Infinity in Mathematics and Physics Emilio Elizalde

Infinity in Mathematics and Physics

• Emilio Elizalde

• ICE/CSIC & IEEC, Barcelona

• Trento, June 13, 2006

Infinities

• The Bible: stars in heaven, sand grains, 70x7

• Zeno’s paradox (Achilles tortoise) & other

• Euclide’s axioms

• Euler: infinite series; zeta 

• Riemann: higher dimensions; zeta 

• Cantor: cardinals; paradoxes

• QFT: Regul./Renorm. (Einstein, Dirac)

• “El Aleph” (Jorge Luis Borges)

Set Theory

• Georg Cantor

• Paradoxes: Bertrand Russell

• Axiomatics

• Bourbaki School

Barber paradox

• In a village there is a barber who shaves every person in the village who does not shave itself

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

Hilbert’s Grand Hotel: has infinite rooms, is full! … and still infinite new hosts arrive… WHAT CAN WE DO!?

The cardinals (Alephs)

• Natural numbers: N א0

Mathematics

• Kurt Gödel’s Incompleteness Theorem

• Crisis of axiomatics

Physics

• Isaac Newton

Recent ideas & trends

• Inflation

• (A. Guth, A. Linde, P. Steinhard, A. Starobinski)

• Strings, Branes, M Theories

• The vacuum energy (H.G.B. Casimir)

• Obs. Cosmology

Understanding the Universe

• Presocratics: substance, number, power, infinity, movement, being, atom, space, time, ...

• Pythagorean School: “all things are numbers”

• 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”

• Eugene Wigner: “the unreasonable effectiveness of mathematics in the natural sciences”

• "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.

• • How does our Universe evolve? And how did structures like stars and galaxies form? Contemporary cosmology for the general reader. T. Padmanabhan.

Thanks so much

• for your attention