Cantor's theorem is a fundamental result in set theory proposed by the mathematician Georg Cantor. It states that for any set \( S \), the set of all subsets of \( S \), known as the power set of \( S \) (denoted as \( \mathcal{P}(S) \)), has a strictly greater cardinality (size) than the set \( S \) itself.
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There is no surjection from a set onto its power set . Consequently has strictly greater cardinality than .
For any candidate function , form . This is a subset of by set comprehension. If , then holds exactly when , a contradiction. Thus is missing from the image of . On the other hand, is an injection from into its power set, establishing the strict comparison of cardinalities. For , the omitted selection is the subset version of Cantor's diagonal argument.