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.
| Table of contents |
|
2 Connection with standard set theory 3 The aim of Zermelo's paper 4 The axiom of separation 5 Cantor's Theorem |
The accepted gold standard for set theory is Zermelo-Fraenkel 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 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.
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 Mo that is not an element of M". Let Mo be the subset of M for which, by AXIOM III, is separated out by the notion "x ∉ x". Then Mo cannot be in M. For
This left the problem of "the domain B" which seems to refer to something. This led to the idea of a proper class.
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':
The Axioms of Zermelo Set Theory
Connection with standard set theory
The aim of Zermelo's paper
The axiom of separation
So Mo 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".Cantor's Theorem
But no element of m' could correspond to M', i.e. such that phi(m') = M'. Otherwise
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.
