Zermelo set theory


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 and original numbering.

The axioms of Zermelo set theory

The axioms of Zermelo set theory are stated for objects, some of which are called sets, and the remaining objects are urelements and do not contain any elements. Zermelo's language implicitly includes a membership relation ∈, an equality relation =, and a unary predicate saying whether an object is a set. Later versions of set theory often assume that all objects are sets so there are no urelements and there is no need for the unary predicate.

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". The empty set axiom is already assumed by axiom of infinity, and is now included as part of it.
Zermelo set theory does not include the axioms of replacement and regularity. The axiom of replacement was first published in 1922 by Abraham Fraenkel and Thoralf Skolem, who had independently discovered that Zermelo's axioms cannot prove the existence of the set where Z0 is the set of natural numbers and Zn+1 is the power set of Zn. They both realized that the axiom of replacement is needed to prove this. The following year, John von Neumann pointed out that this axiom is necessary to build his theory of ordinals. The axiom of regularity was stated by von Neumann in 1925.
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 schema. 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. Zermelo set theory is usually taken to be a first-order theory with the separation axiom replaced by an axiom scheme with an axiom for each first-order formula. It can also be considered as a theory in second-order logic, where now the separation axiom is just a single axiom. The second-order interpretation of Zermelo set theory is probably closer to Zermelo's own conception of it, and is stronger than the first-order interpretation.
In the usual cumulative hierarchy Vα of ZFC set theory, any one of the sets
Vα for α a limit ordinal larger than the first infinite ordinal ω 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 ℵω or larger infinite cardinals, as the model Vω·2 does not contain such cardinals.
The axiom of infinity is usually now modified to assert the existence of the first infinite
von Neumann ordinal ; 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 cannot prove the existence of as a set nor of any rank of the cumulative hierarchy of sets with infinite index.
Zermelo allowed for the existence of urelements that are not sets and contain no elements; these are now usually omitted from set theories.

Mac Lane set theory

Mac Lane set theory, introduced by, is Zermelo set theory with the axiom of separation restricted to first-order 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 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 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. 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 Vω, the construction of Vω+1 is less clear because one cannot constructively define every subset of Vω. 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 possesses at least one subset that is not an element of ". Let be the subset of for which, by AXIOM III, is separated out by the notion "". Then cannot be in. For
  1. If is in, then contains an element x for which x is in x, which would contradict the definition of.
  2. If is not in, and assuming is an element of M, then is an element of M that satisfies the definition "", and so is in which is a contradiction.
Therefore, the assumption that is in 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 . Every set is of lower cardinality than the set of its subsets".
Zermelo proves this by considering a function φ: M → P. 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' . Otherwise we can construct a contradiction:
so by contradiction m' does not exist. Note the close resemblance of this proof to the way Zermelo disposes of Russell's paradox.