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 α were the largest cardinal. Then we would have |𝒫(α)|=|α|. (Here 𝒫(α) denotes the power set
of α.) Suppose f:α→𝒫(α) is a bijection proving their equicardinality. Then X={β∈α∣β∉f(β)} is a subset of α, and so there is some γ∈α such that f(γ)=X. But γ∈X↔γ∉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 |