XI

Transfinit

Platon (427 - 348)

There are many beautiful things. They are transitory.

The idea of beauty is eternal

Platonists: Mathematical items and laws exist. They can be found but not invented.

The numbers are a free creation of man.

... we create a new, an irrational number.

Richard Dedekind (1831 - 1916)

Aristoteles (384 - 322) denies the actually infinite in philosophy and mathematics. Assumes it only in the realm of Gods.

Robert Grosseteste (1168 - 1253)Prof. at Oxford, teacher of Roger Bacon: The actually infinite is a definite number.“The number of points in a segment one ell long is its true measure.”

John Baconthorpe (? - 1346)

The actually infinite exists in number, time, and amount.

Summa theologica I, qu. 7, art. 4There cannot be a finished infinite set.

Thomas of Aquin (1224 - 1274) Saint, Doctor angelicus

Gottfried Wilhelm Leibniz (1646 - 1716)

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

„I am so in favour of the actually 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 ist author. Therefore I believe that there is no single piece of matter that is not – I don‘t say divisible – but actually divided; consequently even the least particle has to be considered as a world filled with an infinity of different creatures.

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).

Bernard de Fontenelle (1657-1757)

Author, philosopher, member of the Académie française, secretary of the Académie des sciences

introduced actually infinite numbersEléments de la Géometrie de l'infini, Paris (1727)

Pater Emanuel Maignan (1601 - 1676)

Minorit, Prof at the university Toulouse.

There can be an actual infinity.

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.

Georg Cantor (1845 - 1918)

1879 professor of mathematics at Halle

The founder of set theory and, together with Möbius and Poincaré founder of topology.

The range of your telescope reaches from 5 m to infinity and beyond.

Dominus regnabit in aeternum et ultra. [Exodus 15, 18]

The completed infinite can appear in different modifications which can be distinguished with extreme sharpness by the so called finite human mind. [Cantor to Lipschitz, 19. 11. 1883]

Georg Cantor (1845 - 1918)Prison sentence for life – with

preventive detention afterwards.

Galileo Galilei (1564 - 1642)

^The infinite should obey another arithmetic than the finite.

Gottfried Wilhelm Leibniz (1646 - 1716)

The rules of the finite remain valid in the infinite and vice versa.

The infinitey small (calculus) and the infinitely large (sum of the harmonic series)

+ 1 =

+ n =

+ = 2 =

=

-

0/0

/

0

Arithmetic of the infinite Undefined expressions

Salviati: Number of squares = number of numbers.Every natural number is the square root of a square.

Bijektive mapping: natural numbers even numbers, n 2n

1 2 3 4 5 6 7 8 9 ... 2 4 6 8 10 12 14 16 18 ...

Galilei: n n2

1 2 3 4 5 6 7 8 9 ... 1 4 9 16 25 36 49 64 81 ...

Bernard Bolzano (1781 - 1848)

Czech Theologian, Philosopher und Mathematician

Creator of the notion: Menge (set)

Different infinities: God (infinite Force, Goodness Wisdom) Numbers, Body, Surface, Line, Space, Time, Digits of 2.

Die Paradoxien des Unendlichen (1851)

Bernard Bolzano (1781 - 1848)

Czech Theologian, Philosopher and 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.

The whole is always larger than ist 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.Corners 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.

A = { x I x2 - 3x + 2 = 0 }

B = { x I x und 0 < x < 3 }

C = { x I a, b, c, x und ax + bx = cx }

D = { 1, 2 }

The infinite set of finite numbers has the smallest infinite cardinal number.

> n for every n .

M countably infinite: Bijection with possible.

Cardinality of every countably infinite set: .

A set is infinite if a bijection with a subset exists.

There are actual infinities:

infinite numbers of different size.

Georg Cantor (1845 - 1918)

Richard Dedekind (1831 - 1916)

1 2 3 4 5 6 7 8 9 ...10 20 30 40 50 60 70 80 90 ...

is countably infinite, has cardinal number.

countable := can be represented by a sequence

I I = 0

Index Equation 1 --- 2 1x1 = 0 0 3 1x2 = 0, 2x1 = 0, 1x1 ± 1 = 0 0, 0, ±1 4 1x3 = 0, 2x2 = 0, 3x1 = 0, 1x2 ± 1 = 0, 2x1 ± 1 = 0, 1x1 ± 2 = 0 0, 0, 0, ±1,±i, ±1/2 ±2

I I = 0

Index = Ia0I + Ia1I + Ia2I +...+ IanI + n

Proof of countability of algebraic numbers

after Dedekind (1873)

p(x) = a0 + a1x1 + a2x2 + ... + anxn = 0

Cantors second diagonal argument

n r(n) ___________________ 1 0,000111199999... 2 0,123456789123... 3 0,555555555555... 4 0,789789789789... 5 0,010000000000... ... ...

n r(n) ___________________ 1 0,000111199999... 2 0,123456789123... 3 0,555555555555... 4 0,789789789789... 5 0,010000000000... ... ...

Cantors second diagonal argument

< 2= C

is uncountable

0,13681...

Cantor's proof of the existence of transcendental numbers There are infinitely many more transcendental than algebraic numbers.

II > 0

n r(n) ___________________ 1 0,000111199999... 2 0,123456789123... 3 0,555555555555... 4 0,789789789789... 5 0,010000000000... ... ...

Cantors second diagonal argument

Power set (M)M = {a,b}({a,b}) = {{ },{a},{b},{a,b}}

Cardinal number of the set M: IMICardinal 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 = 1I({a})I = 21 = 2

M = {a,b,c} has cardinal number I{a,b,c}I = 3I(M)I = 23 = 8I((M))I = 28 = 256I(((M)))I = 2256 1077

I()I = 20 > 0

I()I = IIThere is an actual infinity 0,because it can be surpassed by 20.

Bijection () is impossible:

Let‘s try it: ()

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,...}

Infinite sequence of infinities

Transfinite cardinal numbers: 0 < 20 < 220 < ...

0 + 1 = 0

0 + n = 0

0 + 0 = 0

0 n = 0

0 0 = 0

2 0 > 0

David Hilbert (1862 - 1943)

1892 Professor in Königsberg1895 - 1930 in Göttingen

Hilbert‘s HotelOne guestInfinitely many guestsRoom maid

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 an interval [0,1].

0,x1y1x2y2x3y3x4y4x5y5... .

(x I y) = (0,111 I 0,222) 0,121212

Earl Bertrand Russell (l872 - 1970)

The set of all sets which do not contain themselves: M = X IX X

The set of all sets cannot exist. It would contain ist power set.

Ordinary sets do not contain themselves. 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

Ordinary sets do not contain themselves. 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

Ordinary sets do not contain themselves. 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

Ordinary sets do not contain themselves. 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

Ordinary sets do not contain themselves. 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

Ordinary sets do not contain themselves. 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

Ordinary sets do not contain themselves. 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…

Not self describingNot self describing

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.

Moon consists of white cheese.

Both senteneces in this box are false.

t f t f

t t f f

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 charcters

Berry‘s Paradox

Is the next infinity 1 = 20 ?Or is there an aleph between 0 and 1 = C = 20 ?

Continuum hypothesis

Analogy: Starting from M = {a,b,c} the cardinal number |M| = 100 cnnot be reached

I(M)I = 23 = 8

I((M))I = 28 = 256

I(((M)))I = 2256 1077

In 1900 David Hilbert (1862 - 1943) mentioned the 23 most important problems of mathematics in a talk; no. 1 was the proof of the continuum hypothesis.