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Axiomatic Wiki

HAxioms

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The axioms of ZFC set theory: the assumptions every definition, lemma and theorem is ultimately proved from.

A01ExtensionalitySets with the same members are equal.
The converse, that equal sets have the same members, follows from substitutivity Eq4.
A02PairingAny two objects can be gathered into a set.
The set is the pair of D003.
A03UnionThe union of any family of sets is a set.
The set is the union of D004.
A04Power setThe collection of all subsets of a set is itself a set.
Here is subset, and is the power set of D005.
A05InfinityThere exists a set containing the empty set and closed under successors.
Here is D002 and is the successor of D006.
A06SeparationSubsets carved out by a property exist.
The set is a subset of ; its uniqueness follows from A01.
A07ReplacementThe image of a set under a function is a set.
The set is the image of D007.
A08RegularityEvery non-empty set has a member disjoint from it.
Here is D002.
A09ChoiceA choice function exists for any family of non-empty sets.
Here is D002; is a choice function selecting one member of each .