Zermelo-Fraenkel axioms
Ernst Zermelo and Abraham Fraenkel proposed the following axioms as a for what is now called Zermelo-Fraenkel set theory, or ZF. If this set of axioms are accepted along with the Axiom of Choice, it is often denoted ZFC.
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Equality of sets: If and are sets, and iff , then .
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Pair set: If and are sets, then there is a set containing only and .
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Union (http://planetmath.org/Union) over a set: If is a set, then there exists a set that contains every element of each .
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: If is a set, then there exists a set with the property that iff any element is also in .
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Replacement axiom: Let be some formula. If, for all , there is exactly one such that is true, then for any set there exists a set with the property that iff there exists some such that is true.
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: Let be some formula. If there is some that makes true, then there is a set such that is true, but for no is true.
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Existence of an infinite set: There exists a non-empty set with the property that, for any , there is some such that but .
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: If is a set and is a condition on sets, there exists a set whose members are precisely the members of satisfying . (This axiom is also occasionally referred to as the ).
Title | Zermelo-Fraenkel axioms |
Canonical name | ZermeloFraenkelAxioms |
Date of creation | 2013-03-22 11:47:51 |
Last modified on | 2013-03-22 11:47:51 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 20 |
Author | mathcam (2727) |
Entry type | Axiom |
Classification | msc 03E99 |
Synonym | Zermelo-Fraenkel set theory |
Synonym | ZFC |
Synonym | ZF |
Related topic | AxiomOfChoice |
Related topic | RussellsParadox |
Related topic | VonNeumannOrdinal |
Related topic | Axiom |
Related topic | ContinuumHypothesis |
Related topic | GeneralizedContinuumHypothesis |
Related topic | SetTheory |
Related topic | VonNeumannBernausGodelSetTheory |
Related topic | Set |