xii infinite ¥. david hilbert (1862 - 1943) no one shall drive us from the paradise which cantor...

XII

Infinite

David Hilbert (1862 - 1943)

No one shall drive us from the paradise which Cantor has created for us.

his theory of transfinite numbers; it appears to me as the most admirable blossom of mathematical spirit and one of the supreme achievements of purely intellectual human activity.

Nearly all contemporary mathematicians accept Cantor‘s transfinite set theory as the foundation of mathematics.

Carl Friedrich Gauß (1777 - 1855)

I protest firstly against the use of an infinite magnitude as a completed one, which never has been allowed in mathematics. The infinite is only a mode of speaking, when we in principle talk about limits which are approached by certain ratios as closely as desired whereas others are allowed to grow without reservation.

Leopold Kronecker (1823 - 1891)

Professor of Cantor, later accused him to be a spoiler of youth.

Henri Poincaré (1854 - 1912)There is no actual infinity. The Cantorians forgot this, and so fell into contradiction. Future generations will consider set theory as an illness from which one has recovered.

Luitzen E. J. Brouwer (1881 - 1966)

“De tweede getalklasse van Cantor bestaat niet.“ (Dissertation, 1907)

Hermann Weyl (1885 - 1955)

Successor of Hilbert in Göttingen

Classical logic was abstracted from the mathematics of finite sets and their subsets.Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets.This is the Fall and Original sin of set theory.

Ludwig Wittgenstein (1889 - 1951)

It isn't just impossible "for us men“to run through the natural numbers one by one; it's impossible, it means nothing.You can’t talk about all numbers, because there's no such thing as all numbers.

Set theory is wrong because it apparently presupposes a symbolism which doesn't exist instead of one that does exist (is alone possible). It builds on a fictitious symbolism, therefore on nonsense.

I believe, and I hope, that a future generation will laugh at this hocus pocus.

In the modern intellectual picture of our world the actuaI infinite appears virtually anachronistic.

Paul Lorenzen (1915 - 1994)

Abraham Robinson (1918 - 1974), pupil of Fraenkel, Founder of the Non-Standard-Analysis: Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless.

Walter Felscher (1931 -2000), author of a three-volume text book on set theory: Concerning the application of transfinite numbers in other mathematical branches: the early hopes have been realized only in few special cases …“

Solomon Feferman (1928)

The actual infinity is not necessary for the mathematics of the real world.

At least to that extent the question "Is Cantor necessary?" is answered with a resounding "no".

Edward Nelson (1932)

A construction does not exist until it is made; when something new is made, it is something new and not a selection from a pre-existing collection. If I give you an addition problem like

37460225182244100253734521345623457115604427833+ 52328763514530238412154321543225430143254061105

and you are the first to solve it, you will have created a number that did not exist previously.

A construction does not exist until it is made; when something new is made, it is something new and not a selection from a pre-existing collection. If I give you an addition problem like

37460225182244100253734521345623457115604427833+ 52328763514530238412154321543225430143254061105

89788988696774338665888842888848887258858488938

Sorry. This one exists already.

Edward Nelson (1932)

William Thurston (1946-2012)Fields-Medalist

On the most fundamental level, the foundations of mathematics are much shakier than the mathematics that we do. Most mathematicians adhere to foundational principles that are known to be polite fictions. For example, it is a theorem that there does not exist any way to ever actually construct or even define a well-ordering of the real numbers. There is considerable evidence (but no proof) that we can get away with these polite fictions without being caught out, but that doesn’t make them right. Set theorists construct many alternate and mutually contradictory “mathematical universes” such that if one is consistent, the others are too. This leaves very little confidence that one or the other is the right choice or the natural choice.

Nik Weaver(1969)

Philosophically, it makes sense only in terms of a vague belief in some sort of mystical universe of sets which is supposed to exist aphysically and atemporally (yet, in order to avoid the classical paradoxes, is somehow “not there all at once”). Pragmatically, ZFC fits very badly with actual mathematical practice insofar as it postulates a vast realm of set-theoretic pathology which has no relevance to mainstream mathematics.

Kurt Gödel (1906 - 1978)

Our axioms, if interpreted as meaningful statements, necessarily presuppose a kind of Platonism, which cannot satisfy any critical mind and which does not even produce the conviction that they are consistent.

Your "Paradise“ is a Paradise of Fools, and besides feels more like Hell.

Doron Zeilberger ( 1950):Herren Geheimrat Hilbert und Prof. Dr. Cantor

every statement that starts" for every integer n " is completely meaningless.

Georg Cantor1845 - 1918

There are inconsistent sets. The set of all sets should have a larger cardinality than itself.

Bertrand Russell(l872 - 1970)

The set of all sets that do not contain thermselves (Barber).

Every theory like set theory has a countable model.

Leopold Löwenheim(1878 - 1957)

Thoralf Albert Skolem(1887 - 1963)

Löwenheim-Skolem-Paradox

The Paradox of Banach-Tarski

Stefan Banach(1892 - 1945)

Alfred Tarski(1902 - 1983)

We are, like Poincaré and Weyl, puzzled by how mathematicians can accept and publish such results; why do they not see in this a blatant contradiction which invalidates the reasoning they are using?

Presumably, the sphere paradox and the Russell Barber paradox have similar explanations; one is trying to define weird sets with self-contradictory properties, so of course, from that mess it will be possible to deduce any absurd proposition we please.

The Banach-Tarski paradox amounts to an inconsistency proof.

Émile Borel (1871 – 1956)

Edwin T. Jaynes (1922 – 1998)

Every definition is finite in ist basical nature, i.e., it explains the notion to be defined by a finite number of known notions.

„Infinite definitions" are nonsense.

If the theorem was true that all finitely definable numbers are a set of cardinality 0 then the continuum would be countable, but that is certainly wrong.

Georg Cantor(1845 - 1918)

The constructible numbers like e, oder L are countable.

All possible combinations of finitely many letters belong to a countable set. Since every real number has to be definable by a finite number of words, there can be only countably many real numbers – in contradiction with Cantor‘s theorem and ist proof.

Hermann Weyl (1885 - 1955)

Paul Bernays(1888 - 1977)

It is this absolute platonism which has been shown untenable by the antinomies.

If we pursue the thought that each real number is defined by an arithmetical law, the idea of the totality of real numbers is no longer indispensable.

Kurt Schütte(1909 - 1998)

If we define the real numbers in a strictly formal system, they are countable.

0100011011000…

Every number that can be used belongs to a countable set.

There are only countably many labels.

The constructible numbers like e, oder L are countable.

There are uncountable sets.

Sets are defined by their elements.

We can distinguish elements.

All natural numbers are contained as exponents in finite initial segments.

0,1 = 10-1

0,11 = 10-1 + 10-2

0,111 = 10-1 + 10-2 + 10-3

…

{ 1 } { 1, 2 } { 1, 2, 3 } { 1, 2, 3, 4 }{ 1, 2, 3, 4, 5 } …

{ 1 } { 1, 2 } { 1, 2, 3 } { 1, 2, 3, 4 }{ 1, 2, 3, 4, 5 } …

If numbers exist, then they are in the last column.

If in the last column, then they are in the whole triangle.

Two lines never contain more than one of them.

The set of even numbers is countably infinite: 1 2 3 4 5 6 7 8 9 ...2 4 6 8 10 12 14 16 18 ...

|{2, 4, 6, …, g}| < |{2, 4, 6, …}|

< g

{2}

{2, 4}

{2, 4, 6}

{2, 4, 6, 8}

{2, 4, 6, 8, 10}

{2, 4, 6, 8, 10, 12}

...

Sets of even numbers

|{2, 4, 6, …, g}| < |{2, 4, 6, …}|

< g

Every set of positive even numbers contains numbers that are larger than the cardinal number of the set.

Cantor’s "paradise" as well as all modern axiomatic set theory [AST] is based on the (self-contradictory) concept of actual infinity.

A. Zenkin(1937 –

2006)

It is an intentional and blatant lie, since if infinite sets are potential, then the uncountability of the continuum becomes unprovable.

Cantor emphasized plainly and constantly that all transfinite objects of his set theory are based on the actual infinity.

Modern AST-people try to persuade us to believe that the AST does not use actual infinity.

n r(n) ___________________ 1 0.000111199999... 2 0.123456789123... 3 0.555555555555... 4 0.789789789789... 5 0.010000000000... ... ...

n r(n)00000___________________0000001 0.000...0000002 0.1000...0000003 0.11000...0000004 0.111000...0000005 0.1111000...... ... ...

Cantor‘s diagonal proof is an impossibility-proof

n r(n)00000___________________0000001 0.100...0000002 0.1100...0000003 0.11100...0000004 0.111100...0000005 0.1111100...... ... ...

Without actual completion of 1/9 = 0.111… the diagonal number is in the list. If 1/9 is the diagonal, then also a vertical and a horizontal row must contain it.

Cantor‘s diagonal proof is an impossibility-proof

Percy W. Bridgman (1882–1961)Nobel laureate

The ordinary diagonal Verfahren I believe to involve a patent confusion of the program and object aspects of the decimal fraction, which must be apparent to any who imagines himself actually carrying out the operations demanded in the proof.

In fact, I find it difficult to understand how such a situation should have been capable of persisting in mathematics.

A list with all terminating decimal numbers of the unit interval is possible.

0.3476183

0.34761831


0.347618311


0.347618311312321


0.347618311312321876760760


0.347618311312321876760760

Each one is infinitely often in the list.

dn is infinitely often in the list.


The transfinite numbers stand or fall with the finite irrational numbers. Georg Cantor (1845 - 1918)

Sequence: (1/10n )= 0.1, 0.01, 0.001, …

terms (without limit)

Series: 1/10n = 0.1, 0.11, 0.111, …

terms (without limit)

More than terms are impossible

The limit has no decimal representation.

pairs of parentheses, 2fractions.

Up to the nth pair of parentheses there are 2n fractions.

- + (-) + (- + -) + (- + - + - + -) + ...1 2 3 4 5 6 7 8 1 1 1 1 1 1 1 1

The Life and Opinions of Tristram Shandy, Gentleman

Laurence Sterne(1713-1768)

Adolf Abraham Fraenkel(1891 – 1965)

Tristram Shandy needs one year to write one day of his biography. When living for „countably many“ years, his biography would be finished.

1 1 1 2 1 1 2 3 _,_, _,_, _,_, _,_, ...2 3 4 3 5 6 5 4

Sequence of rational numbers {q I 0 < q < 1}

With an actual infinity of transpositions an ordering by size would be possible.

1 2

345

1 2 3 4 5 6 7

8

2 1

2 1

4 3 2 1

4 3 2 1

6 5 4 3 2 1

6 5 4 3 2 1

8 7 6 5 4 3 2 1

8 7 6 5 4 3 2 1

. . . . . . . . . . . .

0 (set theory)

(mathematics)

1 2 3 4 5 6 7

8

(set theory)

(mathematics)

0 1 2 3 4 5 6 …

Indexing of all positive rational numbers is a supertask:

0 1 2 3 4 5 6 …

in: (0, 1] out 1/1


0 1 2 3 4 5 6 …

in: (0, 1] out 1/1

in: (1, 2] out 1/2


0 1 2 3 4 5 6 …

in: (0, 1] out 1/1

in: (1, 2] out 1/2

in: (2, 3] out 2/1


0 1 2 3 4 5 6 …

Always infinitely many rationals in the intermediate reservoir.

in: (0, 1] out 1/1

in: (1, 2] out 1/2

in: (2, 3] out 2/1

… and so on



0 1 2 3 4 5 6 …

in: (0, 1] out 1/1

in: (1, 2] out 1/2

in: (2, 3] out 2/1

… and so on

The number of „clean“ intervals grows beyond every bound.

0,737342483465448512090030345234985349853493857123554…

0,737342483465448512090030345234985349854493857123554…

Between two irrational numbers there is always a rational number.

II II

0,737342483465448512090030345234985349853493857123554…

0,737342483465448512090030345234985349854000000000000…

0,737342483465448512090030345234985349854493857123554…

Decimal representation of numbers742.25

7 4 2 . 2 5

102 101 100 . 10-1 10-2

Binary representation of numbers 22 21 20 . 2-1 2-2

1 0 1 = 4 + 0 + 1 = 5

1 1 0 . 1 = 4 + 2 + 0 + 1/2 = 6,5

0 . 1 1 = 1/2 + 1/4 = 0,75

0.111111 ... = 1/2 + 1/4 + 1/8 + ... = 1

0.010101... = 1/4 + 1/16 + 1/64 + ... = 1/3

0.

0

1

0 1 0 1

0 1 0 1 0 1 0 1

The Binary Tree

0.

0

1

0 1 0 1

0 1 0 1 0 1 0 1

1 2

3 4 5 6

15

7 118 12

16

13 149 10

The Binary Tree

The elementary cell:

The elementary cell:

2 - 1 - 1 = 0

The path-construction of the Binary Tree

0.

0

0

0


0.

0

1

0 1

0 1


0.

0

1

0 1 1

0 0 1


0.

0

1

0 1 0 1

0 0 0 1


0.

0

1

0 1 0 1

0 0 1 0 1


0.

0

1

0 1 0 1

0 0 1 0 1 1


0.

0

1

0 1 0 1

0 0 1 0 1 0 1


0.

0

1

0 1 0 1

0 1 0 1 0 1 0 1


0.

0

1

0 1 0 1

0 1 0 1 0 1 0 1

= 0


Every single constructed path covers infinitely many nodes.

After every step of the construction the ratio

number of paths

number of nodes

How many natural numbers do exist?

12345678901234567890123456789012345678901234567890123456789012345678901234567890

< 1080 atoms in the universe

10100000

Where exist numbers that cannot be written with 108 digits?

• God• Nature • Mathematis

Where is the infinite realized?

Georg Cantor1845 - 1918

???

There are no different infinities.

There is no finished infinity.

The infinite is direction, not amount.

= + 1 = 2 = 2

< 2?, , ...

John Wallis

(1616 - 1703)

xii infinite ¥. david hilbert (1862 - 1943) no one shall drive us from the paradise which cantor...

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