1. Infinity in Mathematics and Physics Emilio Elizalde ICE/CSIC & IEEC, Barcelona Trento, June 13, 2006

2. Infinities The Bible: stars in heaven, sand grains, 70x7 The Bible: stars in heaven, sand grains, 70x7 Zeno’s paradox (Achilles tortoise) & other Zeno’s paradox (Achilles tortoise) & other Euclide’s axioms Euclide’s axioms Euler: infinite series; zeta  Euler: infinite series; zeta  Riemann: higher dimensions; zeta  Riemann: higher dimensions; zeta  Cantor: cardinals; paradoxes Cantor: cardinals; paradoxes QFT: Regul./Renorm. (Einstein, Dirac) QFT: Regul./Renorm. (Einstein, Dirac) “El Aleph” (Jorge Luis Borges) “El Aleph” (Jorge Luis Borges)

4. 1/2 + 1/4 + 1/8 + 1/16 +...= 1

5. 1/2 +1/4 + 1/8 + 1/16 + … = x 1+ 1/2 +1/4 + 1/8 + 1/16 + … = 2x 1 + x = 2x X = 1 1 – 1 +1 – 1 + 1 – 1 + … = y 1 – (1 – 1 + 1 – 1 + 1 – 1 + … ) = y 1 - y = y 1 = 2y y = 1/2

7. Set Theory Georg Cantor Paradoxes: Bertrand Russell Axiomatics Bourbaki School

8. Barber paradox In a village there is a barber who shaves every person in the village who does not shave itself And the question is: who shaves the barber ?? Since, if he shaves himself he’ll be a person from the place shaving itself, but he is the barber and, as such, he shouldn’t shave this person! But, if he does not shave himself, he’ll be a person in the village who doesn’t shave itself, but he is the barber and must shave such person! Thus: he can neither shave nor remain unshaved !!

9. Bertrand Russell’s Paradox Let’s define the set A = { C | C C } A, C entities The paradox: If A A, then A A But, if A A, then A A

10. Hilbert’s Grand Hotel: has infinite rooms, is full! … and still infinite new hosts arrive… WHAT CAN WE DO!? 12345678..... 1234 A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 8..... A1A1 1A2A2 2A3A3 3A4A4 4.....

11. The cardinals (Alephs) Natural numbers: N א 0 Integer numbers: Z א 0 Rational numbers: Q א 0 Real numbers: R א 1 Cantor Does it exist ? X: Q < X < R Gödel Paul Cohen

12. Mathematics ► Kurt Gödel’s Incompleteness Theorem ► Crisis of axiomatics ► Alan Turing’s machine ► Complexity ► Cryptography ► Quantum Computation ► Peter Shor’s theorem Roger Penrose, The Emperor’s New Mind Douglas R. Hofstadter, Gödel, Escher, Bach

13. Physics  Isaac Newton  Albert Einstein

14. Recent ideas & trends Inflation Inflation (A. Guth, A. Linde, P. Steinhard, A. Starobinski) (A. Guth, A. Linde, P. Steinhard, A. Starobinski) Strings, Branes, M Theories Strings, Branes, M Theories The vacuum energy (H.G.B. Casimir) The vacuum energy (H.G.B. Casimir) Obs. Cosmology Obs. Cosmology DNA & Genome DNA & Genome Codes & Cryptography Codes & Cryptography Computational Biology Computational Biology Quantum Computation Quantum Computation Nanotechnology Nanotechnology

23. Understanding the Universe Presocratics: substance, number, power, infinity, movement, being, atom, space, time,... Presocratics: substance, number, power, infinity, movement, being, atom, space, time,... Pythagorean School: “all things are numbers” Pythagorean School: “all things are numbers” Emmanuel Kant: “the problem is to make inteligible the idea itself of an inteligible Universe” Emmanuel Kant: “the problem is to make inteligible the idea itself of an inteligible Universe” Albert Einstein: “the eternal mystery of the Universe is its comprehensibility”; “the fact that the Universe is so comprehensible is a miracle” Albert Einstein: “the eternal mystery of the Universe is its comprehensibility”; “the fact that the Universe is so comprehensible is a miracle” Eugene Wigner: “the unreasonable effectiveness of mathematics in the natural sciences” Eugene Wigner: “the unreasonable effectiveness of mathematics in the natural sciences” Did you ever think about that?

24. EL ALEPH JORGE LUIS BORGES O God, I could be bounded in a nutshell and count myself a King of infinite space. Hamlet, II, 2. … En la parte inferior del escalón, hacia la derecha, vi una pequeña esfera tornasolada, de casi intolerable fulgor. Al principio la creí giratoria; luego comprendí que ese movimiento era una ilusión producida por los vertiginosos espectáculos que encerraba. El diámetro del Aleph sería de dos o tres centímetros, pero el espacio cósmico estaba ahí, sin disminución de tamaño. Cada cosa (la luna del espejo, digamos) era infinitas cosas, porque yo claramente la veía desde todos los puntos del universo...

25. "It is said that there is no such thing as a free lunch. But the universe is the ultimate free lunch". A. Guth. "It is said that there is no such thing as a free lunch. But the universe is the ultimate free lunch". A. Guth. The fundamentals of the Universe were created in "the first three minutes”. S. Weinberg. The fundamentals of the Universe were created in "the first three minutes”. S. Weinberg. How does our Universe evolve? And how did structures like stars and galaxies form? Contemporary cosmology for the general reader. T. Padmanabhan. How does our Universe evolve? And how did structures like stars and galaxies form? Contemporary cosmology for the general reader. T. Padmanabhan.

30. Thanks so much for your attention