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well-ordering principle implies axiom of choice
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(Theorem)
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Proof. Let  be a collection of nonempty sets. Then
 is a set. By the well-ordering principle,
 is well-ordered under some relation  . Since each  is a nonempty subset of
 , each  has a least member  with respect to the relation  .
Define
 by  . Then  is a choice function. Hence, the axiom of choice holds. 
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"well-ordering principle implies axiom of choice" is owned by Wkbj79.
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Cross-references: choice function, subset, relation, well-ordered, collection, axiom of choice, implies, well-ordering principle
There are 2 references to this entry.
This is version 4 of well-ordering principle implies axiom of choice, born on 2006-07-30, modified 2007-05-30.
Object id is 8200, canonical name is WellOrderingPrincipleImpliesAxiomOfChoice.
Accessed 1921 times total.
Classification:
| AMS MSC: | 03E25 (Mathematical logic and foundations :: Set theory :: Axiom of choice and related propositions) |
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Pending Errata and Addenda
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