InfinityInfinityin in

Mathematics Mathematics and Physicsand Physics

Emilio ElizaldeEmilio ElizaldeICE/CSIC & IEEC, BarcelonaICE/CSIC & IEEC, Barcelona

Trento, June 13, 2006Trento, June 13, 2006

InfinitiesInfinities

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

• ZenoZeno’s ’s paradoxparadox (Achilles (Achilles ttortoisortoisee)) & other & other

• Euclide’s Euclide’s axiomsaxioms

• Euler:Euler: infinite series infinite series; ; zeta zeta • Riemann:Riemann: higher dimensions; higher dimensions; zeta zeta • Cantor:Cantor: cardinals cardinals;; paradoxes paradoxes

• QFT:QFT: Reg Regul.ul./Ren/Renorm.orm. (Einstein, Dirac) (Einstein, Dirac)

• ““El Aleph” (Jorge Luis El Aleph” (Jorge Luis BorgesBorges))

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

1/2 +1/4 + 1/8 + 1/16 + … = x

1+ 1/2 +1/4 + 1/8 + 1/16 + … = 2x1 + x = 2x X = 1

1 – 1 +1 – 1 + 1 – 1 + … = y

1 – (1 – 1 + 1 – 1 + 1 – 1 + … ) = y

1 - y = y 1 = 2yy = 1/2

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

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 !!

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!?

1 2 3 4 5 6 7 8 . . . . .

1 2 3 4 . . . . .

A1 A2 A3 A4 A5 A6 A7 A8 . . . . .

A1 1 A2 2 A3 3 A4 4 . . . . .

The cardinals (Alephs)

Natural numbers: N 0א

Integer numbers: Z 0א

Rational numbers: Q 0א

Real numbers: R 1א Cantor Does it exist? X: Q < X < R Gödel

Paul Cohen

MatMathhememaaticticss

► Kurt GödelKurt Gödel’s ’s Incompleteness Incompleteness TheoremTheorem

► CrisiCrisiss of of axiomaxiomaaticticss

► Alan TuringAlan Turing’s machine’s machine► ComplexitComplexityy► CrCryyptograptographyphy► Quantum ComputationQuantum Computation► Peter ShorPeter Shor’s theorem’s theorem

Roger Penrose, The Emperor’s New Mind

Douglas R. Hofstadter, Gödel, Escher, Bach

PhysicsPhysicsIsaac NewtonIsaac Newton Albert EinsteinAlbert Einstein

r

MmGF 2 2mcE

kmrtot

RReecentcent idideeaas s & trends& trends

• InflaInflattiionon

(A. Guth, A. Linde, (A. Guth, A. Linde, P. SteinhardP. Steinhard, A. , A. StarobinskiStarobinski))

• Strings, Branes, MStrings, Branes, M TThheorieseories

• The vacuum The vacuum energyenergy (H.G.B. (H.G.B. Casimir)Casimir)

• Obs. CosmologyObs. Cosmology

• DNA DNA && Genom Genomee

• CodCodees s && CryptographyCryptography

• ComputaComputattionalional BiologyBiology

• Quantum Quantum ComputationComputation

• NanotechnologyNanotechnology

Understanding the Understanding the UniverseUniverse

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

• Pythagorean School:Pythagorean School: “all things are numbers”“all things are numbers”

• Emmanuel Kant:Emmanuel Kant: “the problem is to make “the problem is to make inteligible the idea itself of an inteligible inteligible the idea itself of an inteligible Universe”Universe”

• Albert Einstein:Albert Einstein: “the eternal mystery of the “the eternal mystery of the Universe is its comprehensibility”; “the fact that Universe is its comprehensibility”; “the fact that the Universe is so comprehensible is a miracle”the Universe is so comprehensible is a miracle”

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

Did you ever think about that?

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 ...

• • "It is said that there is no such thing as a free lunch. "It is said that there is no such thing as a free lunch. But the universe is the ultimate free lunch". But the universe is the ultimate free lunch". A. Guth.A. Guth.

• • The fundamentals of the The fundamentals of the UUniverse were created inniverse were created in "the "the first three minutes”. first three minutes”. S. Weinberg.S. Weinberg.

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

Thanks so muchThanks so much

for your attentionfor your attention