*El cardinal inmediatamente superior a <math>\aleph_0</math>: <math>\aleph_1</math>
Usando los [[axiyttytytytyttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttomasaxiomas de Zermelo-Fraenkel]] (ZF) puede comprobarse que los tres cardinales anteriores cumplen <math>\aleph_0 < \aleph_1 \le c</math>. La [[hipótesis del continuo]] afirma que de hecho <math>c = \aleph_1</math>. [[Kurt Gödel|Gödel]] probó en [[1938]] que esta hipótesis es consistente con los axiomas ZF, y por tanto puede ser tomado como un axioma nuevo para la teoría de conjuntos. Sin embargo, en [[1963]] [[Paul Cohen]] probó que la negación de la hipótesis del continuo también es consistente con los axiomas ZF, lo cual prueba que dicha hipótesis es totalmente independiente de los axiomas ZF. Es decir, pueden construirse tanto "teorías de conjuntos cantorianas" en las que la hipótesis del continuo es una afirmación cierta, como "teorías de conjuntos no cantorianas" en las que la hipótesis del continuo sea falsa. Esta situación es similar a la de las [[Postulados de Euclides|geometrías no euclídeas]].
== Ejemplo de cálculo del cardinal de un conjunto ==
