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Cantor's paradox (Definition)

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 $ \vert\mathcal{P}(\alpha)\vert=\vert\alpha\vert$. (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.



"Cantor's paradox" is owned by Henry.
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Cross-references: logic, set theory, ZF, unbounded, Russell's paradox, argument, paradox, subset, equicardinality, bijection, power set, cardinal, infinite, cardinality
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This is version 3 of Cantor's paradox, born on 2002-09-29, modified 2005-02-14.
Object id is 3488, canonical name is CantorsParadox.
Accessed 7910 times total.

Classification:
AMS MSC03-00 (Mathematical logic and foundations :: General reference works )

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