Every set can be well-ordered, meaning equipped with an ordering under which every nonempty subset has a least element. Proved by Ernst Zermelo, it is logically equivalent to the axiom of choice and was central to the early twentieth-century debates over the foundations of set theory.
Facts
StatementEvery set can be well ordered, meaning a strict total order can be placed on its elements under which every non-empty subset has a least element. In first-order Zermelo-Fraenkel set theory the well-ordering theorem is logically equivalent to the axiom of choice, so either one can be derived from the other together with the remaining axioms. 1 Connections
In Branch
Named After
Derived from the theorem's own name (unambiguous possessive-token match to exactly one live mathematician entity, w-bfill-g5-0924 browse backfill)
Sources
1. Well-Ordering Theorem (Wikipedia)
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In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered.
2. Ernst Zermelo (Wikipedia)
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In 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved the well-ordering theorem (every set can be well ordered).
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