Editing Cardinality (section)

==== Set of all cardinal numbers ====
Similar to Cantor's paradox, the paradox of the set of all cardinal numbers is a result due to unrestricted comprehension. It often uses the definition of cardinal numbers as ordinal numbers for representatives. It is related to the [[Burali-Forti paradox]]. It begins by assuming there is some set <math>S := \{ X \, | X \text{ is a cardinal number}\}.</math> Then, if there is some largest element <math>\aleph \in S ,</math> then the powerset <math>\mathcal{P}(\aleph)</math> is strictly greater, and thus not in <math>S.</math> Conversly, if there is no largest element, then the [[Union (set theory)#Arbitrary union|union]] <math>\bigcup S</math> contains the elements of all elements of <math>S,</math> and is therefore greater than or equal to each element. Since there is no largest element in <math>S,</math> for any element <math>x \in S,</math> there is another element <math>y \in S</math> such that <math>|x| < |y|</math> and <math>|y| \leq \Bigl| \bigcup S \Bigr|.</math> Thus, for any <math>x \in S,</math> <math>|x| < \Bigl| \bigcup S \Bigr|,</math> and so <math>\Bigl| \bigcup S \Bigr| \notin S.</math>