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ZF augmented by the axiom of choice.
CH, GCH, and the existence of inaccessible, measurable, or supercompact cardinals are not theorems of ZFC, under the corresponding consistency assumptions.
wp-zfwp-axiom-choicenlab-zfcsep-set-theorysep-large-cardinalsEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
ZFC is a first-order theory in the language of membership, presented by finitely many axioms together with axiom schemas.
The target is a member or subtype of the broader source class.
wp-zfsep-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Adjoin AC to ZF; the resulting theory is ZFC.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-zfwp-axiom-choicesep-axiom-choiceEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Choice ensures every set is well-orderable, so every infinite set has cardinality for some ordinal .
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
wp-aleph-numbersep-axiom-choiceThis is an authored directed relation from the source endpoint to the target endpoint.
The axiom of choice is one of the axioms of ZFC.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
wp-zfwp-axiom-choiceThis is an authored directed relation from the source endpoint to the target endpoint.
Choice ensures that every power set appearing in the beth recursion has an initial-ordinal cardinal representative.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
wp-axiom-choicewp-beth-numbersep-axiom-choiceThis is an authored directed relation from the source endpoint to the target endpoint.
Choice well-orders , so its equipotence type has a unique initial-ordinal representative .
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
wp-axiom-choicewp-cardinality-continuumsep-axiom-choice