¿Algunos infinitos son más grandes que otros infinitos?

¿Algunos infinitos son más grandes que otros infinitos?

Una pregunta que al parecer carece de coherencia es esa de si existen infinitos más grandes que otros, algunos dirán que es absurdo ya que la idea del infinito no es más que una representación ¡no es un número! Aunque la verdad, intentando filosofar sobre este concepto de manera empírica no llegaremos a nada concluyente, sino estaremos afirmando o negando algo ambiguo.

Antes de dar una respuesta quisiera decirte que no es fácil, no te preocupes si te atacan las dudas, es un problema que superó hasta a uno de los máximos exponentes de las matemáticas en la era del descubrimiento, me refiero a Georg Cantor; tanto que lo envolvió en la locura, según se cuenta vivió sus últimos años aquejado por una depresión ciclo-maníaca por las críticas recibidas tras sus intentos fallidos en la demostración de su conjetura (conocida por la hipótesis del continuo). 

Georg Cantor [imagen tomada de bbvaopenmind.com]

Motivación histórica

En 1900 se realizó en París el II Congreso Internacional de Matemáticos, donde David Hilbert vociferó los 23 problemas más importantes de aquellos tiempos, y cómo no, la hipótesis del continuo encabezaba la lista.

El 10 de agosto de 1904, durante el III Congreso Internacional de Matemáticos que se estaba celebrando en la ciudad alemana de Heidelberg, el matemático ruso Julius König impartió junto con Hilbert y Cantor presentes, una conferencia en la demostró que la hipótesis del continuo era falsa.

Efectivamente, el interés que había suscitado la intervención de König estaba plenamente justificado. En la discusión posterior a la conferencia, el propio Cantor agradeció públicamente a Dios haberle permitido vivir para ver la refutación de su error, pero el susto le duró poco a Cantor: unas semanas después se descubría que la demostración contenía un error (lo descubrió Zermelo), por lo que la hipótesis del continuo continuaba sin demostración. Y seguirá indemostrable.

De hecho, la hipótesis del continuo es independiente o indecidible (mba’e?), significa que partiendo de los axiomas de la teoría de conjuntos no puede probarse ni refutarse ningún enunciado, de manera que no contradiga algún principio lógico (ahhh). La demostración de su consistencia (que no puede refutarse) fue dada por Kurt Gödel en 1940.

Lamentablemente Cantor falleció sin poder saber el resultado de su flamante conjetura. Cabe resaltar en este contexto que Gödel fue uno de los matemáticos más importantes de todos los tiempos; de forma independiente consiguió crear una teoría formal de la lógica y demostró sus teoremas más importantes para el trabajo del día a día del matemático: los teoremas de incompletitud.

Kurt Gödel

Pero más tarde, específicamente en 1963, Paul Cohen demostró que la hipótesis del continuo es independiente (que no puede probarse) de los axiomas de conjuntos de Zermelo-Fraenkel extendida con el axioma de elección, mediante el método de Forcing.

Gödel: «La hipótesis del continuo no puede ser probada»
Cohen: «Tampoco puede ser refutada»
Paul Cohen

Esto fue un gran descubrimiento, puesto que si no puede demostrarse ni puede refutarse ¡debe ser un axioma, no una propiedad de los conjuntos! Así como el postulado de las paralelas, que negando o afirmando cambiamos el enfoque de toda la geometría [las posibles geometrías cambiando el postulado de las paralelas son: la geometría plana, la hiperbólica y la elíptica (como último comentario sobre esto, la geometría )], la hipótesis del continuo puede usarse o negarse en la teoría de conjuntos; una afirmación metamatemática que hace tiritar a cualquier gran pensado de esta área: una amalgama de conjuntos infinitos y de posibles teorías bajo el axioma del continuo.

Para hacernos la idea de lo que en realidad es la hipótesis del continuo, veamos algunos principios intuitivos.

Infinitos comparables

Podemos hacernos la idea de que dos conjuntos infinitos distintos tengan la misma o distinta cantidad de elementos (aunque, claramente, no lo podamos contar). Por ejemplo, si tenemos un conjunto de números naturales N={1, 2, 3, 4, 5, 6, . . .} y otro de números pares P={2, 4, 6, 8, 10, 12, . . . }, ambos de cantidad infinita, pregunto: ¿cuál de los conjuntos posee más elementos?

Piense un momento la respuesta.

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Siga pensando . . .

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De inmediato pueden observar que N contiene estrictamente al conjunto P, pero contestando la interrogante ¡ambos conjuntos tienen la misma cantidad de elementos! Esto es posible porque son conjuntos infinitos. Podemos hacer corresponder para cada elemento de N un sólo elemento de P, así ambos tendrían la misma cantidad de elementos, para visualizar mejor:

N : 1 , 2 , 3 , 4 . . .

↓ , ↓ , ↓ , ↓

P : 2 , 4 , 6 , 8 . . .

Esto se extiende indefinidamente, por tanto la cantidad de elementos de N es igual a la cantidad de elementos de P. Así nacen algunas definiciones importantes.

Definición. La cardinalidad de un conjunto A, denotado por |A|, es la cantidad de elementos del conjunto A. Dos conjuntos A y B tienen la misma cardinalidad si existe una función biunívoca que haga corresponder cada elemento de A a un elemento de B; si es el caso entonces |A|=|B|.

Esta definición es muy útil, ya que si sabemos la cantidad de elementos de B también sabremos la de A. Si nuestro conjunto es A={a,b,c,d,e} podemos tomar el conjunto B={1,2,3,4,5} y la función ƒ:B→A definida como

f(1)=a , f(2)=b , f(3)=c , f(4)=d , f(5)=e ,

así la cantidad de elementos de A coincide con el último elemento de B, que se refiere a su ordinal.

Cardinalidad de los transfinitos

De hecho es imposible saber la cardinalidad de los conjuntos infinitos ¿quién podría contarlos? Si Dios puede contarlos todos entonces no nos podría decir un número, no como interpretamos los humanos, así para facilitarnos la vida usamos la primera letra del alefato (alfabeto hebreo): ℵ (Alef).

Por convención decimos que ℵ-0 es la cardinalidad del infinito más pequeño. El siguiente infinito es
ℵ-1.

Veamos con un ejemplo sencillo todos los posibles subconjuntos de un conjunto cualquiera. Si A={a,b,c}, la partición de A definida como todos los subconjuntos de A es

℘(A) ={ Ø , {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}

entiéndase «Ø» como conjunto vacío. La cardinalidad del conjunto anterior es exactamente 2^3=8. La cantidad de elementos de una partición de un conjunto A se deduce de la combinatoria, y es igual a

2^(cantidad de elementos de A)

Ahora ya estamos en condiciones en interpretar la hipótesis del continuo. Denotemos al cardinal de los números reales R por c (se puede obtener que los conjuntos de números naturales, enteros y racionales son numerables, es decir que todos estos conjuntos son biyectivos o biunívocos con los naturales), existe una prueba en la que R no es numerable y fue el mismo Cantor en probarla, y puede ser usada para saber que precisamente c es el siguiente infinito, es decir ℵ-0 < ℵ-1= c .

Hipótesis del continuo. No existe ningún conjunto A tal que


ℵ-0 <|A|<2 ^ℵ-0

Viendo desde el axioma de elección, la hipótesis del continuo afirma que se cumple que 2^ℵ-0 = ℵ-1.

Respondiendo a la pregunta del título de este artículo, efectivamente hay infinitos que tienen mayor cardinalidad que otros infinitos.

Sin embargo, la hipótesis del continuo sigue allí como un postulado, en la cual debemos aceptar o rechazar. Pero ¿los números no tienen una forma única de comportarse? Depende de cómo definimos conjunto.

Seguir leyendo

Daniel Dávalos

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