3 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 CantorThe word Cantor can mean more than one thing: Cantor is another name for a Hazzan, a member of the Jewish clergy Cantor is the title of a member of a student society who is the main singer at a cantus Famous people named "Cantor" include: Eddie Cantor, si and 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.

4 The axiom of separation

Zermelo comments that Axiom III of his system is the one responsible for elimiating the antinomies. It differs from the original definition by Cantor, as follows.

Sets cannot be independently defined by any arbitrary logically definable notion. They must be separated as subsets of sets already "given". This, he says, eliminates contradictory ideas like "the set of all sets" or "the set of all ordinal numbers".

He disposes of the Russell paradox by means of a Theorem. "Every set M possess at least one subset M_{0 that is not an element of M". Let M0 be the subset of M for which, by AXIOM III, is separated out by the notion "x ∉ x". Then M0 cannot be in M. For}

1. If M_{0 is in M0, then M0 contains an element x for which x is in x (i.e. M0 itself), which would contradict the definition of M0.}
2. If M_{0 is not in M0, M0 is an element of M that satisfies the definition "x ∉ x", and so is in M0.}

So M_{0 cannot be in M, hence not all objects of the universal domain B can be elements of one and the same set. "This disposes of the Russell antinomy as far as we are concerned".}

This left the problem of "the domain B" which seems to refer to something. This led to the idea of a proper class.

5 Cantor's Theorem

Zermelo's paper is notable for what may be the first mention of Cantor's theoremNote: in order to fully understand this article you may want to refer to the set theory portion of the table of mathematical symbols. In Zermelo-Frankel set theory, Cantor's theorem states that the power set ( set of all subsets) of any set A has a strict 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) [the power set of 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

M'  = {m: m ∉ φ(m)} = φ(m' )

so that if m'  in M' , it is in φ(m' ), and thus does not satisfy the definition of being in M' . But if it does not satisfy the definition, it is in M' : contradiction. Note the close resemblance of this proof to the way Zermelo disposes of Russell's Paradox.

• Zermelo, Ernst. "Untersuchungen über die Grundlagen der Mengenlehre I". Mathematische Annalen, 65: 261-281, 1908. English translation, "Investigations in the foundations of set theory" in Heijenoort 1967, pages 199-215.

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