Difference between revisions of "Zermelo theorem"
From Encyclopedia of Mathematics
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Every set can be well-ordered (see [[Well-ordered set|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|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|Axiom of choice]]). | Every set can be well-ordered (see [[Well-ordered set|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|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|Axiom of choice]]). | ||
Latest revision as of 20:26, 19 December 2020
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).
Comments
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: http://encyclopediaofmath.org/index.php?title=Zermelo_theorem&oldid=51021
Zermelo theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zermelo_theorem&oldid=51021
This article was adapted from an original article by V.I. Malykhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article