Infinity in Mathematics and Physics Emilio Elizalde



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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



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





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