Cantor’s paradox

Cantor’s paradoxMathworldPlanetmath demonstrates that there can be no largest cardinality. In particular, there must be an unlimited number of infiniteMathworldPlanetmath cardinalities. For suppose that α were the largest cardinal. Then we would have |𝒫(α)|=|α|. (Here 𝒫(α) denotes the power setMathworldPlanetmath of α.) Suppose f:α𝒫(α) is a bijection proving their equicardinality. Then X={βαβf(β)} is a subset of α, and so there is some γα such that f(γ)=X. But γXγX, which is a paradox.

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

Besides allowing an unboundedPlanetmathPlanetmath number of cardinalities as ZF set theoryMathworldPlanetmath 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