Zermelo theorem

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2020 Mathematics Subject Classification: Primary: 03E25 [MSN][ZBL]

Every set can be well-ordered (see Well-ordered set). This theorem was first proved by E. Zermelo in 1904, starting from the principle of choice, one of the equivalent forms of the axiom of choice (see Zermelo axiom). Later it became clear that Zermelo's theorem is equivalent to the axiom of choice (in the usual system of axioms of set theory), hence also to many other propositions of set-theoretical character (see Axiom of choice).


This result is also commonly known as the well-ordering theorem or Zermelo's well-ordering theorem.

For references see Zermelo axiom.

How to Cite This Entry:
Zermelo theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Malykhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article