Zermelo Set Theory
The first axiomatisation of set theory published by Ernst Zermelo in 1908 [Zermelo08] in response to the antinomies found in informal set theory by Russell and others. Intended to provide a consistent foundation for mathematics, its consistency remains unimpeached, though it has been found necessary to augment the theory with the axiom of replacement (see ZF) to provide an adequate foundation for modern mathematics. Zermelo's system includes the axiom of choice, but the letter "Z" is now normally used to refer to his system with the axiom of choice omitted.
The Z specification language
A language developed by Jean Raymond Abrial and others at the University of Oxford, broadly similar in strength and character to Zermelo set theory (though the etymology seems uncertain), but with a much richer syntax oriented to applications in the specification of software.
Zermelo-Fraenkel set theory, an axiomatisation of set theory consisting of Zermelo set theory (see above) strengthened with the axiom of replacement, due to Abraham Fraenkel, the effect of which is to ensure that any collection of sets which can be shown to be no greater in size than an existing set is itself a set.
Zermelo-Fraenkel set theory augmented by the axiom of choice.

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