case of the ordinal numbers. Comparing the well-ordered natural numbers to the

simply-orderediiii rationals, Cantor stated that, unlike the natural numbers, the rationals were dense,

i.e. for any two rational numbers there is always a third in between them. Furthermore, the real

numbers are also simply-ordered but they were different in ordinal type from both the natural and

rational numbers. Cantor hoped that looking at order types would help him solve the Continuum

Hypothesis.

There were two other approaches that Cantor hoped would help solve the Continuum

Hypothesis. One approach was through cardinality. This allowed Cantor to reduce the Continuum

Hypothesis down to – the real numbers and the second number class have the same cardinality. This

came from showing that the second number class was shown to be uncountable and any of its

infinite subsets were either countable or similar to it. Thus the second number class has the same

properties of the real numbers. However, Cantor was unable to develop this further as he could not

produce a well-ordering for the real numbers.

i Jules Henri Poincaré (1854-1912), French mathematician, he fell only began to doubt Cantor in the late 1890’s.ii As we have seen with Dedekind, the natural numbers are ordinal numbers, and thus the finite ordinal numbers, if they are considered as sets of the numbers that come before them and in their natural order, like the von Neumann natural numbers.

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The other approach was through his definable P sets of real numbers which Cantor had

developed through his work on trigonometry. Cantor came to define perfect sets as those which are

non-empty, closed and containing no isolated points. In 1884, Cantor proved that any uncountable

closed set of real numbers is the union of a perfect set and a countable one and therefore, closed

sets are countable ones. Cantor went on to prove that perfects sets have the power of the

continuum and therefore the Continuum Hypothesis holds for closed sets, i.e. any closed set is

countable or has the power of the continuum. This reduced Cantor’s problem of solving the

Continuum Hypothesis down to determining whether or not there is a closed set of real numbers of

the power of the second number class.

Cantor had thought that he had found a proof for his Continuum Hypothesis in 1884 but the

next day he felt that it contained a mistake. Once the mistake was solved he later rejected the proof

and did not publish it. In May that year, Cantor began to suffer from depression and had a

breakdown that lasted for a few weeks. The cause of Cantor’s breakdown was thought by most at

the time to be the result of his troubles within mathematics and stemming from the difficult

situation that had developed between himself and Kronecker. Once recovered, Cantor decided to

offer Kronecker reconciliation to which Kronecker accepted, but their different philosophical stances

on the infinite would soon again disrupt any possible friendship.

It was about this time that Cantor began to stop corresponding with Mittag-Leffler. Not all

was well in Cantor’s mathematical career; he felt discontent, that his career had failed to take off as

he had hoped and that his work lacked the appreciation that he thought it deserved. Dissatisfaction

began to settle in in the latter half of the 80’s and Cantor began to turn his interests towards other

areas of mathematics. Cantor also branched out to other subjects. With most of his publications

being found in philosophy journals, Cantor expressed his desire to teach philosophy at Halle as

opposed to mathematics, however his request was refused.

This all coincided with releasing a Christianity pamphlet and trying to prove that the work of

Shakespeare was actually rip-off versions of the work by Bacon. Cantor was scared to publish

anything new on his transfinite numbers out of fear of attacks by other mathematicians. In 1887

Cantor is credited with saying “at present of course I observe little interest in my mathematical

research.” However, Cantor only gave up on having his results published as he continued to work in

solidarity on his transfinite numbers.

In 1885 Cantor withdrew a series of papers, Principien einer Theorie der Ordungstypen, from

Mittag-Leffler’s Acta Mathematica. The following year, having been contacted by Mittag-Leffler’s

assistant, Cantor expressed his discontent towards his own work:

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“as for your enquiry concerning my work, I am increasingly convinced that there

is no reason to rush publishing it, since for now there is no great interest in the

transfinite numbers.”

The final memoir of Principien einer Theorie der Ordunstypen was an important one as

Cantor had introduced a couple of new concepts that helped him further expand his set theory. First

of all, Cantor began with a new definition of power which was more abstract and general; it had

been separated from any relation to any specific power already defined on the natural, rational or

real numbers:

“The power of a set M is determined as the concept of that which is common to all sets

equivalent to the set M and only these, and thus also common to the set M itself… I take

it to be the most original (both psychologically and methodologically) and the simplest

basic concept arising by abstraction from all particulars which can represent a set of a

definite class, both with respect to the character of its elements and with respect to the

connections and orderings between the elements, be it with respect to one another or to

objects lying outside the set. Insofar as one reflects only upon that which is common to

all sets belonging to one and the same class, the concept of power or valence arises.”

With this new abstract definition, Cantor enabled himself to develop a more abstract

structure of his set theory. In a similar fashion, Cantor gave a new definition for simple-ordered sets

that was more abstract than any definition he had given for it before:

“Every simply-ordered set has a definite order type… by this I mean that general

concept under which all sets similar to the given set fall, and only those (including the

given ordered set itself).”

I.e. a simply-ordered set is a set S such that, ∀ a ,b∈S ,a<b ,a=b ,or a>b. The new

definition showed that a well-defined set was obviously a simply-ordered set and it allowed Cantor

to define the order type of the rational numbers, which was a huge breakthrough for Cantor; as we

have seen, page 63, Cantor had looked to investigate the difference between the characteristics of

the natural, rational and real numbers and now he was able to represent those differences with his

transfinite numbers.

The ordinal type, and thus the ordinal number, of the natural numbers was given as ω.

However, it was now to be thought of more abstractly, as the order type of any well-ordered set that

is not dense. The ordinal type of the rational numbers was given in Principien as η, however only in

the interval (0 ,1). Cantor deemed this to be their natural order and they were everywhere dense

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here. The concepts of η were generalized, abstract from the rational numbers, as the order type of

simply-ordered sets that are dense. Hence the contrasts between η and ω.

“If M is a simply-ordered set of the first power which has neither a least nor a greatest

element, and which is so constituted that between any two elements e and e ' there is

an infinite number of other elements, then M has the order type of η.”

The order type of the reals numbers, again on the interval (0 ,1), was given as θ, however

Cantor did not go into much more detail on θ as η was given much consideration. The reverse of an

order type was the order type of a set, whose order has been reversed and was denoted by an

adjoined by an asterisk. So for the order type α , its reverse was given as *α . It was noted that *α=α

for all finite ordinal types. However, this is not always the case for infinite sets nor is it ever the case

that ordinal numbers for well-ordered sets are equal.

This again signalled that a well-ordered set is a special kind of simply-ordered set and the

reason why Cantor pays more attention to simply-ordered sets than he did in Grundlagen as they

allowed him to form some foundations for well-ordered sets which is what the set of natural

numbers is, well-ordered. Furthermore, Cantor defined addition and multiplication for order types of

simply ordered types.

It would be three years before Cantor felt ready to publish another paper on his transfinite

numbers; however there were only two new concepts that Cantor introduced to his set theory and

their introduction was only to help distinguish further the differences between cardinal number and

ordinal type. In order to do this, Cantor again mounts a more philosophical construction. The 1888

paper was entitled Theorie der Ordungstypen and was part of Cantor’s Mittheilungen zur Lehre vom

Transfiniten series which did not produce any significant new results, it only expanded slightly on

that which had already been developed.

In the paper, Cantor describes what he meant by a set – it should be thought of as a “thing in

itself”, whether it is of distinct objects or of universal concepts, with or without an ordering. Nothing

new here, but Cantor is still trying to argue his ideas philosophically and without any ambiguity.

Cantor then introduces M to represent the power of a set M , with the double bar implying that

there is a double abstraction to be taken from M in order to think of its power; one from the nature

of its elements and the other from any order of its elements. If only one abstraction is taken, and it is

from the nature of the elements, then Cantor denoted this as M and this was to denote the order of

the set M .

Cantor argues in support of the introduction of these new concepts and the results that they

produce, saying that they are justified by “the logical power of proof”. The new concepts do make it

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easier to visualize the cardinality and order type of a set; they also make it easier to distinguish

between the two.

However, introducing yet again more new concepts was always going to be met by

opposition, especially from those who did not support the earlier introduction of other concepts.

There were some who wanted to know of a relation between the cardinal numbers and order types,

and which one should be given greater importance within mathematics. Cantor emphasized that the

concept of cardinal umber should be independent and separate from that of the order types and

ordinal numbers. But the introduction of the notation above could have led some to think that the

two were indeed connected to one another. This could have only gone against Cantor.

Having again found opposition to his wok, Cantor became discontent with mathematics yet

again. However over the next three years Cantor continued to work on his transfinite numbers and

in 1891, provided his diagonal argument. The year before, Cantor had founded the Deutsche

Mathematiker-Vereinigungi after the death of Rudolf Clebschii; it served as a breakaway by

mathematicians from the Society of German Scientists and Physicians so that they could have their

own independent body and hold their own conferences separate from the other areas of science. It

was at the first meeting of the Society, held in Halle in September 1891 that Cantor presented his

new method (Cantor was also elected as the first president of the Society).

The diagonal method serves as an alternative proof of Cantor’s 1874 proof that the real

numbers were not denumerable. Cantor states that this new proof ignores the existence of the

irrationals in order to avoid the rejection of using the irrational numbers with the real numbers

which was held by some, most notably Kronecker.

A sketch of the method is as follows:

Take two elements m and w . Consider the collection M of elements of the type

E=(x1 , x2 ,…,x p ,…) where each x i is represented by either m or w . Then, for the

simply infinite sequence S=E1 , E2 ,…, Eμ ,… of elements of M , there is always an

element E0 belonging to M which does not correspond to any of the infinite sequence

S.

iii A set is simply-ordered if, for any 2 elements in the set, one precedes the other, based on some relation.ii The German Mathematical Society.

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Figure 17 - Illustration of Cantor's Diagonal Argument.

Cantor used, as an example, the real numbers on the interval (0 ,1) to show how the

method works. This example established that, over this interval, the reals were of a smaller power

than the collection of single-valued functions f ( x ) which map any x∈(0 ,1) to 0 or 1. In showing

this, it was also shown that, for any set M , the collection of all functions of M to a set which only

consists of two elements, is of a higher power than that of M i.e. the cardinality of a set is less than

that of its power set. The argument established that there are distinct powers among infinite sets

and it also provided Cantor with a method of establishing that the ascending sequence of powers for

well-defined sets had no maximum.

The diagonal method was used by Cantor, in 1892, to prove that 2, raised to the power of a

transfinite cardinal number, gives rise to a cardinal number which is greater than the former cardinal

number. This was the basis for Cantor’s method of covering which would massively contribute to

advancing his theory of transfinite numbers.

Now that Cantor had proven that the real numbers were not denumerable without any

consideration of irrational numbers, he may have thought that he could draw the proof to the

attention of Kronecker in order to figure out the true cause of his criticism towards the transfinite

numbers. Cantor even invited Kronecker to address the opening meeting of the Deutsche

Mathematiker-Vereinigung; however Kronecker was unable to attend due to his wife having had a

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serious climbing accident from which she later died from. Cantor would never get the chance to

show his new diagonal argument to his old mentor as Kronecker died soon after the passing of his

wife.

Cantor remained president of the Deutsche Mathematiker-Vereinigung for the next two

years but he again took ill in September of 1893 and gave up his position. The following year Cantor

oddly published a paper listing all the ways which the even numbers up to 10000 can be written as

the sum of two primes; something that had previously been shown by Goldbach’si Conjecture 40

years beforeii.

Due to illness and discontent with mathematics, Cantor made no new significant

publications on his transfinite numbers since the Grundlagen. There was the introduction of new

definitions to replace power and enumeral, and the diagonal method was used as an alternative to

the 1874 proof that the reals were not denumerable, but it was not used to prove anything new.

There was still the issue of the Continuum Hypothesis which remained unsolved. Cantor had still

been working on fine tuning his set theory, releasing little snapshots of some of that progress

through the new definitions, the M and M , and the diagonal argument too. However it still

remained for Cantor to bring all these new post-Grundlagen concepts together and show the overall

progress of the transfinite numbers.

i Rudolf Friedrich Alfred Clebsch (1833-1872), German mathematician who initially had the idea of founding the Deutsche Mathematiker-Vereinigung. ii Christian Goldbach (1690-1764), German mathematician.

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[2.9] Contributions to the Founding Theory of Transfinite Numbers:

[2.2.1] Part I:In 1895, Cantor published one half of his final contribution on transfinite numbers; the

second half coming two years later in 1897. Both were published in Mathematische Annalen under

the title Beiträge zur Begründung der transfiniten Megenlehre. The first part of the Beiträge was

focused on the study of simply ordered sets and cardinality, with the second concentrating on well-

ordered sets and the ordinal numbers. Cantor had completed the second half of the Beiträge six

months after the first was published but he had hoped to include a proof of the Continuum

Hypothesis, but it never materialized and hence the two year gap.

With all the experience of what came before with the infinite and introduced his transfinite

numbers, Cantor was well prepared of any criticism and backlash that could possibly come his way,

so he wrote in the Beiträge in a straightforward structure. In the Beiträge more of the arguments

were mathematical ones as opposed to philosophical ones. With the Grundlagen, Cantor began with

a lot of philosophical justification and explanation of the infinite, whereas in the Beiträge, the basics

of set theory are addressed. There is a more mathematical structure to this piece of work, as if

Cantor was actually writing a mathematical text; there was no need, to Cantor, to justify anything

anymore. At the end of the day, mathematicians will be judged on the mathematics they produce.

Thus Cantor tried to provide a simple and clear presentation of all the fundamental principles of his

transfinite set theory.

The Beiträge begins with the definition of a set and it really highlights how Cantor has

moved on from the trigonometric point sets and towards a whole new general theory of his own:

“By an “aggregate” (Menge) we are to understand any collection into a whole

(Zusammenfassung zu einem Ganzen) M of definite and separate objects m of our

intuition or our thought. These objects are called the ‘elements’ of M”

Cantor reiterated the concept that he introduced in 1887 which allowed for a cardinal

number to be representative of a double abstraction, the M . The concept, with the definition of

cardinal number, now allowed sets to be thought of in a more abstract way, free from concrete

objects and it allowed cardinal number to be thought of as a characteristic of sets; a concept of the

theory of sets. Cantor emphasized in his definition of cardinal number that the act of abstraction

should be “by means of our active thought”. From hence forward, cardinality followed from the

existence of sets. This whole abstract nature and the separation from any other area of mathematics

helped bring transfinite set theory towards an independent mathematical structure and

mathematical acceptance.

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The definition of cardinal number in the Beiträge is given as:

“We will by the name ‘power’ or ‘cardinal number’ of M the general concept

which arises, by means of our active faculty of thought, arises from the aggregate M

when we make abstraction of the nature of its various elements m and of the order in

which they are given.”

Two sets are equivalent if they are in a one-to-one correspondence. For the sets M and N , if

they are equivalent then this relation is denoted by M N in the Beiträge. And as we know, if two

sets can be put into a one-to-one correspondence then they have the same cardinal number, then

for our sets M and N , we denote this by M=N . Since the cardinal numbers were to be thought of

as a new type of number, different from the finite numbers, they were to be given an ordering by

Cantor, so he introduced the concept of greater than and less than for the cardinal numbers.

For the sets M and N , with the cardinal numbers a=M and b=N , the concept of a<b is

defined under the conditions:

(a) “There is no part of M which is equivalent to N”,

(b) “There is a part N1 of N , such that N1 M ”.

This relation was shown to be transitive by Cantor and only one of the relations

a<b ,a=b ,a>b can hold. The latter theorem is “by no means self-evident” and would take “a

survey over the ascending sequence of transfinite cardinal numbers” to prove. However Cantor was

unable to prove that exactly one relation must hold and it gave rise to the question – are the

cardinal numbers strictly comparable? This would later become known as Cardinal Comparability.

With this followed a series of theorems regarding equivalence which can be proven by

corresponding cardinal numbers. But these theorems were all based upon the preceding one which

gave rise to the cardinal comparability problem.

Thus Cantor’s hope of showing a complete piece of work on the transfinite numbers that

could persuade others of its justification within mathematics served up a blemish in the open pages.

With the problem of cardinal comparability remaining unsolved, there would be doubt over whether

or not the cardinals can be put into an ordered sequence. If they could not, then it would be

impossible to know which cardinal number is the greatest out of any two. The ambiguity would have

consequence on the Continuum Hypothesis for how could Cantor compare the cardinality of the real

numbers to that of the second number class (whose cardinal number is comparable as it is defined in

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terms of well-ordered sets) if it turned out that the cardinal number of the real numbers was not

comparable.

Perhaps Cantor knew that there was a possible proof of the Cardinal Comparability problem

and that someone, if not himself, would form one, as Cantor always believed that any set could be

put into a well-ordering. If any set could be put into a well-ordering, then the set of all cardinal

numbers could be well-ordered and thus comparable, proving the theorem. A proof never

materialized from Cantor; nor did he give any more attention to it in the remainder of part one, or at

all in part two, of the Beiträge. Addition and multiplication of cardinal numbers followed.

Addition of a and b resulted in the cardinal number of the union of the sets M and N , only

if they are disjoint, thus a+b ¿ (M∪N ). And since cardinal number has abstracted from the order

of the elements, their addition is commutative, so a+b=b+a. For the same reason, addition of

cardinal numbers is transitive, thus, for another cardinal number c of another set Q, disjoint from

both M and N , (a+b )+c=a+(b+c ). This can then be extended to the addition of several cardinal

numbers to obtain the cardinal number of the union of several disjoint sets.

Addition was defined in the same way that Cantor had defined it in 1888 when he

introduced the double abstraction concept, M . However, the definition of multiplication in the

Beiträge was somewhat different from its 1888 definition. In 1888’s Mittheilungen, multiplication of

a and b for the sets M and N defined a new set whose elements were M n, which correspond to a

set M n M for every n∈N . Then for the set {M n } was denoted by (M ∙N ). Then, in terms of

cardinal numbers, this was denoted by ∙ b ¿ (M ∙ N ).

Cantor defined the multiplication in the Beiträge in terms of “bindings

(Verbindungsmenge)”, pairings of elements m∈M and n∈N of the form (m ,n). So that now,

(M ∙ N )={(m ,n ) :m∈M ,n∈ N }. Then the product of a and b can be defined as the cardinal

number of the set (M ∙N ), thus a ∙b ¿ (M ∙ N ). The new definition is followed by the consideration

that this definition is equivalent to its 1888 counterpart. And, as with addition, multiplication of

cardinal numbers is shown to be commutative and associative as well as distributive.

The reason for Cantor’s alterations to his previous definition of multiplication was to serve as

an aid in defining the exponentiation of cardinal numbers. Exponentiation was defined in terms of

coverings, which Cantor introduced on the back of his diagonal argument. It was not the case that it

was a simple case of extending multiplication, like in the case with finite numbers; it was slightly

more difficult than that.

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For two arbitrary cardinal numbers a and b, if these are both finite, then there is no problem

taking ab, it is simply the same as treating the cardinal numbers as natural numbers. If only b was

finite then this would just be the special case of multiplication of transfinite cardinal numbers.

However, when b is transfinite, the concept of covering has to be applied. Recall, in 1892, when

Cantor introduced his covering, he proved that 2a was a greater cardinal number than a when a is a

transfinite cardinal number.

“By a ‘covering of an aggregate N with elements of the aggregate M ’, we

understand a law by which with every element n of N a definite element of M is bound

up, where one and the same element of M can come repeatedly into application. The

element of Mbound up with n is, in a way, a one-valued function of n, and may be

denoted by f (n ); it is called a ‘covering function of n’. The corresponding covering of N

will be called f (N )…. The totality of different coverings of N with M forms a definite

aggregate with the elements f (N ); we call it the ‘covering-aggregate (Belegungsmenge)

of N with M ’ and denote it by (N|M ¿.”

Two covering functions f 1 and f 2 are equal if f 1 (n )=f 2 (n ) for all n∈N . Cantor gives two

simple examples of covering functions of N to highlight how they can differ. If, for example, m0∈M ,

then f ( n )=m0 for all n∈N is a covering function of N with M . Then we can define another

covering function by taking m0 ,m1∈M such that m0≠m1 and defining f (n0 )=m0 for a specific

n0∈N and f ( n )=m1 for every other n≠n0 in N .

Thus, (N|M ¿={f (N ) }. This resulted in ab ¿ (N∨M ) . From this definition and the use of

cardinal multiplication, Cantor says that the same laws of finite number exponentiation can be

applied to cardinal number exponentiation. Thus for three cardinal numbers a ,b and c the following

three laws can be proven by cardinal multiplication:

1. ab ∙ ac=ab+c,

2. ac ∙ bc=(a∙b )c,

3. (ab )c=ab ∙c.

Exponentiation was a major breakthrough for Cantor. It completed the principle arithmetic

of the cardinal numbers and, serving as the foundations of his work, allowed Cantor to begin to

tackle the Continuum Hypothesis by building up his theory of transfinite sets. Cantor showed an

example of what the arithmetic of cardinal numbers will allow him to obtain, even though he was

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yet to define certain terms that he was usingi. The example essentially showed that the cardinal

number of the real numbers is 2ℵ 0 where ℵ 0 is the cardinal number of the natural numbers. The

example is also extendable to show that (2ℵ 0 )ν=2ℵ 0 for a finite cardinal number ν, as well as

(2ℵ 0 )ℵ 0=2ℵ0. This represents a previous result of Cantor that said that the ν-dimensional space has

the same cardinality of the continuum and can be extended to say that the ℵ 0-dimensional space has

the same cardinality of the continuum.

Now working towards the introduction of such terms in the example previously stated,

Cantor addressed finite cardinal numbers which are simply the natural numbers, defining them in a

similar manner as the natural numbers of von Neumann that were stated in a previous section. Since

the cardinal numbers represent a double abstraction, Cantor is vague in naming the elements of the

sets that he uses to define the natural numbers.

Referring to the “thing” e0, if we consider the set E0consisting only of this thing, E0={e0 }. Then we can think of the number 1 as the cardinal number of this set. Then, adding another, new

thing e1 to the set E0, whatever this thing may be, its characterises were irrelevant to Cantor, we

obtain the set E1=(e0 , e1) and the number 2 is the cardinal number of this set, and any other set

that is similar to it. Continuing in this manner, we obtain ν+1=Eν=Eν−1+1, and by induction, the

finite cardinal numbers, as an infinite sequence of ascending, distinct terms.

Here the order of the elements in the sets E ν is irrelevant too since we have abstracted from

any possible ordering but the ordering of the finite cardinal numbers themselves must not be

abstracted from. Cantor continued to prove that, for finite ν, v<ν+1 and there does not exist a

finite cardinal μ, such that ν<μ<ν+1 and thus, any set that is made up of distinct finite cardinal

numbers must have a least amongst them. Consequently, any set of finite cardinal numbers can be

put into a well-ordering.

Like with the ordinal number ω, Cantor found no problem in introducing a new cardinal

number, one that was greater than all the finite cardinal numbers and one that would be the first

transfinite cardinal number that is less than all other transfinite cardinal numbers. This first

transfinite cardinal number was to be the cardinal number of the transfinite set made up by the

totality of all the finite cardinal numbers and it was to be defined as Aleph-zero and denoted by ℵ 0.

There was no case for arguing its existence in the Beiträge due to the new mind-set that Cantor has

towards how he wanted to present this piece of work as a mathematical piece.

i In 1742, Golbach conjectured that any even integer greater than 2 can be written as the sum of two prime numbers. The conjecture still has no proof but had been proven for the even numbers up to 10000 by Desboves in 1855.

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The Beiträge was the first time that Cantor had considered powers as possible transfinite

numbers, for he had always referred to powers of infinite sets; the infinite identity was to be held

with the set and the cardinal number was just a characteristic.

For every infinite set S, it will always have a subset whose cardinal number is ℵ 0, hence

ℵ 0≤ S. This can be compared to Dedekind’s theorem that any infinite set has the natural numbers as

a subset. Thus, since any infinite set S has a subset T that is equivalent to it, if S=ℵ0, then T=ℵ 0.

Consequently, ℵ 0 can be considered as the least transfinite cardinal number.

Also, since ℵ 0= {ν } where ν is a finite cardinal number, then adjoining a new thing to the set

of all finite cardinal numbers will just produce the same set. Thus ℵ 0=ℵ 0+1=ℵ0+2=… and

consequently, ℵ 0+ℵ 0=ℵ 0 and extending further, it is obtained that ℵ 0 ∙ℵ 0=ℵ 0. This again can be

extended to get (ℵ 0 )ν=ℵ0.

Cantor states that he will produce higher transfinite cardinal numbers and show how they

succeed ℵ 0 such that they form a well-ordering based upon their magnitude. From ℵ 0 can be

produced the next transfinite cardinal number, ℵ 1 and from it ℵ 2, etc. to produce the unlimited

sequence of transfinite cardinals ℵ 0 ,ℵ1 ,ℵ2 ,…,ℵ ν ,… . In a similar form as with the natural

numbers, we can introduce ℵ ω which is greater than all the ℵ ν, and from this, a next greatest ℵ ω+1

and so on, without any ending. But how is one aleph obtained from the preceding aleph?

To do this, Cantor required the use of ordinal numbers and it is with this that Cantor

directed his attention to the order types of imply-ordered sets. The need to do so highlights the fact

that the ordinals were of more importance than the cardinals, hence their earlier development

ahead of the cardinal numbers, but the fact that Cantor was to pay attention to order types, showed

that he knew that they served as a foundation to the specific case of the well-ordered ordinal

numbers and thus needed explaining. The Beiträge was to hold Cantor’s first full presentation of

order types; before any explanation had been sporadic and some results were withdrawn from

publication by Cantor.

Recall a set M , is simply ordered if its elements can be put into a definite, transitive,

ordering: m1 ,m2 ,m3∈M ,m1<m2<m3 and therefore m1<m3. A set M can also be ordered in

different ways depending on different laws and remain simply-ordered. Cantor uses the set of

positive rational numbers to show this. This is then followed by the introduction of the 1887 concept

of abstracting from the nature of the elements alone. Cantor says that every set that has an ordering

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placed upon it, has an order type, and this is denoted by M which is itself an ordered set whose

elements are ordered by the same ordering of M .

The definition of similarity between two sets M and N was defined as before – if they can

be put into a bijective correspondence that preserves the order of both sets. The relation is now

denoted as M≃N and is said to be transitive and reflexive. It is briefly noted by Cantor that

M≃N⇔M=N .

From the order type, one more abstraction from the ordering of the elements gives the

cardinal number of a set. Thus, the cardinal number of a set with the order type α was denoted by α

with the new bar above representing one more step of abstraction. For the finite order type a ,a=a.

Thus the finite cardinal numbers are the same as the finite order types, and since they are well-

ordered, the ordinal numbers. However, in the case of the transfinite cardinal numbers this is

different, for a cardinal number can be representative of different order types of simply ordered

sets. These order types can then be put into a class of types (Typenclasse) corresponding to this one

transfinite cardinal number. If the cardinal number in question is a, then this type class will be

denoted by [a].

At this stage of the Beiträge Cantor is making the connection between cardinal number and

order types to highlight how one is obtained from the other and to distinguish between their

characteristics; thus they will have different arithmetic in the transfinite case.

An important concept that Cantor introduced in Principien was the reverse of an order type.

In the Beiträge this has been renamed as the inverse. So for our set M with order type α , when its

elements are arranged in the opposite order (the least becomes the greatest and vice-versa), then

the result is denoted by *M with order type *α . For the order type of the rational numbers, η, it

happens that *η=η. This is not the case with ω but *ω+ω is the ordinal type of the set of integers.

Addition and multiplication of ordinal types are then defined. This was done by considering

the sets M and N in the same manner as addition and multiplication was defined for cardinal

numbers. However, when adjoining the sets M and N it is important to coincide the order of the

elements in the union of the sets with the order of the summand.

For M=α and N=β:

α+β=(M , N ).

Where (M ,N )={M∪N :mν<nμ ,∀mν∈M ,∀nμ∈ N }.

Addition of order types is associative, thus, for another order type γ:

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α+(β+γ )=(α+β )+γ .

Cantor verifies that 1+ω is equal to ω and not ω+1. Thus, and from the definition of

addition itself, addition is not commutative: α+β≠ β+α .

Multiplication is defined by producing a new set S which takes every n∈N and replaces it

with a simply-ordered set M n which has the same order type α . Then S={M n} has a unique order

that satisfies the two characteristics:

1. For every pair of elements in S that belong to the same M n, then they are ordered by the

ordering of M n, i.e. α .

2. If the pair of elements belong to two different M i and M j, they have the same relation as i

and j have to one another in N .

Then multiplication is defined as:

α ∙ β=S .

Multiplication of order types is associative and distributive over addition.

The final two sections of part one of the Beiträge focus on the order types of the rational

numbers, η, and the real numbers, θ. As we have seen, η represents the class of all sets that are

everywhere dense and have no greatest or least elements (i.e. are without endpoints). Cantor

adjoins the criteria that these sets must have the same cardinal number of the natural numbers, ℵ 0.

Thus, the rational numbers and the natural numbers have the same cardinal number but a different

order type, a real example of what Cantor had already proven.

The order type η possessed some unique properties:

η+η=η ,

ηη=η ,

(1+η )=η ,

(η+1 )η=η ,

(1+η+1 ) η=η ,

Also, for finite ν, η ∙ ν=η and ην=η. Moreover, η ,1+η ,η+1 , ν ∙η ,1+η+1 are all different

order types but η+1+η=η. Finally, *η=η.

When Cantor had defined η in 1883 as the order type of the rational numbers, he was able

to supply characteristics of η, whereas he only stated that θ would be the order type of the real

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numbers, he was unable to give any characteristics of what type of sets would be represented by the

order type θ as with η. The Beiträge served as the first occasion on which Cantor would define θ

explicitly. It was clearly something Cantor needed to do in order to complete his main objective and

solve the Continuum Hypothesis.

Defining the characteristics of θ would not be as straight forward as η. In order to define θ,

Cantor had to look at the fundamental series contained in a transfinite simply-ordered set – these

were the subsets of an infinite set M that had the order type ω or *ω. Cantor introduced coherent

sequences and defined their limits, principal elements, in a similar manner to Dedekind’s cuts. If

every fundamental sequence in M has a limit that belongs to M , then M was called closed. If every

element of M is a principal element, then M was said to be dense in itself. Then, if a set is both

dense in itself and closed, then it was said to be perfect.

As an example, Cantor says that η is, as well as being dense, dense in itself. However it is not

closed – think of the series of increasing rationals approaching √2 as we seen earlier. Examples of

closed sets are given by Cantor as ω+ν and ν+¿*ω since they both have a principal element. The

type ω∙3 has two principal elements but is not closed whereas the type ω∙3+ν is closed as it has

three principal elements.

Once this is established, Cantor defined θ as the order type of the set M such that:

1. It is perfect;

2. There is the subset S such that S=ℵ0and, for any two given elements of M , elements of S

can occur.

The definition of θ brought Beiträge Part I to a strong and significant end. With a definition

of an order type for the real numbers, Cantor had finished and left his readers wondering what use

was to be made of it, knowing that this was only one half of the full instalment. The characterization

of the real numbers was also a larger extension of what he had previously given before in 1874 when

he just proved that the reals were not denumerable and it showed the progress that Cantor had

made over the previous two decades and how far he had distanced himself from point sets and

trigonometric functions. The definition of θ produced the most detailed account of the continuum

that was available for Cantor. Today, its definition has merely been simplified.

Part I had only given an introduction to the basics of transfinite set theory. It did not come

without its problems – there were the paradoxes of a largest ordinal and cardinal numbers which

shall be discussed later. However it is believed that Cantor had already come across these problems

in 1895 and was seeking to address them. This could have been another factor for the delay in

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publishing Part II. Furthermore , it still remained to define a cardinal number for the power of the

real numbers, which Cantor hoped to have included in Part II.

However, Part I still remains an exceptional, ground-breaking piece of mathematical text. It

would inevitably have its problems being such a young area of mathematics that had only really

been investigated and expanded by one sole person. Part I would sway a lot of previous doubters

and invite others in to try and solve any problems that had arisen. Overall, the main accomplishment

of Part I was its generality of ordering.

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[2.2.2] Part II:

The second instalment of the Beiträge was meant to build up to a proof of the Continuum

Hypothesis which at this point, because of the use of coverings, could soon be stated as:

The power of the continuum is ℵ 1=2ℵ 0 .

As we know, the proof was never to come, hence the delay to this publication. With the

foundations of transfinite set theory laid down in Part I, Cantor was able to expand his theory to the

most important results and go beyond ℵ 0 to the first non-denumerable transfinite cardinal number,

ℵ 1. In Part II Cantor was to only deal with well-ordered sets whose order types are the finite and

transfinite ordinal numbers. The special attention to well-ordered sets would lead to their theory

being applied to determine ℵ 1.

Cantor begins Part II with highlighting the fact that well-ordered sets are special cases of

simply-ordered sets whose order types are the ordinal numbers and then follows the definition of a

well-ordered set:

“We call a simply ordered aggregate F ‘well-ordered’ if its elements f ascend in

a definite succession from a lowest f 1 in such a way that:

I. There is in F an element f 1 which is lowest in rank.

II. If F ' is any part of F and if F has one or many elements of higher rank

than all the elements of F ' , then there is an element f ' of F which

follows immediately after the totality F ' , so that no elements in rank

between f ' and F ' occur in F.”

The definition of well-ordered sets is followed by three examples which highlight how we

can adjoin well-ordered infinite and finite sets together to produce different well-ordered sets:

(a1 , a2 ,…,aν ,…)

(a1 , a2 ,…,aν ,…;b1 , b2 ,…,bμ ,…)

(a1 , a2 ,…,aν ,…;b1 , b2 ,…,bμ ,…;c1 , c2 , c3 ) ,

where aν<aν+1<bμ<bμ+1<c1<c2<c3.

Notice that, in the second example there has been an infinite, well-ordered set adjoined to

the first infinite well-ordered set. Here, the aν will always be less than any bμ which in turn are

always less than c1 in the third, where c1 is less than c2 and every element is less than c3. Thus c3 is

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the greatest element of the third example and a1 the least of all three. The first and second

examples then have no greatest elements. There are an infinite number of aλ between any aν and b1

in the second and third examples and similarly, there are an infinite number of bκ between any bμ

and c1 in the third example.

This is followed by a theorem which states that every subset of a well-ordered set has a least

element and consequently, every simply-ordered set, whose subsets all have a least element, is well-

ordered. Under this important theorem, Cantor was able to prove a succession of theorems that

followed on well-ordered sets. And with the definition of similar, which was defined in Part I, Cantor

states that, if a well-ordered set is similar to another set, then that set is also well-ordered. However,

before it was possible to define a satisfactory definition of ordinal numbers, Cantor needed the

concept of segments of well-ordered sets.

A segment defined by f of a well-ordered set is a proper subset made up of all the elements

which precede f such that f is different from the initial elementi. Every segment leaves behind a

remainder set which contains all the other elements which do not belong to the segment. Both the

segment and the remainder are well-defined. The main result that Cantor produced on segments

was the following theorem:

“If F and G are any two well-ordered aggregates, then either;

(a) F and G are similar to one another, or

(b) There is a definite segment B1of G to which F is similar, or

(c) There is a definite segment A1of Fto which G is similar;

and each of these three cases excludes the other two.”

Using segments, Cantor was able to state the relations that were possible for two well-

ordered sets under his similar relation. Resultantly, any subset of a well-ordered set is either similar

to a segment of the parent set or similar to the well-ordered set itself. Cantor would go on to define

the ordinal numbers of the second number class using segments and once they were defined, the

above theorem was used in developing results concerning the order of any two ordinal numbers.

“We call the ordinal type of a well-ordered aggregate F its ‘ordinal number’.”

The previous theorem above ensured that only one of a possible three relations between

the ordinal numbers of two well-ordered sets, F=α ,G=β, can occur:

(a) F≃G and thus α=β , or

i Alternatively defined as b>a.

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(b) G contains a definite segment B1 such that F≃B1and thus α<β, or

(c) F contains a definite segment A1 such that G≃A1 and thus α>β.

As we seen with the cardinal numbers, Cantor had not found a solution to his Cardinal

Comparability problem, however, the comparability of segments allowed Cantor to compare any

number of ordinal numbers. Thus, α<β ,α=β or α>β. This relation is reflexive and thus, the set of

ordinal numbers arranged by magnitude is simply-ordered.

Arithmetic of ordinal numbers is just the same as that of order types. This follows from the

union of two well-ordered sets also being a well-ordered set. Since F is a segment of (F ,G),

α<α+ β. But, since G is the remainder of (F ,G ) , it is possible that G is similar to it by the previous

two theorems. Thus Cantor notes that β≤α+β.

Multiplication is defined in the same manner as order types, by substituting a well-ordered

set Fg, which has the ordinal number α , for every element g∈G, whose order type is β. This

defines a new set H whose ordinal number is α ∙ β. Cantor concluded that the product of any two

ordinal numbers always produces an ordinal number and the concept of multiplication is associative

and distributive over addition.

Subtraction of ordinal numbers was not straightforward because addition is not

commutative. Subtraction had been defined in the Grundlagen but it was to be defined again, in the

same careful manner in the Beiträge in order to avoid any ambiguity. For two ordinalsα ,β such that

α<β, Cantor showed with the use of segments, that there was always an ordinal β−α that satisfies

the equation α+x=β. Then, for any ordinal number γ, subtraction can be governed by the

following two laws:

I. ( γ+β )−( γ+α )=β−α ,

II. γ (β−α )=γβ−γα.

With addition, subtraction and multiplication defined, Cantor sought to define the second

number class, which was already defined in the Grundlagen. However, in the Beiträge, Cantor

wanted to define it in a different, more mathematical manner by introducing limits for sequences of

ordinal numbers. Cantor defined the ordinal number ω as a limit of the natural numbers, i.e.

ω=Lim νν where ν is the sequence of natural numbers, which are also the finite ordinal numbers,

beginning from 1.

For Cantor, there were two different kinds of ordinal numbers: the first kind were those

obtained by the recursive method of addition, α ν=αν−1+1. The second kind were those obtained by

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taking limits of sequences of ordinal numbers, as above with ω; however in the Beiträge, Cantor

wanted to generalize these limits and abstract from the case of ω.

Take the well-ordered set G=(G1 ,G2 ,G3 ,…. ,Gν ,…) where Gi=β i for all i, and G=β .

Then we can define

α ν=(G1,G2,…,Gν )=β1+β2+…+βν .

Then we have α ν<α ν+1. This provides the fundamental sequence α 1 , α 2 ,…,α ν ,… of

succeeding ordinals, which is an infinite sequence. But, for all ν ,α ν<β, thus β is the limit of the

fundamental sequence α 1 , α 2 ,…,α ν ,… and this is defined as

β=Limνα ν.

Cantor went on to show that, for a given ordinal :

I. Limν (γ+α ν )=γ+Limν αν,

II. Limν γ ∙α ν=γ ∙ Limν αν.

Cantor was now prepared to define the second number class. The first number class,

consisting of the finite ordinal numbers, corresponded to their finite cardinal number; they were

equal. However, in the case of the transfinite ordinal numbers, this was not the case, as was seen in

Part I – there is an infinity of order types corresponding to any transfinite cardinal number; the same

applies to the ordinal numbers with them just being the special case of order types. For the

transfinite cardinal number a, its infinity of corresponding transfinite ordinal numbers make up a

“unitary and connected system” which Cantor called the number class Z(a) which is a subclass of

the type class [a] defined before in Part I, the class of all order types with corresponding cardinal

number a.

The second number class was loosely defined as Z(ℵ 0) with the first number class being

defined again as “the totality {ν } of finite ordinal numbers”. Cantor was now looking to define the

second transfinite cardinal number ℵ 1 through the second number class, thus the bulk of the

remainder of the Beiträge was devoted to laying the foundations and developing fully the true

extent of the second number class. In doing so, Cantor would further his transfinite set theory far

beyond any of the earliest foundations he had previously set. The final parts of the Beiträge would

highlight the significance that Cantor’s set theory had in mathematics; through its generality and

clear presentation, set theory’s future was sown.

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In the Grundlagen Cantor defined the second number class through various principles of

generation, however, in the Beiträge, the definition would consist simply in terms of ordinal

numbers:

“The second number-class Z(ℵ0) is the totality {α } of ordinal types α of well-ordered

aggregates of the cardinal number ℵ 0.”

For the second number class, Cantor defines ω as the smallest and least number of the class.

There is no longer any need to justify the introduction of ω or to imagine ω as a non-finite number

that has the certain property of being greater than all the finite numbers; the mathematically based

foundations that have come before and thinking of ω as this infinite set is the cause of such

development.

The numbers of the second number class is then defined recursively, beginning from ω, by

the theorem – if α belongs to the second number-class, then so does its successor ordinal α+1; and

this is the next greatest ordinal after α .

Some important characteristics of the second number classare, for finite ordinal ν and

transfinite ordinal α :

ν+α=α ,i

α−ν=α ,

ν ∙ω=ω

(α+ν ) ∙ω=αω

{1 ,2 ,…,ν ,…;ω ,ω+1 ,ω+2 ,…,α−1 } has ordinal number α .

Recall the firstii and secondiii principles of generation which had been defined in the

Grundlagen. Cantor defines both the first and second principles of generation for the second

number class together and in a lot more generality than before when he used the natural numbers

and the ω’s as examples in order make his definitions. The first principle of generation was now

defined to produce numbers of the first kind; α=α−1+1 where α−1 is the unique ordinal preceding

α . The second principle of generation was then defined to consist of numbers α of the second kind,

i This was due to the fact that the Beiträge was submitted for publishing in several stages. ii ℵ is the first letter of the Hebrew alphabet. Cantor choose Hebrew as he thought other ancient languages had been overused and his transfinite language needed its own identity. The choice of aleph was because it also represented one in Hebrew and each transfinite cardinal number were, to Cantor, separate unities. Cantor also believed that his transfinite numbers signalled a new beginning, hence the aleph, the first letter of a language that had not previously been used in mathematics. iii In the definition of a well-ordered set, this is f 1.

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such that they have no next smaller number α−1 but are the limit of a fundamental series {α ν};

α=Limνα ν . Here, α is the next largest in magnitude after all the numbers α ν.

The previous two definitions are clear examples of how Cantor has abstracted and thus

generalized his theory. Cantor had realized that dependence on examples would not win him

plaudits, or acceptance. The Beiträge was meant to be independent of any particulars apart from its

application of specific cases which justified its value. This is highlighted through defining ω as the

order type of any denumerable well-ordered transfinite set, thus requiring no connection to the

natural numbers and allowing for the definitions of generation to exclude them from any specific

cases as well.

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[2.10] Aleph-One and Transfinite Induction:

Having explored and investigated the ordinal numbers, Cantor was now ready to define ℵ 1.

Chapter 16 of the Beiträge is entitled “The Power of the Second Number Class is Equal to the Second

Greatest Transfinite Cardinal Number Aleph-One.” Before Cantor determined the power of the

second number class, he showed that Z (ℵ 0 ) is a well-ordered set. This allowed him to apply all the

previous results he had established at the beginning of the Beiträge. Thus, every set of different

numbers from the first and/or second number classes has a least number and, if arranged according

to magnitude, is a well-ordered set.

From this, Cantor was able to state a fundamental theorem that would propel him ever

closer to defining the second transfinite cardinal number:

“The power of the totality {α } of all numbers α of the second number-class is not equal

to ℵ 0.”

Consequently, the set of transfinite ordinal numbers of the second number class is not

denumerable, as, if it was, then it could be put into a one-to-one correspondence with the natural

numbers. But the natural numbers have the cardinal number ℵ 0, therefore this would be the

cardinal number of the second number class, which is contradictory. This was the essence of

Cantor’s proof of the above theorem. Since ℵ 0 is the smallest transfinite cardinal number, it remains

that the cardinal number of the second number-class is larger than ℵ 0 and in fact, Cantor proved

that it is the next largest transfinite cardinal number, ℵ 1.

Three quarters of the Beiträge had been devoted to obtaining the first two transfinite

cardinal numbers and when the second one was finally obtained and defined, it was mentioned once

more in the remainder of the paper. Instead Cantor turns his attention away from cardinality almost

as soon as he had returned to it. The remainder of the Beiträge would focus again on the second

number class of transfinite numbers; for Cantor had not even covered all the theory that he had

previously introduced in the Grundlagen, nor had he defined any higher number classes. So the

remainder of the Beiträge was devoted to the arithmetic of the second number class and Cantor’s

interests first turned to defining exponentiation of ordinal numbers. To do this however, Cantor had

to introduce transfinite induction.

It is possible to represent some numbers of the second number class with a unique

polynomial of a finite number of terms in the form:

ϕ=ωμ ν0+ωμ−1 ν1+…+ν μ .

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where μ , ν0 are finite and non-zero and νi 1≤i ≤μ are finite, not necessarily non-zero.

For ψ=ωλ ρ0+…+ρ λ, the addition of ϕ and ψ follows these requirements:

(a) If μ<λ, then ϕ+ψ=ψ ;

(b) If μ= λ, then ϕ+ψ=ωλ (ν0+ ρ0 )+ωλ−1ρ1+…+ ρλ;

(c) If μ>λ, then ϕ+ψ=ωμ ν0+ωμ−1 ν1+…+ωλ+1νμ−λ−1+ωλ (ν μ−λ+ ρ0 )+ωλ−1ρ1+…+ ρλ ¿.

The result of multiplication also gets trickier when higher powers are concerned but it is of a

similar fashion to that of addition. For ϕ and ψ as above, Cantor noted that ϕωλ=ωμ+ λ, thus, similar

to addition, all the smaller terms get ‘absorbed’ by the new highest term. Then by the distributive

law for multiplication, it happens that

ϕ ∙ψ=ϕωλ ρ0+ϕωλ−1ρ1+…+ϕω ρ λ−1+ϕ ρ λ .

Therefore under certain parameters:

(a) If ρ λ=0 ,then ϕ ∙ψ=ωμψ

(b) If ρ λ≠0 ,then ϕ ∙ψ=ωμ+ λ ρ0+…+ωμ+1 ρ λ−1+ωμ ν0 ρλ+ω

μ−1 ν1+…+ν μ.

However, Cantor could not expand this to terms of the second number class that are only

representable by polynomial consisting of a sequence of terms that include terms of a transfinite

degree. For example ωω could not be included in Cantor’s transfinite theory until there was a

satisfactory method of introducing a product of transfinite number of ordinal numbers. To do this,

Cantor introduced the method of transfinite induction for ordinal numbers which was similar to the

mathematical induction for the natural numbers. Cantor was able to extend the procedure to

provide a similarly inductive method of counting infinite sets in a successive manner.

For a statement P(n), if this is assumed to be true for a least value of n, the new method of

transfinite induction looked to show that, when P(n) was assumed true for all n<α, then it was also

true for all ordinal numbers α . The new method was applicable for all α of different cardinalities,

thus transfinite induction is valid for ordinal numbers of higher number classes than the second and

first. Having introduced his limiting process for ordinal numbers in terms of fundamental sequences,

transfinite induction made it possible for Cantor to develop multiplication of a transfinite number of

factors and to establish transfinite exponentiation of ordinal numbers.

In Cantor’s terms, let ξ be a variable from the domain consisting of numbers from both the

first and second number classes, including zero. Then define two constants γ and δ from the same

domain such that δ>0 , γ>1. Then we can define the theorem of transfinite induction as follows:

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There is a wholly determined single-valued function f (ξ) of the variable ξ which satisfies the

following conditions:

(a) f (0 )=δ ;

(b) If ξ ' and ξ ' ' are two values of ξ such that ξ '<ξ ' ' , then f (ξ' )< f (ξ ' ');

(c) For every value of ξ we have f ( ξ+1 )=f (ξ ) γ ;

(d) If {ξν} id any fundamental series, the so is f ¿. And if ξ=Limνξν, then

f ( ξ )=Limν f (ξν).

As Cantor had stated the principle of transfinite induction as a theorem, it remained to be

proven. First, Cantor established that the function f ( ξ ) was completely determined for all ξ<ω.

Then, assuming that the theorem was true for all possible values of ξ<α , for α an ordinal of Z(ℵ0),

Cantor proved the validity of the theorem for all ξ ≤α , splitting the proof up for the two different

kinds of ordinal numbers, before showing that, for all values ξ ≤α , the function that satisfied all the

terms of the theorem was entirely determined.

With transfinite induction established, Cantor was now able to define the exponentiation of

ordinal numbers to transfinite degrees. By taking δ=1 and f ( ξ )=γ ξ, Cantor specified the theorem

of induction in order to define exponentiation of transfinite ordinal numbers:

For any constant γ>1 of the second number class, there is an wholly determined function γξ of ξ ,

such that:

(a) γ0=1;

(b) If ξ '<ξ ' ' , then γξ'

<γξ' '

;

(c) For every value of ξ , γ ξ+1=γ ξ γ;

(d) If {ξν } is a fundamental series, then so is {γξ ν }, and if ξ=Limνξν, then γξ=Limνγξ ν.

Standard laws of exponentiation followed from this definition and proven by the induction

theorem: γα+ β=γ α γ β , γ αβ=( γ α )β for any α ,β of the first and second number classes.

With transfinite exponentiation, Cantor sought to establish his system of ϵ -numbers, which

he had only mentioned once before in the past, in a letter to Franz Goldscheider in 1886. The ϵ -

numbers were transfinite numbers ϵ of the second number class such that ωϵ=ϵ . The ϵ -numbers

would conclude the Beiträge and bring to an end Cantor’s foundations of the transfinite ordinal

numbers. The introduction of transfinite induction was the only way that Cantor could define

exponentiation of the transfinite ordinal numbers and thus conclude his foundations.

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Since transfinite exponentiation allowed Cantor to define ordinal numbers of transfinite degree,

he was now able to define sequences of the form ωω ,ωω+1 ,ωω+2 ,…. And consequently,

sequences of the form ωω ,ωω2

,ωω3

,… . It became clear to Cantor that the limit of the sequence

would be Limν ωων

=ωωω

. Cantor considered the sequence of ordinal numbers that could not be

considered as a finite polynomial of numbers of lower order: 1 ,ω,ωω ,ωωω

,ωωωω

,… . A limit can be

assigned to this well-ordered sequence and Cantor denoted it by ϵ 0. Thus, by definition of transfinite

exponentiation

ωϵ 0=Limν (ω ,ωω ,ωωω

,ωωωω

,…)=ϵ 0.

The remainder of the Beiträge is devoted to exploring the unique properties of these ϵ -

numbers. Using limits of sequences, transfinite induction and exponentiation, allowed Cantor to

formulate higher ordered ϵ ν’s and arranged in order of magnitude, these formed a well-ordered set

whose order type was the same as that of the second number class which Cantor denoted as Ω, and

whose cardinal number was ℵ 1. Some properties of the ϵ -numbers bring Cantor’s last major piece of

work on his transfinite set theory to a conclusion.

Having rigorously constructed the foundations of the transfinite numbers of the second

number class, it seemed inevitable that Cantor would now fulfil his wish of Part I and establish the

ascending sequence ℵ 0 ,ℵ1 ,ℵ2 ,…,ℵ ν ,… of transfinite cardinal numbers and prove that the cardinal

ℵ ω exists. However it never materialized. Nor did Cantor find a solution for his Continuum

Hypothesis.

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[2.11] Post-Beiträge:

The Beiträge was a real breakthrough for Cantor and his transfinite set theory. Its

abstractness led to a generality that was well presented and mathematically sound. Its concepts had

been ground-breaking; its results valuable. The infinite had been accepted and inducted into

mathematics. Most doubters had been swayed and the theory of transfinite numbers was ready to

be studied, mostly by younger mathematicians who were willing to defend its creator. For Cantor, it

was a huge success; it must have been a great sense of relief to finally be free of most of the ridicule

and to have produced such an expansive mathematical theory, free of philosophical justification,

would have been greatly satisfying. Not long after its publication, the Beiträge was translated into

both Italian and French and sent out to mathematicians around the world.

However, the Beiträge did not form sound foundations for Cantor’s theory. The theory of

transfinite cardinal numbers was based upon the Cardinal Comparability problem which Cantor

failed to answer. Similarly, there was a lack of a proof that every set can be well-ordered, a theorem

that Cantor only assumed to be true in the Beiträge. The Beiträge also came with its problems. There

are two famous cases of paradoxes that were found in Cantors set theory, one being Russell’s

Paradox and the other courtesy of Burali-Forti. These paradoxes would be the cause of Pioncaré

doubting Cantor’s work and labelling it a “disease”.

Cesare Burali-Forti (1861-1931) was an Italian

mathematician who found a paradox in Cantor’s set theory. In

February 1897, Burali-Forti wrote the paper A Question on

Transfinite Numbers which he presented at the Circolo

mathematico di Palermo meeting of the 28th March that same

year and was published in the Circolo’s Rendiconti afterwards. In

this paper, Burali-Forti highlighted the need for a re-examination

of the foundations of set theory. Based upon the foundations laid

out by Cantor, the set of ordinal numbers is well-ordered and

therefore has an ordinal number itself. This is what became

known as the Burali-Forti Paradox of Cantor’s set theory.

As was stated earlier, Cantor had already realized such a problem in 1895 and had begun to

work on finding a solution. In a letter to Dedekind, whom he again became acquainted with, in 1899,

Cantor addressed the Burali-Forti paradox, turning it into a positive and using it to define his idea of

The Absolute. Cantor considered the collection Ω of all ordinal numbers and he defined an absolute

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Figure 18 - Cesare Burali-Forti.


infinity or an inconsistent multiplicity as one in which Ω can be embedded into under an injective

correspondence and these were deemed not to be sets by Cantor.

This solution only appeared in 1932, long after Cantor had died, when Ernst Zermelo

released some of Cantor’s works. Zermelo had found in these works, much anticipation of the

structure of set theory which was to later support the subject. The letter to Dedekind just mentioned

is an example of this; in it, Cantor again argued that any set can be well-ordered. Cantor did this by

contradiction, for if a set cannot be well-ordered, Cantor argued that Ω can be injected into it,

making it an absolute infinite. Cantor also stated some set existence principles which some thought

were necessary as axioms in order to make his theory consistent. The principles included a union

axiom and forms of separation and replacement. These would later be used by Ernst Zermelo in his

axiomatization of Cantor’s set theory.

The second paradox was discovered in 1901 by Welsh

mathematician and logician Bertrand Russell (1872-1970) whilst

working on his Principia Mathematica. Russell’s argument was, if

W is the set whose elements are the sets x, such that x is not a

member of x. Then if W is a member of W , then, by definition,

W is not a member of W . However, if W is not a member of W ,

then by definition, W is a member of W . A solution to this

paradox never came from Cantor, but it was later solved by

Zermelo through his axioms.

The axioms of Zermelo, with contribution from Frankel,

would solve the remaining problems of Cantor’s set theory and

come to be the most popular set of axioms of set theory amongst a handful of others.

The biggest flaw of the Beiträge was its failure to solve the Continuum Hypothesis; even

today it remains unsolved but its truth is assumed. After the turn of the century Cantor suffered

from depression after the death of his youngest son and the remainder of his own life would be

plagued with physical and mental illnesses. Honours did come his way, notably the awarding of an

honorary degree of Doctor of Laws from the University of St. Andrews in 1912, but Cantor was

unable to collect this in person due to ill health. Retirement came in 1913 and due to the war at the

time, Cantor was constantly ill in his final years through limited supplies of food. In 1917, Cantor was

admitted to a sanatorium where he later died of a heart attack.

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Figure 19 - Bertrand Russell.


[2.12]: Axiomatic Set Theory and Transfinite Recursion:

The set theory of Cantor and that of Dedekind too, is often called naïve set theory due to its

foundations on philosophical grounds. Most mathematicians felt that the foundations of the theory

of sets should be laid out in an axiomatic system, like that of Peano, in order to consider which

statements are ‘obviously’ true of the universe of sets. In mathematics, we do not want to study any

theories that produce contradictions. Axioms serve as a foundation for what is allowed in

mathematical terms. Therefore there is no need to rely wholly on logical arguments. Induction is an

example of the benefit of having axioms as they allow for finite proofs to infinite arguments.

After the success of the Beiträge, a handful of mathematicians took up the task to formulate

their own set of axioms for set theory and they produced a varied collection. These include the

Kripke-Platek Axioms, the Von Neumann-Bernays-Gödel Axioms, the Morse-Kelley Axioms, the

Tarski-Grothendieck Axioms and the Zermelo-Frankel Axioms.

Figure 20 - Ernst Zermelo.

Ernst Zermelo (1871-1953) was a German logician and mathematician whose attention was

drawn to the Continuum Hypothesis by David Hilbert whilst at Göttingen. Hilbert had listed the

problem as one of his 23 problems in 1900. At the Congress of Mathematics in Paris where Hilbert

gave his list of problems, he suggested that Cantor’s main difficulty in being unable to find a solution

was in suggesting that a definable well-ordering of the reals should be foundi. One specific axiom of

Zermelo allowed for a well-ordering of the reals to be assumed and allowed Zermelo to overcome

i Notice that each Gi is well-ordered. If they were not, then Gwould not be well-ordered.

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the difficulty that Hilbert suggested Cantor had. This axiom was the Axiom of Choice. Zermelo

formulated his axiom in 1904, followed by his Well-Ordering Theorem.

The Axiom of Choice: Let Σ be a collection of disjoint non-empty set Sv , then there exists a

set R , the Representative Set of Σ, consisting of the elements xv from each Sv such that, this is the

only element of Sv which can be found in R. This element is called the Representative Element of Sv .

The Well-Ordering Theorem: If S is any set, then there exist a well-ordering of S.

It may seem strange to consider the Axiom of Choice as an axiom since it is not constructed

in the normal axiomatic manner found in mathematics which uses undefined terms, along with other

axioms, and puts them under interpretation; it seems to be more like a basic principle. There does

exist a Choice Principle which says – for a collection Σ of non-empty sets Sv , disjoint or not, there

exists a set R whose elements are pairs (Sv , xv), such that xv∈Sv and each Sv occurs in one and

only one pair. Thus, if we declare the sets Sv to be disjoint, then we obtain the Axiom of Choice.

The Axiom of Choice does seem trivial, since the sets are disjoint and non-empty, we can

obviously just take one element from each as a representative. However the axiom does not provide

a precise definition of the representative set R and how it is obtained. This led to a paradox which

was again found by Russell. The paradox was phrased in terms of shoes and socks – imagine that an

infinite number of shoes exist, then we can just take one shoe, say the left, as a representative of

each pair and add that shoe to a representative set. However, what if there were an infinite number

of socks instead of shoes, how do we choose which sock to take as a representative from each pair

considering that socks are identical and do not have a left or a right one.

There is justification in assuming that there exists a representative set for socks, thus the

Axiom of Choice is applied here. However there is no need to refer to the axiom for shoes. The

Axiom of Choice is only needed when there is insufficient information about the characteristics of

the specific sets in question. Such an axiom can be said to exceed the principles available in the

classic forms of logic; the development of mathematics gave way for the need of such an axiom.

Although the choice axiom seems trivial, it has many consequences.

With the Axiom of Choice and the Well-Ordering Theorem, Zermelo was able to solve two

problems left behind in Cantor’s set theory. Firstly Zermelo was able to prove that every transfinite

cardinal number was an aleph and secondly, he secured the Cardinal Comparability problem. Both

results depended on the fact that the Axiom of Choice provides a well-ordering for every set, which

Cantor simply believed to be true but was unable to prove. The Well-ordering Theorem shifted the

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weight from Cantor’s numbering through successive choices in his well-orderings, to an arbitrary

function that would make simultaneous choices instead.

The Well-Ordering Theorem was proven by Zermelo using the Axiom of Choice, however, in

the same year, he also showed that the two were actually equivalent. There are many theorems in

mathematics that cannot be proven without the use of the Axiom of Choice, but there were a lot of

mathematicians who did not believe in its validity and tried to formulate their own variation of the

axiom, hence Zermelo introducing the Well-Ordering Theorem; the next, most popular equivalent of

the Axiom of Choice would be Zorn’s Lemmai.

The real power of the Well-Ordering Theorem was in its application of uncountable sets,

such as the real numbers, where it led to new methods that were not necessary for finite or

denumerable problems, such as transfinite induction, which we seen Cantor restricted to his ordinal

numbers. Now, since the Well-Ordering Theorem allowed for every set to be well-ordered,

transfinite induction could be extended to any transfinite set, (that is if a well-ordering can

realistically by found). So assume we have found a well-ordering for an arbitrary set S, with S well-

ordered, we can prove that all the elements of S have a property P if it can be proven that:

(a) The least element of S has the property P;

(b) For s∈S, and for all t∈S ,t<s, if t has the property P, then so does s.

However, the use of an arbitrary function had its critics. In 1908, Zermelo responded with a

second proof the Well-Ordering Theorem. This was followed up in the same year with more axioms

for set theory, in response to pressure for a new mathematical context, separate from the naivety of

Cantor and Dedekind’s set theoretic foundations. Zermelo looked to use his axioms to support the

Well-Ordering Theorem by making explicit its underlying set existence assumptions.

The axioms introduced by Zermelo were:

1. Axiom of Extensionality: two sets are equal if, and only if, they have the same elements.

2. Pairing Axiom: for all sets X ,Y , then {X ,Y }, is a set, consisting of the elements X and

Y .

3. Union Axiom: if X is a set, then X is a set too.

4. Axiom Schema of Separation: for a set X and a property ϕ, {x∈X : ϕ(x ) holds} is a set.

This is a schema as there is a different axiom for each ϕ .

5. Axiom of Infinity: there is a set having zero as an element which is closed under the

succession function.

i Notice that this implies the natural numbers can begin with either 0 or 1, or any other number for that matter, so long as the succession under addition of the number 1 is infinite.

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6. Power Set Axiom: if X is a set, then so is (X ).

With the axioms, Zermelo was able to solve Russell’s paradox i and led set theory into further

abstraction and a greater depth of generalization. The Axiom of Choice, along with the Continuum

Hypothesis, were no longer seen as an underlying axiom and a primitive principle; they became the

centre of set theory and were explored merely as a part of mathematics. With the Zermelo-Fraenkel

axioms, the Continuum Hypothesis can neither be proven, nor disproven; otherwise the axioms

would become inconsistent. This was first proven by Gödel in 1940.

Figure 21 - Abraham Fraenkel.

Abraham Fraenkel (1891-1965) was an Israeli mathematician who also worked on set theory.

In 1922 Fraenkel, and independently Skolem ii in 1923, pointed out to Zermelo the need for an extra

axiom as his axioms alone could not establish the set {ω, (ω ) , ((ω)),… }. Both Fraenkel and

Skolem felt the need for an axiom that asserted that the image of a set under a function is also a set.

The axiom became known as:

7. Axiom Schema for Replacement: for a set W and a well-defined function/statement ϕ,

there exists a set S such that S={y :∃ x∈W ,ϕ( x , y) holds}.

The axiom of replacement can also be credited to Von Neumann who worked on the

developing the axiom between 1923 and 1328. Von Neumann had noticed that Zermelo’s axioms

alone did not allow him to define 2ℵ 0=ℵ1. To eradicate this problem, von Neumann believed that an

i Cantor used the natural numbers as an example of this. Also ω ,ω+1 ,ω+2 ,… was an example for the second number class.ii Here the example given was ,2ω,3ω,… .

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axiom of replacement, or substitution, was a necessary addition to the Zermelo Axioms. In

formulating his version of the axiom, von Neumann saw it necessary to establish transfinite

recursion:

Transfinite Recursion: let H be any set and α any ordinal. Then there exists a unique

function ϕ which is a function on α and for every β<α , ϕ (β )=H (ϕ¿β).

The most popular form of set theory is the Zermelo-Fraenkel set theory which is axiomatized

by Zermelo’s six axioms of 1908 with the addition of the Axiom Schema for Replacement, the Axiom

of Choice, as well as one other axiom for which Mirimanoff i (1917) and von Neumann (1925)

developed:

The Axiom of Regularity: every non-empty set X contains an element Y such that X and Y

are disjoint sets.

The regularity axiom implies that for all sets , X∉ X. Thus, the axiom of regularity solves

Russell’s paradox. This additional axiom can also be related to transfinite induction as it permits that

there are no infinite sequence of descending sets. The first six axioms given by Zermelo are often

regarded as elementary principles of set theory; some believe that there is no need to have them as

axioms, but they do serve as an underlying truth that all proofs can be referred back to and they

avoid a dependence upon logical arguments. The Axiom of Choice, the Axiom of Regularity and the

Replacement Axiom Schema are the key axioms of the Zermelo-Fraenkel set theory; they brought

Cantor’s Naïve set theory into a mathematical rigour and ensured its existence within mathematics.

i Philip Jourdain (1879-1919) would prove in 1918 that every set can be well-ordered. In the proof, Jourdain makes claim that Zermelo’s axiom is merely a consequence of logical principles – P. E. Jourdain, A Proof That Any Aggregate Can Be Well-Ordered, Mind (Journal), New Series, Vol. 27, No. 107, July 1918, P386-388.

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[2.13] Summary:

“no one can drive us from the paradise Cantor created for us”

- David Hilbert, 1925.

Cantor’s struggle to develop the transfinite cardinal and ordinal numbers changed the

perception of the principles of mathematics and brought mathematics into a higher rigour than

Dedekind or Peano could dare dream of themselves. An imaginative, yet controversial figure in the

history of mathematics, Cantor’s study of the continuum and his adoption of the infinite, forced him

to depart from the standard interpretations of mathematics, dragging the subject out of its ancient

restrictions. The fame which he thought he had deserved within the mathematical community would

not come Cantor’s way whilst alive; however his name today is held in high regard with other greats

such as Gauss and Euler.

The axioms of Zermelo, with the addition of Fraenkel, helped to resolve any unsolved issues

and paradoxes that were found in Cantor’s set theory, as well as bring the Cantorian set theory out

of its naïve foundations and into more mathematical rigour. The Zermelo-Fraenkel axiom system is

now the most accepted, and studied, axioms that were developed for set theory.

Mathematics began as a means of counting objects before it was brought into a

philosophical view by the early Babylonians and ancient Greeks. The subject gradually grew and

developed into its’ own science before Euclid sought to axiomatize mathematics when geometry

began to develop. Over the next two centuries mathematics came under the influence of great

Indian and Islamic mathematicians who abstracted the subject with the introduction of symbols,

European mathematicians of the Renaissance and the Science Revolution who highlighted that

mathematics should not be without proof, before being modernized into a fully abstract form in the

19th and 20th Century’s.

It would be more than 2000 years before mathematics would add axioms to the subject as

the need for that higher rigour that Peano and Dedekind sought, became apparent. The young

mathematicians who followed these two great mathematicians were the ones who bought into the

idea of a need for axioms for transfinite set theory; the axioms of Peano could only serve for the

potential infinite of the natural numbers and not the transfinite of the second number class.

As regard for the course of this investigation is the challenge of representing the infinite

through induction and recursion. We have seen in section one how both processes represent the

potential infinite of the natural numbers in finite terms. With larger infinities to deal with in

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transfinite set theory we would have always hoped to have a similar principle to deal with these

different infinites and with sets containing a new kind of number. The transfinite ordinal numbers do

not share the same characteristics of their finite counterparts. The natural numbers are obtained

through successive addition and there are ordinal numbers which are obtained through this process;

however we also know that there is a different kind of ordinal too – the limit ordinals.

Thus, mathematical induction and recursion do not have the criteria to satisfy this jump,

from the first kind of numbers obtained from successive addition, to the second kind of numbers,

obtained from taking the limit of an infinite, fundemental sequence of ordinals, which we say are of

a lower magnitude. New transfinite principles were required in order to cover the limit ordinals

whilst at the same time covering the first kind of ordinals. To do so required the addition of an extra

step in the proof process.

We had seen the introduction of transfinite induction by Cantor who introduced the

principle as a theorem, based on the existence of a function, in order to give a general definition of

ordinal exponentiation, one which allowed for ordinals of a transfinite degree, for example ωω.

Induction was critical for Cantor to develop this exponentiation as it allowed him to expand further

his set theory to include the ϵ -numbers; this all hinged on his theorem which I stated as:

There is a wholly determined single-valued function f (ξ) of the variable ξ which satisfies the

following conditions:

(a) f (0 )=δ ;

(b) If ξ ' and ξ ' ' are two values of ξ such that ξ '<ξ ' ' , then f (ξ' )< f (ξ ' ');

(c) For every value of ξ we have f ( ξ+1 )=f (ξ ) γ ;

(d) If {ξν} id any fundamental series, the so is f ¿. And if ξ=Limνξν, then

f ( ξ )=Limν f (ξν).

We can look at this theorem and say that the function f (ξ) is simply the property which we

would use transfinite induction to prove. Then step (a) would resemble the base case, step (c) the

inductive step for the first kind of ordinal numbers, with (d) the inductive step for the second kind of

ordinal numbers. The second step can be removed from the procedure as it merely asserts that the

function f ( ξ ) is continuous; specifically ascending in Cantor’s case. Thus we can rephrase the

procedure of transfinite induction in its modern form as follows:

Transfinite induction: for a property P(α), where α is an ordinal, P(α) is true for all

ordinals α if it satisfies the following three criteria:

(a) P(0) is true;

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(b) If P(β ) is true for all ordinals β<α , then P (α ) is true;

(c) If P (β ) is true for all ordinals β of fundamental sequence with limit ordinal λ, then P(λ)

is true.

The principle of transfinite induction can come in different variations; one such variation is

to simply show that ‘if P (α )holds for all α<β, then P (β )holds’. If this is can be shown, then P (β )

holds for all ordinals β. The above definition, like that of Cantor’s theorem, simply breaks the

method of transfinite induction up into the three cases necessary to prove that the property P(α)

holds for every possible kind of ordinal number.

The origin of transfinite induction came out of necessity. Cantor did not see the need to

even define his theorem under transfinite induction, nor does he feel the need to mention the word

induction even though he clearly shows his awareness of the principle in the Beiträge. Cantor’s

theorem is merely included in the Beiträge to further his foundations to his transfinite set theory.

However Cantor’s theorem would be adopted into other areas of set thory.

The Axiom of Choice is usually used to extend the principle of transfinite induction, and that

of transfinite recursion too, to any transfinite set by assigning a well-ordering to those sets which are

not, in their normal form, well-ordered. The key aspect of using the Axiom of Choice is that it will

assign a least element to a set and from there transfinite induction simply follows. Of course, the

Well-Ordering Theorem permits that such a well-ordering does exist.

Transfinite induction severs the ordinal numbers in the same manner to which mathematical

induction does for the natural numbers, condensing the unattainable down into a couple of finite

steps. Without transfinite induction a lot of results would remain unproven, especially those results

obtained through transfinite recursion.

We have seen that transfinite recursion was developed by von Neumann so that he could

obtain his version of the Axiom of Replacement. The recursion that we saw earlier in section one

developed as a result of the progression of mathematics; it was just adopted into mathematics as a

natural concept, one without definition, distinction or procedure. Although we do now in modern

mathematics have a specified procedure to which we follow when applying recursion, it still seems a

natural process. So like transfinite induction, recursion in transfinite set theory was introduced by

one man in order to obtain something else; the magnitude of the new principle is overlooked.

Like everything in mathematics, transfinite recursion has progressed and developed in its

brief history and has a more modern structure which can be stated as:

(Recall that Ω is the set of all ordinal numbers).

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Transfinite Recursion: let α 0∈Ω and let f :Ω→Ω and g a function that assigns every

segment an ordinal. Then we can define a function F by

(a) F (0 )=α 0;

(b) F (β+1 )=f (F (β ) ) for all successor ordinals β;

(c) F ( λ )=g (F ¿λ ) for all limit ordinals λ.

Notice the extra step to deal with the new kind of number that is the limit ordinal. The

introduction of the function g allows the new step to contrast from the definition from f and defines

the limit ordinal λ in terms of the fundamental sequence of ordinals that approach it from below.

The recursion principle of the transfinite numbers is then supported and verified by its

induction counterpart. Transfinite recursion provides the same shortcut as induction in squashing

the larger task of tackling large, unmanageable quantities into a finite process.

The concepts of induction and recursion in mathematics have allowed for a simplification of

mathematical proof and definition. They remain probably the most important and most popular

concepts within mathematics due to the simplicity in themselves and the simplicity they provide to

their results. Their natural inclusion into and progression within mathematics mirrors how valuable

they are to mathematics. Their adoption into all areas of mathematics emphasizes how much of a

necessity they are. But their overall main attribute is in how they make large, infinite, transfinite

quantities obtainable and manageable; it was inevitable that transfinite set theory would create its

own variations of these concepts.

105


[2.14] Bibliography: 1. H.C. Kennedy, Peano, Life and Works of Giuseppe Peano, Reidel Publishing Company, 1980 P1-

46.2. For some references on the life on Peano, the following website:

http://www.peano2008.unito.it/crono.php .3. E. T. Bell PH.D., Men of Mathematics, 1937, Victor Gollancz LTD., London. P572-575.4. R. Dedekind, Essays on the Theory of Numbers, translated by W. W. Beman, 1909, The Open

Court Publishing Company, Chicago.5. Van Heijenhoort, Frege to Godel, A Source Book in Mathematical Logic, 1879-1931, 3rd Printing,

1977, Harvard University Press.6. G. T. Kneebone, Mathematical logic and the Foundations of Mathematics, An Introductory

Survey, 2001, Dover Publications, INC., New York. P142-146.7. A. Badiou, Number and Numbers, translated by R. Mackay, 2008, Polity Press.8. S. M. Srivastava, Resonance, Transfinite numbers, March 1997, Volume 2, Issue 3, pp 58-68.9. F Cajori, History of Mathematics, 5th Edition, 1991, Vol. 2, Chelsea Publishing Company, P446-

447.10. E. Dubinsky, K. Weller, M.A. McDonald, A. Brown, Some Historical Issues and Paradoxes

Regarding the Concept of Infinity: An Apos-Based Analysis: Part 1, Educational Studies in Mathematics, 2005, Vol. 58, No. 3, P335-359.

11. J.W. Dauben, Georg Cantor, His Mathematics and the Philosophy of the Infinite, 1979, Harvard University Press.

12. J.W. Dauben, Georg Cantor and the Battle for Transfinite Set Theory, lecture of the 9th ACMS Conference, 1993, Westmount College, Santa Barbara, California, P1-22. Published in the Journal of the ACMS in 2004.

13. D. Berlinski, Infinite Ascent, A Short History of Mathematics, 2006, The Orion Publishing Group Ltd., London.

14. P.E. Jourdain, Transfinite Numbers and the Principles of Mathematics Part 1, The Monist, Vol.20, No.1, January 1910, P93-118.

15. G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, translated and introduced by P.E. Jourdain, 1955, Dover Publications, INC., New York.

16. H.M. Edwards, Kummer and Kronecker, Mathematics in Berlin, P61-69, 1998, Birkhäuser Basel. 17. J.J. O’Connor, E.F. Robertson, MacTutor History of Mathematics archive, University of St.

Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Cantor.html . This website we also used for some biographical notes on other mathematicians.

18. Encyclopedia.com, Georg Ferdinand Ludwig Philip Cantor, 2008, http://www.encyclopedia.com/topic/Georg_Ferdinand_Ludwig_Philip_Cantor.aspx .

19. A. Kanamori, The Mathematical Development of Set Theory From Cantor to Cohen, The Bulletin of Symbolic Logic, Vol.2, No.1, March 1996.

20. L.C. Young, Mathematicians and their Times, 1981, North-Holland Publishing Company, Amsterdam, P227-233.

21. R.L. Wilder, Introduction to the Foundations of Mathematics, 2nd Edition, 1965, John Wiley & Sons INC., New York.

22. P. Suppes, Axiomatic Set Theory, 1960, van Nostrand Reinhold Company, New York, P195-205.

106

http://www.encyclopedia.com/topic/Georg_Ferdinand_Ludwig_Philip_Cantor.aspx

http://www-history.mcs.st-andrews.ac.uk/Biographies/Cantor.html

http://www.peano2008.unito.it/crono.php


23. J. Roitman, Introduction to Modern Set theory, 1990, John Wiley & Sons INC., New York, P23-57.

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