##  [Cantor's Theorem](/cantors-theorem-0) 

 Definition

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

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

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

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

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

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

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

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

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.