Ernst Zermelo

Ernst Zermelo (1871–1953) was a German mathematician who formulated an influential axiomatization of set theory and proved the well-ordering theorem.

Zermelo–Fraenkel set theory: An axiomatic set theory built around the Zermelo–Fraenkel axioms, usually abbreviated ZF. Zermelo’s 1908 axioms were the starting point for this later, expanded foundation.
Axiom of separation: The axiom schema allowing subsets defined by a property to be formed from an existing set. Zermelo’s 1908 system restricted set formation to avoid unrestricted comprehension.
Georg Cantor: A German mathematician who founded set theory and developed the theory of infinite cardinalities. Cantor’s work on well-ordering and infinite sets framed problems Zermelo sought to resolve.
Intuitionism: A philosophy of mathematics that treats mathematical objects as constructions and rejects some classical existence principles. Its standards of proof conflict with the unrestricted classical use of choice in Zermelo’s results.
Axiom of choice: The principle that every family of nonempty sets has a choice function. Zermelo used this principle to prove that every set can be well-ordered.
Axiom of pairing: The axiom asserting that any two sets belong together in some set. It is one of the basic set-construction rules in Zermelo’s axiomatization.
Richard Dedekind: A German mathematician whose work helped establish modern algebra and rigorous treatments of number systems. Dedekind’s set-based foundations formed part of the mathematical context for Zermelo’s axiomatic program.
Constructive mathematics: Mathematics that requires existence claims to provide constructions or algorithms for the objects asserted to exist. A well-ordering guaranteed by choice need not come with an explicit way to construct the order.
Well-ordering theorem: The theorem that every set can be given a well-order, in which every nonempty subset has a least element. Zermelo’s 1904 proof made this theorem a central consequence of the axiom of choice.
Axiom of union: The axiom asserting that the union of a set of sets is itself a set. It supplies a basic operation for building sets within Zermelo’s system.

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