MATHEMATICAL LOGIC*

R.A. Bull

(received 1 June, 1971)

The purpose of this talk is to give a sketch-map of mathematical

logic, with highlights on some points which have amused me in my recent

reading. It is very much an amateur performance. The flow chart in

the abstract for the talk (see page 66) may help you to follow where I

am going.

l. Philosophical studies

There are two opposing tendencies in mathematical logic: asking

logical questions about mathematics; and giving mathematical answers in

logic. Of course this split is an illuminating exaggeration and the

counter-examples are important. For example, to answer a logical ques­

tion about mathematics one must often formalise the reasoning involved

in that mathematics; and some technical results in mathematics are of

great logical significance.

The basic form of logical questions about mathematics is: 'What

am I doing when I do this piece of mathematics?' If this question is

really hurting, then one needs some kind of consolation; and there seem

to be two kinds of consolation available, that offered by the philos­

ophers, and that offered by the mathematicians. This split is another

illuminating exaggeration.

One kind of answer to the question, 'What am I doing when I do

this piece of mathematics?' is the reductionist answer, 'You are not

doing mathematics, you are doing X '- where X is something nice and

* Invited address delivered at the sixth New Zealand Mathematics

Colloquium, held at Wellington, 17-19 May, 1971.

Math. Chronicle 2(1972), 17-27.

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unproblematic. This is an unfortunate answer, because in fact mathe­

matics is not X, it is mathematics. The earlier years of this century

were notable for three large-scale reductionist answers to the basic

question. Russell said that mathematics is really a branch of formal

logic. Brouwer said that mathematics ought to be the science of mental

constructions (intuitionism). Thus one cannot say 'either a is true or

a is false', until one has constructed a proof for one of these assert­

ions. Hilbert hoped that mathematics was the science of manipulating

symbols according to formal rules. Thus even though the content of

mathematics was problematic, the results could be obtained by finitary

methods. All these answers are obviously silly. This is yet another

illuminating exaggeration: if X is a good approximation to mathematics

then a study of X will throw light on mathematics; and while X is not

all of existing mathematics, it can be a very interesting new branch of

mathematics, as these three X's have turned out to be.

But what turned mathematical logicians from reductionist answers

to the basic question, was GOdel's proof that if X was adequate for the

theory of the natural numbers under addition and multiplication, then X

was not nice and unproblematic. This meant that substituting X for

mathematics could not be the knock-out answer which was hoped for then.

To give a later, more general variant of GOdel's result: Let arithmetic

be the theory with the natural numbers and variables over them, addit­

ion and multiplication, equality, logical operations including quanti­

fication. A satisfactory formalisation of arithmetic would include

numerals n and a derivability predicate |— with the properties:

f- is effectively defined;

for each statement P in arithmetic, there is a formula a with

Pin) true iff f-a(n).

There is no formal theory with these properties.

GOdel's work has been crucial to the history of mathematical

logic, and I should give three of his main results (in rather stretched

variants). These are classic instances of technical results which are

of philosophical significance. He showed that any theory formalised as

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a predicate calculus is model-complete with respect to suitable models;

so that adding predicate constants and axioms to the predicate calculus,

the formulas which are derivable are those which are valid in some class

of models. He showed that any formalisation of arithmetic is not model-

complete with respect to the intended model alone; so that any formal

mathematical theory of interest involves us in non-standard models. He

constructed a non-standard model for basic set theory plus the axiom of

choice plus the continuum hypothesis, within basic set theory, showing

that the axiom of choice and the continuum hypothesis are consistent

relative to basic set theory.

It seems to me that there is still room for a philosophical

discussion of mathematics which is not reductionist, but which helps to

remove the problematic feeling about topics such as the mode of exist­

ence of mathematical entities. Wittgenstein, who gave helpful discus­

sions of meaning in ordinary language, left rough notes on the nature

of mathematics; but these are generally considered to be unsatisfactory.

Otherwise very little work of this 'therapeutic' kind has been done for

mathematics. I want discussions of 'ordinary' mathematics rather than

of formalised mathematics; I envisage much of the discussion being

itself mathematical. The only work which fits my needs are the very

brilliant analyses of Kreisel - except that they soon get out of my

depth!

2. Mathematical studies

Another kind of answer to 'What am I doing when I do this piece

of mathematics?' is to continue the mathematical analysis of the intui­

tive foundations of that piece of mathematics. In an ordinary mathe­

matical theory we have formal mathematics down to a certain level, and

below that we rely on our understanding of the terms. This level tends

to get shifted down. Much work has been done on the mathematical analy­

sis of mathematical concepts; the importance of mathematical analysis of

mathematical proofs is now being realised. The 19th century was notable♦

for its mathematical clarification of the concepts of the calculus. In

this century, the main targets of mathematical clarification have been

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the concepts of sets, of computable functions, and of models of a for­

mal mathematical theory. These are very large subjects; I shall only

comment on some details.

The set theorists have arrived at a much better intended inter­

pretation for the concept of a set. Earlier in the century, the inten­

ded interpretation was that a set was the domain of a predicate, that

is a class, with some suitable restriction to avoid paradoxes. Four

different restrictions led to the four main set theories: Russell res­

tricted the formulas which could be called predicates: Quine restric­

ted the predicates from which classes could be formed; Zermelo et al

restricted sets to being the intersections of classes and given sets;

von Neumann et al restricted sets to being the classes which were memb­

ers of classes. The new method is a construction of the collection of

sets, with stages typed by ordinals. Writing E(a) for the collection

of sets available at the a th stage, we take:

£(0 ) = {<*>}

Z(a + 1) = {1(a)}

E(a) = U E($), for limit ordinal a.$<a

With this intended interpretation the theories ZF and NGB are true,

together with the axiom of choice. The ordinal a which types the col­

lection Z(a) of all sets of a set theory is characteristic; for example

if a Zermelo-type set theory is E(a) then the corresponding von-Neumann-

type set theory is E(a + 1). Much work is being done on E(a) for very

large ordinals a, but the question of the continuum hypothesis remains

open.

Take a mathematical theory; form the set of sentences which are

valid in this theory; form the class of models in which this set of

sentences is valid. Model theory considers this closure of a given

mathematical theory. I shall illustrate the extent to which models can

differ from the intended model, and the extent to which non-standard

models are relevant to the intended model. If a theory has a model

with cardinality H for any infinite X? , then it has a model with

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cardinality V . This is easy to show, but the axiom of choice is

equivalent to the converse, namely: If a theory has a model with

cardinality q then it has a model with cardinality X , for each

infinite X . So if a theory has a model with any infinite cardinality

then it has models with every infinite, cardinality. This shows that a

formal set theory has models which are very non-standard as regards card­

inality. However non-standard models can be easier to handle than stan­

dard ones. Abraham Robinson has shown that analysis has non-standard

models with members h such that 0 < h < x, for each standard positive

real number x. The derivative f'(a) of / at a is precisely the standard

part of the fraction

f(a + h) - f(a) for such non-standard h.

h

It can be shown that the results which hold for the standard members

of this model are precisely the results which hold in the standard

model.

It is in the analysis of mathematical proofs that intuitionist

mathematics comes into its own, because its intended interpretation is

in terms of proofs rather than in terms of sets, as for classical math­

ematics. The real numbers of intuitionist mathematics are so weak that

there are reals x and y with none of x < y3 x = y3 x > y. However,

proofs of arithmetic and large fragments of classical analysis can be

translated into proofs in intuitionist analysis. Further, distinctions

made in intuitionist mathematics, but lost in the intended interpreta­

tion of classical mathematics, can be used to get a better analysis of

proofs than can be obtained in classical mathematics. See the works

of Kreisel, and the references given there.

3. Formal logics

I now turn from the very large subject of foundations of mathe­

matics to the smaller subject of the mathematics of formal systems of

logic. The central system is the classical first-order predicate cal­

culus. Here we have propositions which are true or false, individuals

from some domain, predicates (that is, mappings from individuals to

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propositions), and the logical operations, A for 'and', ~ for 'not',

Vx for 'for all x', v for 'or', -* for 'implies', 3x for 'for some x'.

It is in terms of this calculus that mathematical theories are usually

formalised. Given a mathematical theory we have the set of formulas

which are valid when that theory is used as a model; and given a set

of formulas we have the class of models in which all the formulas are

valid. A mathematical theory can be studied by setting up an applied

predicate calculus with a derivability predicate |— in terms of an eff­

ectively defined set of axioms and finitary derivation rules, so chosen

that the derivable formulas in the predicate calculus approximate to the

true statements in the mathematical theory. The applied predicate cal­

culus itself can be studied formally where the original theory cannot,

because j- is defined effectively while the original theory is based on

intuitive concepts.

The concept of validity requires some set theory, but the concept

of derivability is effective and finitary. So although model-theoretic

methods are usually stronger, proof-theoretic methods can be more inter­

esting, and I shall talk about them on their own first.

The theory of derivable formulas )- a is difficult to analyse

directly; instead derivable sequents o ,,...,01̂ (— are stud­

ied. The semantics for sequents is a ,...,0^ f= 3 .»•••.> 3g iff

(^A.-.Aop * (s^.-.vep.

Most of the derivation rules for sequents introduce logical operators

into sequents; for example, the rules for A - introduction are:

r , a , 3 f ~ A r |— A , a r (— A , 3 «

T , a A 3 } - A r ( - A , o t A 3

Note that in these rules each formula of an antecedent sequent is a

subformula in the consequent sequent. A derivation which has this sub-

formula property is called a direct derivation.

Gentzen showed that there is a derivation of |— a in the ordinary

predicate calculus iff there is a direct derivation of |— a in the seq­

uent calculus. Typing the sequents in a direct derivation with ordinals,

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he obtained a finitary consistency proof for predicate calculus rela­

tive to the existence of suitable ordinals. This argument has been

extended to give finitary consistency proofs for formal arithmetic and

formal analysis relative to the existence of suitable ordinals.

A full formalisation of many statements requires quantifiers

over predicates, as well as over individuals; for example, the induction

principle refers to all properties of the integers. Adding such quanti­

fiers to the first-order predicate calculus gives the second-order pre­

dicate calculus. Stronger mathematical theories can make use of logi­

cal objects of higher type again, although these can be avoided by using

sets instead. The simple theory of types due to Church can handle pre­

dicates and functions of all finite types, in a very elegant manner.

We have types 0,1, and 018 where a and 8 are types. A formula of type 0

is a proposition; a formula of type 1 is an individual; and a formula

of type a8 is a function from objects of type 3 to objects of type a.

The apparatus of simple type theory consists of: variables x , where

a is a type, etc; B ̂ (of type a), the value of the function

with argument B \ x ^ A^ (of type aB) the function determined by the

formula by taking

k n A : y a -* B 3 8 a y 3 a

where B is obtained from A by substituting y Q for x ;a a ° 8 8

quantifiers CT0(0a) (given a predicate forming propositions from

objects of type a, n , .A is the proposition 'for all x , A of x J o(oa) oa r r a oa ais true1); a choice function l r a finite number of suitable axioms

J a(oa)and derivation rules. This calculus is sufficient for all known mathe­

matics.

I now return to model-theoretic considerations. To study the

relation between a mathematical theory and its formalisations it is

necessary to study the class of models with respect to which model­

completeness holds for the formal system, that is, the definitions of

(= for which (- a iff f= a. The relations between these three systems

are liable to be complicated.

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Applied first-order predicate calculi are known to be model-

complete with respect to some models. However, if the intended model

is characterised by higher-order properties then completeness will

involve non-standard models, as with arithmetic. Henkin has proved

model-completeness for higher-order predicate calculi, including simple

type theory. However, for higher-order predicate calculi this complet­

eness involves secondary models, where ranges over a subset of all

objects of type a on the domain. So here again model-completeness will

involve non-standard models, as with arithmetic. Even in arithmetic,

model-completeness must involve models which are non-standard in some

way. To summarise: for many theories both first-order and higher-

order predicate calculi permit non-standard models; the higher-order

calculi have greater logical elegance; but first-order predicate calc­

ulus is easier to use for most practical purposes.

Appendix A. In tu itio n is m

To see that intuitionism is reasonable, remember that its inten­

ded interpretation of a statement a is 'a is provable1, rather than

'a is (set-theoretically) true'. Set theory makes the truth of

a v ~ a trivial, but if one is interested in proofs then it is not

trivial, as intuitionism observes. Kripke has given a semantics for

intuitionism, in terms of a mathematician passing through a tree of

situations with increasing information at each node. He knows what

information is available at future trees, but not which path he will

actually take through the tree.

The intuitionist theory of sequents is as for the classical the­

ory, except that the right hand side must have at most one formula.

In classical semantics a v 3 is ~(~a A ~3), a -* 3 is ~(a A ~ 3 ) , 3a;a is

~ V x ~ a ; if r 1 |- A 1 is obtained from r (— A by making these replacements

then r (— A in classical logic iff r f (— A ' in intuitionist logic.

Godel used an extension of this argument to embed classical arithmetic

in intuitionist arithmetic.

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Appendix B. Non-classical logic

In classical logic it is assumed that each proposition is either

true or false, but for many purposes this assumption does not worki

Kripke and Hintikka have given semantics for a number of logical

operators, such as 'it is necessarily the case that a', 'it ought to be

the case that a', 'A knows it is the case that a1, M believes it is the

case that a'. This is done by taking a set K of possible worlds, and

a relation R on K, with the intended interpretation that ctRb means

'world b is compatible with world a', under an appropriate criterion

of compatibility, for example, under necessary truth, or under a moral

code, or under A's knowledge, or under 4's beliefs. The formal logical

operator o is then defined by taking Da to be true in world a iff a is

true in every world which is compatible with world a, that is,

V(uaLtd) = T iff Vb(dRb + 7(ct,B) = T).

Unlike almost all formal mathematical theories, these logics with­

out quantifiers are decidable. A good method of proving decidability

through semantics is the finite model property, due to Harrop. He show­

ed that if a calculus is model-complete with respect to each member of a

set of finite models, then the calculus is decidable. Using this result,

proving decidability becomes a matter of algebraic manipulation of the

models of a logic.

When quantifiers are introduced in non-classical logics, con­

siderable logical and technical difficulties arise. This can be expres­

sed in terms of the Kripke semantics, by saying that the different pos­

sible worlds may have different languages, making it difficult to work

out 7(a,a) when a contains both □ and V. Several mathematical logicians

are working on these problems at present.

Appendix C. Henkin's method

Proofs of model-completeness can be obtained by defining a model

which satisfies a in terms of the direct proof of |— a, giving a fini-

tary proof of model-completeness. A more elegant method due to Henkin

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takes the set of derivable formulas of the given calculus and uses its

maximal consistent extensions.

First, some notation:

A set of formulas r is inconsistent iff I— ~(a A...A a ) for some1 r

{a ,...,a } c r.1 r -

r is consistent iff it is not inconsistent, r |— a ('a is derivable

from r 1) iff r U {'mj} is inconsistent; that is r [— a iff

f- (a A •. Aa ) a for some {a , a } c r.1 i r j* ' v —~ -completeness, T (— ~a iff not r |— a.

V-completeness, r |— Vxa(x) iff r (— a(t) for each variable t.

r is a theory iff r is consistent and r |— a implies a € r. (Thus for

a theory T we have: ~ -completeness, ~a € r iff not a € T;

V-completeness, Vxa(x) € r iff a it) € r for each variable t.)

Henkin's method of constructing a model satisfying given 6 with

{6) consistent now runs: there is a ~ -complete, V-complete theory A

with 3 € A; define 7(a) = T iff a € A, then 7 is an evaluation and

7(3) = T. The trouble with Henkin's construction is that the V-complete

theory obtained has a bigger language than the given set of formulas.

The following result, due to Thomason, obtains a V-complete theory in the

same language as the given set of formulas. I would be prepared to call

it the fundamental theorem of formal logic, at any rate for the formal

logics with countable languages. Given consistent, V-complete r there

is a ~ -complete, V-complete theory A with r £ A. The proof can be obta­

ined through the following lemmas:

1. r (- a + 3 iff r U {a} (- 3.

2. If r is V-complete then r U {a} is V-complete.

3. Given V-complete r, for each formula a(x) we have

r |— 3xa(x) -»• a it), for some variable t.

4. Enumerate the formulas with x free as a (x), a (x), a (x),... .1 2 3

Given consistent, V-complete r there is a theory r f with r £ r'

and 3xa^(#) € r 1, for some variable t^, for i = 1,2,3, ..

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5. Given a theory P with 3xa. (x) ■+ a. (t.) € r', for i - 1,2,3,...is 1s 1s

there is a ~ -complete, V-complete theory A with T'£ A .

University of Canterbury

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