2. XI
Transfinite

3. Plato ( )
There are many beautiful things.
They are transitory.
The idea of beauty is eternal.
Platonists: Mathematical items and laws exist independently.
They can be found but not be invented.
Richard Dedekind ( )
The numbers are a free creation of man.
... we create a new, an irrational number.

4. Aristoteles ( ) denies the actual infinite in philosophy and mathematics. Assumes it only in the realm of Gods.
Robert Grosseteste ( )
Prof. at Oxford, teacher of Roger Bacon
The actual infinite is a definite number.
"The number of points in a segment one ell long is its true measure."
John Baconthorpe (?-1346)
The actual infinite exists in number, time, and amount.

5. Thomas of Aquin ( )
Saint, Doctor angelicus
Summa theologica I, qu. 7, art. 4
There cannot be a finished infinite set.
Gottfried Wilhelm Leibniz ( )
Three degrees of infinity:
1) Greater than every nameable magnitude
(like the mathematical )
2) The largest of its kind: the whole space, eternity.
3) God

6. "I am so in favour of the actual Infinite
"I am so in favour of the actual Infinite. I believe that nature instead of abhorring it, as usually is assumed, uses it everywhere frequently in order to show better the perfection of its author."
Leibniz, in a letter to Dangicourt 1716, said he did not believe in the real existence of the "grandeurs veritablement infinitesimales". They are only "fictions utiles"; but he had been asked by his followers not to publish this opinion in order not to betray their idea (i.e., actually infinitely small magnitudes).

7. Pater Emanuel Maignan (1601-1676)
Minorit, Prof. at the university Toulouse.
There can be an actual infinity.
Bernard de Fontenelle ( )
Philosopher, member of the Académie Française, secretary of the Académie des sciences,
introduced actually infinite numbers in
Eléments de la Géometrie de l'infini, Paris (1727).

8. Georg Cantor ( )
1872 Prof. of mathematics at Halle
The founder of set theory and, together with Möbius and Poincaré founder of topology.
I distinguish an "Infinitum aeternum increatum sive Absolutum", referring to God and his properties, and an "Infinitum creatum sive Transfinitum", referring to infinity in the created nature like, e.g., the actually infinite number of created beings in the universe as well as on our Earth and, very probably, in each not vanishing part of space.

9. The completed infinite can appear in different modifications which can be distinguished with extreme sharpness by the so called finite human mind.
Dominus regnabit in aeternum et ultra. [Exodus 15,18]
The range of your telescope reaches from 5 m to infinity - and beyond.
Prison sentence for life – with preventive detention afterwards.

10. Galileo Galilei ( )
The infinite should obey another arithmetic than the finite.
Gottfried Wilhelm Leibniz ( )
The rules of the finite remain valid in the infinite and vice versa:
in the infinitely small (calculus) and the infinitely large (sum of the harmonic series).

11. Arithmetic of the infinite
Undefined expressions

12. Bijective mapping: n  2n1 2 3 4 5 6 7 8 9
... 10 12 14 16 18
Galilei: n  n2 1 2 3 4 5 6 7 8 9
... 16 25 36 49 64 81
Salviati: Number of squares = number of numbers.
Every natural number is the square root of a square.

13. Bernard Bolzano ( )
Czech Theologian, Philosopher, Mathematician
Creator of the notion: Menge (set)
Die Paradoxien des Unendlichen (1851)
Different infinities: God (infinite Force, Goodness, Wisdom) Numbers, Body, Surface, Line, Space, Time, Digits of Ö2.

14. Bernard Bolzano ( )
Czech Theologian, Philosopher, Mathematician
Creator of the notion: Menge (set)
Die Paradoxien des Unendlichen (1851)
A bijective mapping
y = 2x
does not prove the same number of points.

15. The whole is always larger than its proper part.
There are different degrees of infinity.
There are as many circles as circumferences.
There are infinitely more diameters of a circle.
Focal points to centers of ellipses = 2:1.
Vertices to sides of cubes = 8:6.
There are more natural numbers than squares, more squares than cubes.
An interval is finite with respect to ist length, infinite with respect to ist points.

16. A = {x I x2 - 3x + 2 = 0}
B = {x I x   and 0 < x < 3}
C = {x I a, b, c, x   and ax + bx = cx}
D = {1, 2}

17. There are actual infinities: infinite numbers of different size.
The infinite set of finite numbers ô has the smallest infinite cardinal number À0.
À0 > n for every n e ô.
M countably infinite: Bijection with ô is possible.
Cardinality of every countably infinite set: À0.
A set is infinite if a bijection with a proper subset exists.
Georg Cantor ( )
Richard Dedekind ( )

18. Ð is countably infinite, has cardinal number À0.
countable := can be represented by a sequence
 II = 0

19. Proof of countability of the set of algebraic numbers
Dedekind (1873)
p(x) = a0 + a1x1 + a2x anxn = 0
Index = Ia0I + Ia1I + Ia2I IanI + n
 II = 0

20. Cantor's (second) diagonal argument
n r(n)
___________________

21. Cantor's (second) diagonal argument
n r(n)
___________________

22. Cantor's (second) diagonal argument
n r(n)
___________________
 II > 0
Cantor's proof of the existence of
transcendental numbers:
There are infinitely many more transcendental than algebraic numbers.
Ñ is uncountable
À0 < 2À0 = C

23. Power set (M)
M = {a, b}
({a, b}) = {{ }, {a}, {b}, {a, b}}
Cardinal number of the set M: IMI
Cardinal number of the power set (M): 2IMI
({a,b,c}) = {{ }, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b, c}}
I({ })I = 20 = 1
I({a})I = 21 = 2
M = {a, b, c} has cardinal number I{a, b, c}I = 3
I(M)I = 23 = 8
I((M))I = 28 = 256
I(((M)))I = 2256  1077
I()I = 20 > 0
I()I = II
There is an actual infinity 0
because it can be surpassed by 20.

24. A bijection ô « P(ô )) is impossible:
Let‘s try it: ô « P(ô )(
1 ® {1}
2 ® {2, 4, 6, ...}
3 ® {1, 2}
4 ® {3}
5 ® {1, 3, 5, ...}
...
M = {3, 4 , ...} = Set of "non-generators": n not in the image set.
What number is mapped on M?
4711 ® {3, 4, ..., 4711, ...}

25. Infinite sequence of infinities
Transfinite cardinal numbers: 0 < 20 < 220 < ...

26. David Hilbert ( )
1892 Professor at Königsberg
at Göttingen
Hilbert's Hotel
One guest
Infinitely many guests
Room maid

27. Cantor: Je le vois, mais je ne crois pas:
The cardinal number of points of a square [0,1]2 is equal to the cardinal number of points of an interval [0,1].
0.x1y1x2y2x3y3x4y4x5y5... .
(x I y) = (0.111 I 0.222) 

28. Gottlob Frege ( )
Logician
Unrestricted comprehension allows to form all definable sets. This is an axiomatization of Cantor's original ideas but leads to a severe antinomy.
Earl Bertrand Russell (l )
Logician
The set of all sets which do not contain themselves: M = {X I X Ï X }
(1903)
The set of all sets already cannot exist.
It would contain its power set.

29. Ordinary sets do not contain themselves.
ô is not a natural number.
Extraordinary sets contain themselves.
The set of all objects except cars is not a car.
The set of abstract notions is an abstract notion.
The set of all ordinary sets is impossible.
As an ordinary set it would contain itself (together with all ordinary sets) but then it would be extraordinary and would not belong to the set of all ordinary sets – and would not contain itself – and would be ordinary – and would belong to the set of all ordinary sets – and …

30. Ordinary sets do not contain themselves. ô is not a natural number.
Extraordinary sets contain themselves.
The set of all objects except cars is not a car.
The set of abstract notions is an abstract notion.
The set of all ordinary sets is impossible.
As an ordinary set it would contain itself (together with all ordinary sets) but then it would be extraordinary and would not belong to the set of all ordinary sets – and would not contain itself – and would be ordinary – and would belong to the set of all ordinary sets – and…
Self describing
Not self describing
frequent
seldom

31. Ordinary sets do not contain themselves. ô is not a natural number.
Extraordinary sets contain themselves.
The set of all objects except cars is not a car.
The set of abstract notions is an abstract notion.
The set of all ordinary sets is impossible.
As an ordinary set it would contain itself (together with all ordinary sets) but then it would be extraordinary and would not belong to the set of all ordinary sets – and would not contain itself – and would be ordinary – and would belong to the set of all ordinary sets – and…
Self describing
Not self describing
frequent
seldom
abstract
happy

32. Ordinary sets do not contain themselves. ô is not a natural number.
Extraordinary sets contain themselves.
The set of all objects except cars is not a car.
The set of abstract notions is an abstract notion.
The set of all ordinary sets is impossible.
As an ordinary set it would contain itself (together with all ordinary sets) but then it would be extraordinary and would not belong to the set of all ordinary sets – and would not contain itself – and would be ordinary – and would belong to the set of all ordinary sets – and…
Self describing
Not self describing
frequent
seldom
abstract
happy
old
new

33. Ordinary sets do not contain themselves. ô is not a natural number.
Extraordinary sets contain themselves.
The set of all objects except cars is not a car.
The set of abstract notions is an abstract notion.
The set of all ordinary sets is impossible.
As an ordinary set it would contain itself (together with all ordinary sets) but then it would be extraordinary and would not belong to the set of all ordinary sets – and would not contain itself – and would be ordinary – and would belong to the set of all ordinary sets – and…
Self describing
Not self describing
frequent
seldom
abstract
happy
old
new
comprehensible
incomprehensible

34. Ordinary sets do not contain themselves. ô is not a natural number.
Extraordinary sets contain themselves.
The set of all objects except cars is not a car.
The set of abstract notions is an abstract notion.
The set of all ordinary sets is impossible.
As an ordinary set it would contain itself (together with all ordinary sets) but then it would be extraordinary and would not belong to the set of all ordinary sets – and would not contain itself – and would be ordinary – and would belong to the set of all ordinary sets – and…
Self describing
Not self describing
frequent
seldom
abstract
happy
old
new
comprehensible
incomprehensible
short
supershort

35. Ordinary sets do not contain themselves. ô is not a natural number.
Extraordinary sets contain themselves.
The set of all objects except cars is not a car.
The set of abstract notions is an abstract notion.
The set of all ordinary sets is impossible.
As an ordinary set it would contain itself (together with all ordinary sets) but then it would be extraordinary and would not belong to the set of all ordinary sets – and would not contain itself – and would be ordinary – and would belong to the set of all ordinary sets – and…
Self describing
Not self describing
frequent
seldom
abstract
happy
old
new
comprehensible
incomprehensible
short
supershort
Not self describing
Not self describing
Not self describing

38. Protagoras' Student
Socrates: I know that I don't know.
Epimenides: All Cretans lie.
No rule without exception.
This theorem is not provable.
The next sentence is wrong. The preceding sentence is true.
The moon consists of white cheese.
Both sentences in this box are false.
t f t f
t t f f

39. Berry's Paradox
The set of all numbers which cannot be defined with a finite number of characters contains
the smallest number that cannot be defined with a finite number of characters.
= 77 characters

40. Ernst Zermelo ( )
Adolf A. Fraenkel ( )
Zermelo and Fraenkel invented an axiom system for set theory including the axiom of choice. By restricted comprehension antinomies could be excluded: Only subsets of existing sets are possible.
ZFC: Zermelo-Fraenkel-Choice (from the Axiom of Choice, 1904).
Both claimed that mathematics, except most elementary arithmetic, cannot exist without the actual infinite of set theory.

41. Ernst Zermelo ( )
Those who are really serious about rejection of the actual infinite in mathematics should ... do without the whole modern analysis.

42. Adolf A. Fraenkel ( )
The attitude ... that there do not exist altogether non-equivalent infinite sets is consistent, though almost suicidal for mathematics.
If the attack on the infinite will succeed ... only remnants of mathematics will remain.

43. Continuum hypothesis
Is the next infinity 1 = 20 ?
Or is there an aleph between 0 and C = 20 ?
Analogy: Starting from M = {a, b, c} the cardinal number |M| = 100 cannot be reached.
I(M)I = 23 = 8
I((M))I = 28 = 256
I(((M)))I = 2256  1077
In 1900 David Hilbert ( ) listed the most important problems of mathematics; no. 1 concerned the proof of the continuum hypothesis.

44. Continuum hypothesis
Is the next infinity 1 = 20 ?
Or is there an aleph between 0 and C = 20 ?
Kurt Gödel ( )
1. There are undecidable statements.
2. The consistency of a mathematical theory cannot be proved by this theory.
Gödel proved in 1937: The continuum hypothesis is not in contradiction with ZFC.
Paul J. Cohen ( )
proved in 1963: The continuum hypothesis is undecidable. Also its negation is not in contradiction with ZFC.
Both Gödel and Cohen doubt the continuum hypothesis: The cardinality of the continuum is probably much larger.