# Cantor’s paradox

*Cantor’s paradox ^{}* demonstrates that there can be no largest cardinality. In particular, there must be an unlimited number of infinite

^{}cardinalities. For suppose that $\alpha $ were the largest cardinal. Then we would have $|\mathcal{P}(\alpha )|=|\alpha |$. (Here $\mathcal{P}(\alpha )$ denotes the power set

^{}of $\alpha $.) Suppose $f:\alpha \to \mathcal{P}(\alpha )$ is a bijection proving their equicardinality. Then $X=\{\beta \in \alpha \mid \beta \notin f(\beta )\}$ is a subset of $\alpha $, and so there is some $\gamma \in \alpha $ such that $f(\gamma )=X$. But $\gamma \in X\leftrightarrow \gamma \notin X$, which is a paradox.

The key part of the argument strongly resembles Russell’s paradox, which is in some sense a generalization^{} of this paradox.

Besides allowing an unbounded^{} number of cardinalities as ZF set theory^{} does, this paradox could be avoided by a few other tricks, for instance by not allowing the construction of a power set or by adopting paraconsistent logic.

Title | Cantor’s paradox |
---|---|

Canonical name | CantorsParadox |

Date of creation | 2013-03-22 13:04:39 |

Last modified on | 2013-03-22 13:04:39 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 6 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 03-00 |