2. XII
Infinite

3. Nearly all contemporary mathematicians accept Cantor's transfinite set theory as the foundation of mathematics.
No one shall drive us from the paradise which Cantor has created for us.
His theory of transfinite numbers appears to me as the most admirable blossom of mathematical spirit and one of the supreme achievements of purely intellectual human activity.
David Hilbert
( )
Arguing in favour means carrying coals to Newcastle.
But ...

4. Jacob Friedrich Fries (1773-1843)
Professor of mathematics and physics at Heidelberg.
An infinite magnitude or smallness must never be considered as a given entity.
Carl Friedrich Gauß ( )
Professor of mathematics and director of the observatory at Göttingen.
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.

5. Cantor's professor in Berlin
later accused him to be a spoiler of youth.
Had much correspondence with Hermite in order to demonstrate to him that all set theory was "only humbug".
Leopold Kronecker
( )
I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there.

6. There is no actual infinity
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.
Henri Poincaré
( )
Whatever be the remedy adopted, we can promise ourselves the joy of the doctor called in to follow a fine pathological case.

7. De tweede getalklasse van Cantor bestaat niet. (Dissertation, 1907)
The second number class and the higher power sets are examples of meaningless word play.
Luitzen E.J. Brouwer
( )
Theorem of well-ordering also turns out to be illusory.

8. 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.
Hermann Weyl
( )
Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers.

9. You can’t talk about all numbers, because there's no such thing as all numbers.
Set theory is wrong.
There is no path to infinity, not even an endless one.
I believe, and I hope, that a future generation will laugh at this hocus pocus.
Ludwig Wittgenstein
( )

10. 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.
Kurt Gödel ( )

11. In the modern intellectual picture of our world the actuaI infinite appears virtually anachronistic.
We introduce numbers for counting. This does not at all imply the infinity of numbers. For, in what way should we ever arrive at infinitely-many countable things?
Paul Lorenzen
( )
In arithmetic there does not exist a motive to introduce the actual infinite.

12. Pupil of A. Fraenkel
Founder of 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.
Abraham Robinson ( )

13. The actual infinite 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".
No set-theoretically definable well-ordering of the continuum can be proved to exist from the Zermelo-Fraenkel axioms together with the axiom of choice and the generalized continuum hypothesis.
Solomon Feferman
( )
The platonism which underlies Cantorian set theory is utterly unsatisfactory as a philosophy of our subject

14. 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 …"
Walter Felscher
( )

15. If I give you an addition problem like
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.
Edward Nelson
( )
If I give you an addition problem like
and you are the first to solve it, you will have created a number that did not exist previously.

16. If I give you an addition problem like
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.
Edward Nelson
( )
If I give you an addition problem like
Sorry. This one exists already.

17. 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.
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.
William Thurston
( )

18. Herren Geheimrat Hilbert
und Prof. Dr. Cantor
Your "Paradise" is a Paradise of Fools, and besides feels more like Hell.
Every statement that starts
"for every integer n"
is completely meaningless.
Doron Zeilberger
(*1950)

19. If you have an elaborate theory of "hierarchies upon hierarchies of infinite sets", in which you cannot even in principle decide whether there is anything between the first and second "infinity" on your list, then it's time to admit that you are no longer doing mathematics.
Norman J. Wildberger
(*1956)
A "sequence'" which is not generated by such a finite rule? Such an object would contain an "infinite amount" of information, and there are no concrete examples of such things in the known universe. This is metaphysics masquerading as mathematics.

20. 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.
Nik Weaver
(*1969)

21. There are inconsistent sets
There are inconsistent sets. The set of all sets should have a larger cardinality than itself.
Bertrand Russell
(l )
Georg Cantor
The set of all sets that do not contain thermselves (Barber).

22. Löwenheim-Skolem paradox
Leopold Löwenheim
( )
Thoralf Albert Skolem
( )
Every theory like set theory has a countable model.

23. Banach-Tarski paradox
Stefan Banach
( )
Alfred Tarski
( )

24. The Banach-Tarski paradox amounts to an inconsistency proof.
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.
Émile Borel (1871–1956)
Edwin T. Jaynes (1922–1998)

25. The constructible numbers like e, p oder L are countable.
Every definition is finite in its 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
( )

26. 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 its proof.
Hilbert's successor
at Göttingen
Hermann Weyl
( )

27. 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.
Hilbert's co-author
Paul Bernays
( )

28. An uncountable set of relation symbols - such a system of notations can not exist.
Hilbert's co-author
Wilhelm F. Ackermann
( )

29. If we define the real numbers in a strictly formal system, they are countable.
Hilbert's last student
Kurt Schütte
( )

30. The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking.
Hilbert himself
David Hilbert
( )

31. The constructible numbers like e, p oder L are countable.
There are only countably many labels. 1 00 01 10 11
000 …
Every number that can be used belongs to a countable set.

32. We can distinguish À0 elements.
There are uncountable sets.
Sets are defined by their elements.

33. 0,1 = 10-1
0,11 =
0,111 = …
All natural numbers are contained as exponents in finite initial segments.
{ 1 }
{ 1, 2 }
{ 1, 2, 3 }
{ 1, 2, 3, 4 }
{ 1, 2, 3, 4, 5 } …

34. If À0 numbers exist, then they are in the last column.
If À0 numbers are in the last column, then they are in the whole triangle.
Two lines never contain more than one of them.
{ 1 }
{ 1, 2 }
{ 1, 2, 3 }
{ 1, 2, 3, 4 }
{ 1, 2, 3, 4, 5 } …

35. The set of even numbers is countably infinite: À0

36. |{2, 4, 6, …, 2n}|
< 2n
< |{2, 4, 6, …}|
= À0

37. |{2, 4, 6, …, 2n}|
< 2n
< |{2, 4, 6, …}|
= À0
Sets of even numbers
{2}
{2, 4}
{2, 4, 6}
{2, 4, 6, 8}
{2, 4, 6, 8, 10}
{2, 4, 6, 8, 10, 12}
...
Every set of positive even numbers contains numbers that are larger than the cardinal number of the set.

38. Cantor’s "paradise" as well as all modern axiomatic set theory [AST] is based on the (self-contradictory) concept of actual infinity.
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.
Alexander Zenkin
(1937–2006)
It is an intentional and blatant lie, since if infinite sets are potential, then the uncountability of the continuum becomes unprovable.

39. n r(n)
___________________

40. Cantor‘s diagonal proof is an impossibility-proof
n r(n)
00000___________________

41. Cantor‘s diagonal proof is an impossibility-proof
n r(n)
00000___________________

42. Cantor‘s diagonal proof is an impossibility-proof
n r(n)
00000___________________
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.

43. 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.

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

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

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

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

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

49. A list with all terminating decimal numbers of the unit interval is possible.
Each one is infinitely often in the list.
dn is infinitely often in the list.

50. The transfinite numbers
stand or fall with the
finite irrational numbers.
Georg Cantor ( )
Sequence: (1/10n )
= 0.1, 0.01, 0.001, …
À0 terms (without the limit)
Series: S 1/10n
= …
À0 terms (without limit)
More than À0 terms are impossible.
The limit has no decimal representation.

51. n pairs of parentheses yield 2n fractions.
À0 pairs of parentheses yield 2À0 fractions.

52. Adolf Abraham Fraenkel
The Life and Opinions of Tristram Shandy, Gentleman
Laurence Sterne
( )
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.

53. Sequence of rational numbers {q | 0 < q < 1}
With an actual infinity of transpositions
an ordering by size would be possible.

54. 1 2 4 5 3 6

55. 1 2 5 1 5 2 3 4 4 6 3

56. 1 8 2 7 6 3 5 4

57. 0 (set theory)   (mathematics) 2.1 .2 1 4 3.2 1 4.3 2 1 6 5 4.3 2 1
0 (set theory)
  (mathematics)

58. 1 8 2 7 6 3 5 4
 (set theory)
  (mathematics)

59. Exhaustion of infinite sets?

60. Indexing of all positive rational numbers is a supertask:…

61. Indexing of all positive rational numbers is a supertask:…
in: (0, 1] out 1/1

62. Indexing of all positive rational numbers is a supertask:…
in: (0, 1] out 1/1
in: (1, 2] out 1/2

63. Indexing of all positive rational numbers is a supertask:…
in: (0, 1] out 1/1
in: (1, 2] out 1/2
in: (2, 3] out 2/1

64. Indexing of all positive rational numbers is a supertask:…
in: (0, 1] out 1/1
in: (1, 2] out 1/2
in: (2, 3] out 2/1
… and so on
Always infinitely many rationals in the "intermediate reservoir".

65. Indexing of all positive rational numbers is a supertask:…
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.

66. 0, …
0, …

67. 0, …
0, …
0, …
Between two irrationals there is always a rational number.
I×I  IÐI

68. Decimal representation of numbers
742.25
Binary representation of numbers
= = 5
= /2 = 6,5
= 1/2 + 1/4 = 0,75
= 1/2 + 1/4 + 1/ = 1
= 1/4 + 1/16 + 1/ = 1/3

69. The Binary Tree 0. 1

70. The Binary Tree 0. 1

71. The Binary Tree 0. 2 1 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16

72. The elementary cell:

73. The elementary cell: 2
- 1
- 1
= 0

74. The path-construction of the Binary Tree

75. New Game
We conquer the Binary Tree.
Start with one cent.
Pay one cent for each path.
Earn one cent for each newly covered node.
Get bankrupt if set theory is right.

76. The path-construction of the Binary Tree0.

77. The path-construction of the Binary Tree0. 1

78. The path-construction of the Binary Tree0. 1

79. The path-construction of the Binary Tree0. 1

80. The path-construction of the Binary Tree0. 1

81. The path-construction of the Binary Tree0. 1

82. The path-construction of the Binary Tree0. 1

83. The path-construction of the Binary Tree0. 1

84. The path-construction of the Binary Tree0. 1

85. The path-construction of the Binary Tree0. 1

86. The path-construction of the Binary Tree0. 1

87. The path-construction of the Binary Tree0. 1

88. The path-construction of the Binary Tree0. 1

89. The path-construction of the Binary Tree0. 1

90. The path-construction of the Binary Tree0. 1

91. The path-construction of the Binary Tree0.
Every single constructed path covers infinitely many nodes.
After every step of the construction the ratio
number of paths
number of nodes 1
= 0

92. How many natural numbers do exist?
< 1080 atoms in the universe.
Where exist numbers that cannot be written with 1080 digits?

93. ??? God Nature Mathematics Where is the infinite realized?
Georg Cantor

94. There are no different infinities.
There is no finished infinity.
The infinite is a direction, not an amount.
¥ = ¥ + 1 = 2¥ = 2¥
À0 < 2À0 = À1?, À2, ...
John Wallis
( ) À0

95. By Cantor was - it is well known - as biggest beast alephant grown.
Fritz Fischer

96. By Cantor was - it is well known -
as biggest beast alephant grown.
But aleph is a number that
turned out too large - thus ant

97. By Cantor was - it is well known -
as biggest beast alephant grown.
But aleph is a number that
turned out too large - thus ant fell dead.
(After Christian Morgenstern)

98. End