¥ more than any other question the infinite always has moved so deeply the human soul. more than...
TRANSCRIPT
More than any other question the infinite always has moved so deeply the human soul. More than most other ideas the infinite has effected so inspiringly and fruitfully on the human mind. But more than any other notion the infinite is in need of elucidation.
David Hilbert (1862 - 1943)
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.
Aristoteles (384 - 322)
I Naturally infiniteII Towards infinityIII AlogosIV InfinitesimalV UnlimitedVI MicroscopicalVII CosmicalVIII EternalIX TheologicalX TranscendentalXI TransfiniteXII Infinite
The History of the Infinite
I
Naturally infinite
Four categories of being
finite
unlimited
potentially infinite 1, 2, 3, ...
actually infinite ?
Aristotle (384 - 322)
The infinite exists potentially.
There is no actual infinity.
There are only finite numbers. The finite would be eliminated and destroyed by the infinite if this existed.
Medieval Scholastics: Infinitum actu non datur.
John Wallis (1616 - 1703)
Used the symbol for the first time 1655 in his Arithmetica Infinitorum.
Latin: 100 millions
Greek: hippopede
Speyer cathedral
Blaise Pascal (1623 - 1662)
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.
God made the integers. The rest is man-made.
Leopold Kronecker (1823 - 1891)
da, deux, , due, duo, dvi, (dve),два, to, tva, twa, two, zwei (zwo)
Leonardo de Pisa (1170 - 1240)
= Fibonacci
Gentle and content and kindly disposed toward every friend of mathematics.But: “Hand the boy a coin!“
The elements (), about 1500 printed editions.All former text books lost without trace - later unknown.Euclidean form: definition, theorem, proof, final clause.
Euclid (325 - 275)
Lighthouse of Alexandria 130 m hight, 280 BCOne of 7 Wonders of the World
Books I-VI: plane geometryBooks VII-X: arithmeticBooks XI-XIII: spatial geometry
Museion of Alexandria (600.000 books)
There are more than any given number of primes.
Let P = P1P2P3...Pn be the product of all primes.
(P ± 1) cannot be divided by one of these primes.
Therefore it is a prime itself or contains another prime Pn+1.
23571113= 30031 = 59509
23571113 = 30029
If b > a, then there exists n with na > b.Euklid (325 - 275)
quod erat demonstrandum.
Archimed (287 - 212)
Greatest mathematician, physicist, technician of the ancient world
buoyancy force (heureka)rule of leverblock and tackle (give me a fixed point ...)calculating the center of masswater screwexhaustionparabolaspiral
Machinery of war (block and tackle, catapult, concave mirror).He defended Syracuse two years long nearly alone against the Romans. Defeat by betrayal.
Archimedes (287 - 212)
Many people believe, o my King Hieron, 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.
A myriad (= 10.000) grains of sand are due to the size of a poppy seed …
64*1057 grains of sand can be placed in the universe known at his times.
There are numbers up to 1063 „and we can move on!"
Archimed could move on, but without powers:
ai myriakismyriostas periodou myriakismyrioston arithmon myriai myriades
= 108*1016
a 1 with 80000 trillions of zeros
Axiom of Archimed (unlimitedness of numbers):
For every number a larger natural number can be found.
Myriad 10.000 = 104 Asankbyeya 10140
Myriad 10.000 = 104 Asankbyeya 10140 Googol 10100
Googolplex 1010100
Carl Friedrich Gauß (1777 - 1855)
"measurable infinity" 9999
9999
999999
S. Skewes (1933): The first change happens below eee79,122 10101034
Number of primes < x:
li(x) = 2xdu/lnu
Supplies too large values for small x.
Wotan‘s ring Draupnir
t/s Anzahl
t/s Anzahl
10 Roman public debts (Vespasian)
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
34 Bacteria in the terrestrial oceans
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
34 Bacteria in the terrestrial oceans
38 Greatest prime determined by pencil and paper 2127-1
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
34 Bacteria in the terrestrial oceans
38 Greatest prime determined by pencil and paper 2127-1
59 Sand grains of Archimed surpassed
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
34 Bacteria in the terrestrial oceans
38 Greatest prime determined by pencil and paper 2127-1
59 Sand grains of Archimed surpassed
80 Protons in the universe
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
34 Bacteria in the terrestrial oceans
38 Greatest prime determined by pencil and paper 2127-1
59 Sand grains of Archimed surpassed
80 Protons in the universe
43 min 1000! surpassed
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
34 Bacteria in the terrestrial oceans
38 Greatest prime determined by pencil and paper 2127-1
59 Sand grains of Archimed surpassed
80 Protons in the universe
43 min 1000! surpassed
? 999
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
34 Bacteria in the terrestrial oceans
38 Greatest prime determined by pencil and paper 2127-1
59 Sand grains of Archimed surpassed
80 Protons in the universe
43 min 1000! surpassed
11 a 263 d 999
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
34 Bacteria in the terrestrial oceans
38 Greatest prime determined by pencil and paper 2127-1
59 Sand grains of Archimed surpassed
80 Protons in the universe
43 min 1000! surpassed
11 a 263 d 999
2,5 109 a 1081016
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
34 Bacteria in the terrestrial oceans
38 Greatest prime determined by pencil and paper 2127-1
59 Sand grains of Archimed surpassed
80 Protons in the universe
43 min 1000! surpassed
11 a 263 d 999
2,5 109 a 1081016
31092 a Googolplex 1010100
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
34 Bacteria in the terrestrial oceans
38 Greatest prime determined by pencil and paper 2127-1
59 Sand grains of Archimed surpassed
80 Protons in the universe
43 min 1000! surpassed
11 a 263 d 999
2,5 109 a 1081016
31092 a Googolplex 1010100
Million-illion-illion 1000.0001000.0001000.000
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
34 Bacteria in the terrestrial oceans
38 Greatest prime determined by pencil and paper 2127-1
59 Sand grains of Archimed surpassed
80 Protons in the universe
43 min 1000! surpassed
11 a 263 d 999
2,5 109 a 1081016
31092 a Googolplex 1010100
Million-illion-illion 1000.0001000.0001000.000
“Measurable infinity“ 9999
t/s Anzahl
10 Roman public debts (Vespasian)
11 Stars of the galaxy
14 Human intestinal bacteria
20 Combinations of the Rubik-cube 41019 = 8!12!21137/2
22 Stars in the universe
34 Bacteria in the terrestrial oceans
38 Greatest prime determined by pencil and paper 2127-1
59 Sand grains of Archimed surpassed
80 Protons in the universe
43 min 1000! surpassed
11 a 263 d 999
2,5 109 a 1081016
31092 a Googolplex 1010100
Million-illion-illion 1000.0001000.0001000.000
“Measurable infinity“ 9999
999! > 9999
lglg 999
! = 369 693 108,2
lglg 9999
= 369 693 099,6
z = 10lgz 100 = 102
z = 1010lglgz 10 000 000 000 = 10101
