Zermelo Set Theory - Cantor's Theorem

Cantor's Theorem

Zermelo's paper is notable for what may be the first mention of Cantor's theorem explicitly and by name. This appeals strictly to set theoretical notions, and is thus not exactly the same as Cantor's diagonal argument.

Cantor's theorem: "If M is an arbitrary set, then always M < P(M) . Every set is of lower cardinality than the set of its subsets".

Zermelo proves this by considering a function φ: M → P(M). By Axiom III this defines the following set M' :

M' = {m: m ∉ φ(m)}.

But no element m' of M could correspond to M' , i.e. such that φ(m' ) = M' . Otherwise we can construct a contradiction:

1) If m' is in M' then by definition m' ∉ φ(m' ) = M' , which is the first part of the contradiction

2) If m' is in not in M' but in M then by definition m' ∉ M' = φ(m' ) which by definition implies that m' is in M' , which is the second part of the contradiction.

so by contradiction m' does not exist. Note the close resemblance of this proof to the way Zermelo disposes of Russell's paradox.