（参考）
突然ですが、Zermelo set theory がヒットしたので貼る（＾＾
https://en.wikipedia.org/wiki/Zermelo_set_theory
Zermelo set theory
Zermelo set theory (sometimes denoted by Z-), as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo-Fraenkel set theory (ZF) and its extensions, such as von Neumann?Bernays?Godel set theory (NBG). 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
1 The axioms of Zermelo set theory
2 Connection with standard set theory
3 Mac Lane set theory
4 The aim of Zermelo's paper
5 The axiom of separation
6 Cantor's theorem

The axioms of Zermelo set theory
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 axiom of infinity is usually now modified to assert the existence of the first infinite von Neumann ordinal ω; 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_{ω} 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.

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