Zermelo set theory
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Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences to its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering.
Contents 
The axioms of Zermelo set theory
 AXIOM I. Axiom of extensionality (Axiom der Bestimmtheit) "If every element of a set M is also an element of N and vice versa ... then M = N. Briefly, every set is determined by its elements".
 AXIOM II. Axiom of elementary sets (Axiom der Elementarmengen) "There exists a (fictitious) set, the null set, ∅, that contains no element at all. If a is any object of the domain, there exists a set {a} containing a and only a as element. If a and b are any two objects of the domain, there always exists a set {a, b} containing as elements a and b but no object x distinct from them both." See Axiom of pairs.
 AXIOM III. Axiom of separation (Axiom der Aussonderung) "Whenever the propositional function –(x) is definite for all elements of a set M, M possesses a subset M' containing as elements precisely those elements x of M for which –(x) is true".
 AXIOM IV. Axiom of the power set (Axiom der Potenzmenge) "To every set T there corresponds a set T' , the power set of T, that contains as elements precisely all subsets of T".
 AXIOM V. Axiom of the union (Axiom der Vereinigung) "To every set T there corresponds a set ∪T, the union of T, that contains as elements precisely all elements of the elements of T".
 AXIOM VI. Axiom of choice (Axiom der Auswahl): "If T is a set whose elements all are sets that are different from ∅ and mutually disjoint, its union ∪T includes at least one subset S_{1 having one and only one element in common with each element of T". }
 AXIOM VII. Axiom of infinity (Axiom des Unendlichen) "There exists in the domain at least one set Z that contains the null set as an element and is so constituted that to each of its elements a there corresponds a further element of the form {a}, in other words, that with each of its elements a it also contains the corresponding set {a} as element".
Connection with standard set theory
The accepted standard for set theory is ZermeloFraenkel set theory. The links show where the axioms of Zermelo's theory correspond. There is no exact match for "elementary sets". (It was later shown that the singleton set could be derived from what is now called "Axiom of pairs". If a exists, a and a exist, thus {a,a} exists. By extensionality {a,a} = {a}.) The empty set axiom is already assumed by axiom of infinity, and is now included as part of it.
The axioms do not include the Axiom of regularity and Axiom of replacement. These were added as the result of work by Thoralf Skolem in 1922, based on earlier work by Adolf Fraenkel in the same year.
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 Cantor 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.
The axiom of separation
Zermelo comments that Axiom III of his system is the one responsible for eliminating 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 }
 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. }
 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.
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) [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: 261281, 1908. English translation, "Investigations in the foundations of set theory" in Heijenoort 1967, pages 199215.