Editing Cardinality (section)

==== Cantor's paradox ====
{{Main|Cantor's paradox}}
[[Cantor's theorem]] state's that, for any set <math>A,</math> possibly infinite, its [[powerset]] <math>\mathcal{P}(A)</math> has a strictly greater cardinality. For example, this means there is no bijection from <math>\N</math> to <math>\mathcal{P}(\N) \sim \R.</math> [[Cantor's paradox]] is a paradox in [[naive set theory]], which shows that there cannot exist a "set of all sets" or "[[Universe (mathematics)|universe set]]". It starts by assuming there is some set of all sets, <math>U := \{x \; | \; x \,\text{ is a set} \},</math> then it must be that <math>U</math> is strictly smaller than <math>\mathcal{P}(U),</math> thus <math>|U| \leq |\mathcal{P}(U)| .</math> But since <math>U</math> contains all sets, we must have that <math>\mathcal{P}(U) \subseteq U,</math> and thus <math>|\mathcal{P}(U)| \leq |U|.</math> Therefore <math>|\mathcal{P}(U)| = |U|,</math> contradicting Cantor's theorem. This was one of the original paradoxes that added to the need for a formalized set theory to avoid these paradoxes. This paradox is usually resolved in formal set theories by disallowing [[unrestricted comprehension]] and the existence of a universe set.