Cantor's paradox

Cantor proved that the set of real numbers has a larger cardinality than the set of integers, paradoxically meaning that the infinite set of real numbers is bigger than the infinite set of integers. More generally, Cantor's paradox begins by stating that the set of all sets (call it set B) is its own power set, where a power set is the set of all subsets of a given set A. Power sets are always bigger than the sets associated with them. The paradox concludes that given set B, the cardinality of set B must be bigger than itself. To understand the paradox, one must consider Cantor's Theorem, which states that the cardinality of any set is lower than the cardinality of all of its subsets. The paradox is that if the set B is the set of all sets, then the cardinality of the subset of B would be bigger than set B; however, the cardinality of set B should be the same since set B and the subset of B are the same.