Zermelo Set Theory - The Aim of Zermelo's Paper

The Aim of Zermelo's Paper

The introduction states that the very existence of the discipline of set theory "seems to be threatened by certain contradictions or "antinomies", that can be derived from its principles – principles necessarily governing our thinking, it seems – and to which no entirely satisfactory solution has yet been found". Zermelo is of course referring to the "Russell antinomy".

He says he wants to show how the original theory of Georg Cantor and Richard Dedekind can be reduced to a few definitions and seven principles or axioms. He says he has not been able to prove that the axioms are consistent.

A non-constructivist argument for their consistency goes as follows. Define V_α for α one of the ordinals 0, 1, 2, ...,ω, ω+1, ω+2,..., ω·2 as follows:

• V₀ is the empty set.
• For α a successor of the form β+1, V_α is defined to be the collection of all subsets of V_β.
• For α a limit (e.g. ω, ω·2) then V_α is defined to be the union of V_β for β<α.

Then the axioms of Zermelo set theory are consistent because they are true in the model V_ω·2. While a non-constructivist might regard this as a valid argument, a constructivist would probably not: while there are no problems with the construction of the sets up to V_ω, the construction of V_ω+1 is less clear because one cannot constructively define every subset of V_ω. This argument can be turned into a valid proof in Zermelo–Frenkel set theory, but this does not really help because the consistency of Zermelo–Frenkel set theory is less clear than the consistency of Zermelo set theory.