## You are here

HomeCantor's paradox

## Primary tabs

# 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 $f:\alpha\rightarrow\mathcal{P}(\alpha)$ is a bijection proving their equicardinality. Then $X=\{\beta\in\alpha\mid\beta\not\in f(\beta)\}$ 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.

## Mathematics Subject Classification

03-00*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag

Aug 24

new question: Scheduling Algorithm by ncovella

new question: Scheduling Algorithm by ncovella