It bears certain differences from its descendants, which are not always understood, and are frequently misquoted.
Zermelo's language implicitly includes a membership relation ∈, an equality relation = (if it is not included in the underlying logic), and a unary predicate saying whether an object is a set.
The links show where the axioms of Zermelo's theory correspond.
(It was later shown that the singleton set could be derived from what is now called the "Axiom of pairs".
Zermelo set theory does not include the axioms of replacement and regularity.
The following year, John von Neumann pointed out that the axiom of regularity is necessary to build his theory of ordinals.
The notion of "first order formula" was not known in 1908 when Zermelo published his axiom system, and he later rejected this interpretation as being too restrictive.
, the existence of an uncountable strong limit cardinal is not satisfied in such a model; thus the existence of ℶω (the smallest uncountable strong limit cardinal) cannot be proved in second-order Zermelo set theory.
(where L is the constructible universe) forms a model of first-order Zermelo set theory wherein the existence of an uncountable weak limit cardinal is not satisfied, showing that first-order Zermelo set theory cannot even prove the existence of the smallest singular cardinal,
Within such a model, the only infinite cardinals are the aleph numbers restricted to finite index ordinals.
The axiom of infinity is usually now modified to assert the existence of the first infinite von Neumann ordinal
[2] Zermelo's axioms (original or modified) cannot prove the existence of
In any formulation, Zermelo set theory cannot prove the existence of the von Neumann ordinal
, despite proving the existence of such an order type; thus the von Neumann definition of ordinals is not employed for Zermelo set theory.
Mac Lane set theory, introduced by Mac Lane (1986), is Zermelo set theory with the axiom of separation restricted to first-order formulas in which every quantifier is bounded.
It is strong enough to carry out almost all ordinary mathematics not directly connected with set theory or logic.
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".
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.
Define Vα for α one of the ordinals 0, 1, 2, ...,ω, ω+1, ω+2,..., ω·2 as follows: Then the axioms of Zermelo set theory are consistent because they are true in the model Vω·2.
This argument can be turned into a valid proof with the addition of a single new axiom of infinity to Zermelo set theory, simply that Vω·2 exists.
Zermelo comments that Axiom III of his system is the one responsible for eliminating the antinomies.
For example, they can be constructed by taking powersets, or they can be separated as subsets of sets already "given".
He disposes of the Russell paradox by means of this Theorem: "Every set
By Axiom III this defines the following set M' : But no element m' of M could correspond to M' , i.e. such that φ(m' ) = M' .
Note the close resemblance of this proof to the way Zermelo disposes of Russell's paradox.