2. More than any other question the infinite always has moved so deeply the human soul. More than most other ideas the infinite has affected so inspiringly and fruitfully the human mind. But more than any other notion the infinite is in need of elucidation.
David Hilbert ( )
But the consideration of the unlimited has a difficulty. For, there result a lot of impossibilities, may we assume that it does not exist or that it does exist.
Aristotle ( )

3. History of the Infinite
I Naturally infinite
II Towards infinity
III Alogos
IV Infinitesimal
V Unlimited
VI Microscopical
VII Cosmical
VIII Eternal
IX Theological
X Transcendental
XI Transfinite
XII Infinite

4. I
Naturally infinite

5. Four categories of being
finite
unlimited
potentially infinite , 2, 3, ...
actually infinite {1, 2, 3, … }

6. Aristotle ( )
The infinite exists only in potential form.
There is no actual infinity (except the divine).
There are only finite numbers. The finite would be eliminated and destroyed by the infinite if this existed.
Medieval Scholastics: Infinitum actu non datur.

7. John Wallis ( )
Used the symbol  for the first time 1655 in his Arithmetica Infinitorum.
Latin: 100 millions
Greek: Hippopede

8. Speyer cathedral

9. Octogon

10. We recognize that there is infinity but don't know anything about its nature.
It is not even, it is not odd.
By +1 it is not changed.
Blaise Pascal
( )

11. God made the integers. The rest is man-made.
Leopold Kronecker ( )

12. Son of an obviuously good-natured Pisa consular officer nicknamed Bonaccio.
Called himself Fils Bonacci (Son of Bonaccio).
Merchant, travelled to Egypt, Syria, Greek, Sicily, Provence, became acquainted with Arabic (= Indian) numerals.
Later at the court of Friedrich II showed exhibition mathematics:
Leonardo de Pisa
( )
Fibonacci
Solution of x3 + 2x2 + 10x = 20
x = 1°22'7''42'''33''''4'''''40'''''' =
Mathematica gives

13.    Banking began to flourish.
Negative numbers interpreted as debts.
  

14. Leonardo de Pisa
( )
Fibonacci
Fibonacci-sequence
First recurrent sequence (outside India)

15. Gentle and content and kindly disposed toward every friend of mathematics.
But: "Hand the boy a coin!"
Museion of Alexandria ( books)
Euclid ( )?
Lighthouse of Alexandria
280 BC, 130 m hight. One of seven Wonders of the World.

16. The Elements (stoiceia), about 1500 printed editions.
All former text books lost without trace - later unknown.
Books I-VI: plane geometry
Books VII-X: arithmetic
Books XI-XIII: spatial geometry
Euclidean form: definition, theorem, proof, final clause.
quod erat demonstrandum.
oper edei deixai.
There are more than any given number of primes.
Let P = P1P2P3Pn be the product of all primes.
(P + 1) divided by one of these primes leaves remainder 1.
Therefore P is a prime itself or it contains another prime Pn+1.
235711 = = 59509
235711 = 30029

17. But there are gaps between primes larger than any number n.
n! + 2, n! + 3, n! + 4, ... , n! + n
All terms are divisible.
n! is divisible by all natural numbers 2, 3, ..., n.
n! + 2 is divisible by 2, n! + 3 is divisible by 3, ...
There are more than any given number of primes.
Let P = P1P2P3Pn be the product of all primes.
(P + 1) divided by one of these primes leaves remainder 1.
Therefore P is a prime itself or it contains another prime Pn+1.
235711 = = 59509
235711 = 30029

18. Archimedes
( )
Related to king Hieron II and his son Gelon. Educational journey to Egypt, Alexandria, Museion
Correspondence with Eratosthenes.
Greatest mathematician, physicist, technician of the ancient world.

19. buoyancy force (heureka)
rule of lever
block and tackle ("give me a fixed point ...")
calculating the center of mass
water screw
exhaustion p
parabola
spiral
Machinery of war (block and tackle, catapult, concave mirror).
He defended Syracuse over two years nearly alone against the Romans. Defeat by betrayal.
Archimedes
( )

20. Archimedes (287-212) Many people believe, o my King
Gelon, the number of sand grains
be infinite. Others think that this
number is not unlimited but that
never such a large number could
be named. But I will try to show
that among the numbers that I
have determined already there
are numbers surpassing the
number of sand grains not only
in a heap of sand as large as
our earth but even if the whole
universe was filled with sand.
Archimedes
( )

21. The Sand Reckoner
Less than a myriad (10000) grains of sand are due to a poppy seed.
Less than 40 poppy seeds laid side by side make up more than a finger-breads (2 cm).
1 stadion (185 m) < 1 myriad finger-breadths (200 m).
The diameter of the earth < 100 myriad stadia.
The diameter of universe < 1 myriad earth diameters.
641057 grains of sand can be placed in the universe known at his times, 1063 inside the sphere of fixed stars.
"And we can move on."
Archimedes could move on, but without powers:
ai myriakismyriostas periodou myriakismyrioston arithmon myriai myriades
= 1081016

22. Unboundedness of numbers
"The Axiom of Archimedes"
first stated by Eudoxus of Cnidus
also given by Euclid:
If a < b, then there exists n with na > b.
For every number a larger natural number can be found.
Eudoxus ( )

23. Great numbers
Saul has slain his thousands, and David his tens of thousands.
He live 1000 years (Mandarin). He live years (Emperor).
Myriad = 10000
Asan(k)byeya
Googol
Googolplex

24. Great numbers
Saul has slain his thousands, and David his tens of thousands."
He live 1000 years (Mandarin). He live years (emperor).
Myriad = 10000
Asan(k)byeya
Googol
Googolplex
The number p(x) of primes less than x
The number of possible chess games is between
10120 (games adhering to the rules)
and (including games allowing for rule violations).
For go games is the upper limit.

25. The number p(x) of primes less than x

26. The number p(x) of primes less than x is about
li(x) = 2òxdt/lnt

27. The number p(x) of primes less than x is about
li(x) = 2òxdt/lnt
li(x) > p(x)

28. The number p(x) of primes less than x is about
li(x) = 2òxdt/lnt
li(x) > p(x)
for small x.

29. The number p(x) of primes less than x is about
li(x) = 2òxdt/lnt
li(x) > p(x)
for small x.
The first change is before

30. Stanley Skewes ( )
The number p(x) of primes less than x is about
li(x) = 2òxdt/lnt
li(x) > p(x)
for small x.
The first change is before
eee79,122 »
Skewes (1933)

31. The largest number with three digits?
Carl Friedrich Gauß ( )
5! = 12345

32. Wotan‘s ring Draupnir

44. ?

53. Appendix

54. A computable but not primitive recursive function (Wilhelm Ackermann):{
The function starts innocuously:
But then it gets furious:

55. Tetration is the next (fourth) shorthand after exponentiation (Reuben L. Goodstein):
The up-arrow notation (Donald Knuth):

56. Comparing some values:

57. Graham's number G
Frank Plumpton Ramsey devised a theory in combinatorics. A special problem therein with upper bound estimated by Ronald Graham is as follows: Consider an n-dimensional hypercube. Connect every pair of its 2n vertices by edges. Colour the edges in an arbitrary way red or blue. What is the smallest value of n for which every such colouring contains at least one single-coloured complete subgraph of 6 coplanar edges?
In the graph opposite a three-dimensional cube is shown with only two vertices connected, by 7 edges each, to all others. A planar subgraph with its 6 blue edges is marked. Not for every three-dimensional coloured cube this is possible. The smallest cube which has at least one single-coloured coplanar subgraph is at most G-dimensional where G is calculated from:

58. 64 arrows-levels, the first and smallest of which is}
64 floors