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 X and Y are sets, and x∈X iff x∈Y, then X=Y.
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Pair set: If X and Y are sets, then there is a set Z containing only X and Y.
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Union (http://planetmath.org/Union) over a set: If X is a set, then there exists a set that contains every element of each x∈X.
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: If X is a set, then there exists a set 𝒫(x) with the property that Y∈𝒫(x) iff any element y∈Y is also in X.
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Replacement axiom: Let F(x,y) be some formula
. If, for all x, there is exactly one y such that F(x,y) is true, then for any set A there exists a set B with the property that b∈B iff there exists some a∈A such that F(a,b) is true.
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: Let F(x) be some formula. If there is some x that makes F(x) true, then there is a set Y such that F(Y) is true, but for no y∈Y is F(y) true.
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Existence of an infinite set
: There exists a non-empty set X with the property that, for any x∈X, there is some y∈X such that x⊆y but x≠y.
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: If X is a set and P is a condition on sets, there exists a set Y whose members are precisely the members of X satisfying P. (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 |