The theorem that for any set S, the power set P(S) (the set of all subsets of S) has strictly greater cardinality than S itself; in particular there is no surjection from S onto P(S).

Principle

Diagonalization: construct a subset of S that differs from each subset in any proposed list by choosing elements on the diagonal, showing no listing can exhaust all subsets and hence no surjection exists.

Demonstration

Given any function f: S → P(S), define D = {x in S : x ∉ f(x)}; D is a subset of S but cannot be f(x) for any x because membership of x in D contradicts the defining condition, so f is not surjective.

Misapplication

Confusing Cantor's result with statements about particular cardinal equalities (e.g., mistakenly concluding from Cantor that the power set of an infinite set must be 'much larger' in a specific measurable sense) or applying the diagonal argument to proper classes without care.

Consequence

Establishes an infinite hierarchy of strictly increasing cardinalities (S, P(S), P(P(S)), ...), proves the uncountability of the real numbers (via reals ≈ P(N)), and limits possible bijections and surjections between sets and their power sets.

Reversal

The inverse claim—that a set can be in bijection with its power set—is impossible for sets; considering inversions highlights distinctions between sets and proper classes (where different phenomena may occur).

Boundary

Applies to sets in ZF; the theorem does not directly apply to proper classes and does not by itself decide the size of P(S) relative to other specific infinite cardinals (it only guarantees strict inequality).

Semantic Tension

Interacts with the Cantor–Bernstein–Schroeder theorem: CB–S gives conditions for equality of cardinalities from mutual injections, while Cantor's theorem gives a one-sided strict inequality that prevents any surjection from S onto P(S).

Synthesis

Cantor's Theorem uses a diagonal construction to show that the collection of all subsets of a set cannot be listed or matched by the set itself, producing a provable strict increase in cardinality whenever one passes to the power set.