Well-order

Quick meaning: Well-order is a noun meaning (set theory, order theory) A total order of some set such that every nonempty subset contains a least element.

Meaning

(set theory, order theory) A total order of some set such that every nonempty subset contains a least element.

More Meanings

1. (set theory, order theory, transitive) To impose a well-order on (a set).

Example Sentences

The set of positive integers is well-ordered by the relation ≤.

Then #92;le#95;C is a well defined order on C, and (C,#92;le#95;C) belongs to #92;mathcal#123;X#125; (that is, #92;le#95;C well orders C) and is an upper bound for #92;mathcal#123;C#125;.

1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, 2006, Dover (Dover Phoenix), page 111, Starting from these special well-ordered subsets, it is then possible to well-order the entire set.

1975 [The Williams & Wilkins Company], Dennis Sentilles, A Bridge to Advanced Mathematics, Dover, 2011, page 182, To carry the analogy a bit further, the axiom of choice implies the ability to well order any set.

Word Forms

• well-orders
• well-ordering
• well-ordered

Synonyms for well-order

Depending on context, synonyms include well-ordering.

Related Words

Related dictionary entries include regular, order, chain, poset, antichain, well-quasi-ordering.

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