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 from 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 1

Connection with standard set theory 2

Mac Lane set theory 3

The aim of Zermelo's paper 4

The axiom of separation 5

Cantor's theorem 6

See also 7

References 8
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 \equiv N. Briefly, every set is determined by its elements."

AXIOM II. Axiom of elementary sets (Axiom der Elementarmengen) "There exists a 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 an 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 most widely used and accepted set theory is known as ZFC, which consists of Zermelo–Fraenkel set theory with the addition of the axiom of choice. 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 Abraham Fraenkel in the same year.
In the modern ZFC system, the "propositional function" referred to in the axiom of separation is interpreted as "any property definable by a first order formula with parameters", so the separation axiom is replaced by an axiom scheme. The notion of "first order formula" was not known in 1904 when Zermelo published his axiom system, and he later rejected this interpretation as being too restrictive. Zermelo set theory is usually taken to be a firstorder theory with the separation axiom replaced by an axiom scheme with an axiom for each firstorder formula. It can also be considered as a theory in secondorder logic, where now the separation axiom is just a single axiom. The secondorder interpretation of Zermelo set theory is probably closer to Zermelo's own conception of it, and is stronger than the firstorder interpretation.
In the usual cumulative hierarchy V_{α} of ZFC set theory (for ordinals α), any one of the sets V_{α} for α a limit ordinal larger than the first infinite ordinal ω (such as V_{ω·2}) forms a model of Zermelo set theory. So the consistency of Zermelo set theory is a theorem of ZFC set theory. Zermelo's axioms do not imply the existence of ℵ_{ω} or larger infinite cardinals, as the model V_{ω·2} does not contain such cardinals. (Cardinals have to be defined differently in Zermelo set theory, as the usual definition of cardinals and ordinals does not work very well: with the usual definition it is not even possible to prove the existence of the ordinal ω2.)
The axiom of infinity is usually now modified to assert the existence of the first infinite von Neumann ordinal \omega; the original Zermelo axioms cannot prove the existence of this set, nor can the modified Zermelo axioms prove Zermelo's axiom of infinity. Zermelo's axioms (original or modified) cannot prove the existence of V_{\omega} as a set nor of any rank of the cumulative hierarchy of sets with infinite index.
Mac Lane set theory
Mac Lane set theory, introduced by Mac Lane (1986), is Zermelo set theory with the axiom of separation restricted to firstorder formulas in which every quantifier is bounded, Mac Lane set theory is similar in strength to topos theory with a natural number object, or to the system in Principia mathematica. It is strong enough to carry out almost all ordinary mathematics not directly connected with set theory or logic.
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 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 nonconstructivist 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:

V_{0} 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 nonconstructivist 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.
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 constructed in some way from previously constructed sets. For example they can be constructed by taking powersets, or they can 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 this Theorem: "Every set M possesses at least one subset M_0 that is not an element of M ". Let M_0 be the subset of M for which, by AXIOM III, is separated out by the notion "x \notin x". Then M_0 cannot be in M. For

If M_0 is in M_0, then M_0 contains an element x for which x is in x (i.e. M_0 itself), which would contradict the definition of M_0.

If M_0 is not in M_0, and assuming M_0 is an element of M, then M_0 is an element of M that satisfies the definition "x \notin x", and so is in M_0 which is a contradiction.
Therefore the assumption that M_0 is in M is wrong, proving the theorem. 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 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 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.
See also
References

Mac Lane, Saunders (1986), Mathematics, form and function, New York: SpringerVerlag,

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