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つづき
URLﾘﾝｸ(en.wikipedia.org)
Zermelo?Fraenkel set theory
Contents
1 History
2 Axioms
2.1 1. Axiom of extensionality
2.2 2. Axiom of regularity (also called the axiom of foundation)
2.3 3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension)
2.4 4. Axiom of pairing
2.5 5. Axiom of union
2.6 6. Axiom schema of replacement
2.7 7. Axiom of infinity
2.8 8. Axiom of power set
2.9 9. Well-ordering theorem
3 Motivation via the cumulative hierarchy
4 Metamathematics
4.1 Virtual classes
4.2 Von Neumann?Bernays?Godel set theory
4.3 Consistency
4.4 Independence
4.5 Proposed additions
5 Criticisms
Criticisms
For criticism of set theory in general, see Objections to set theory URLﾘﾝｸ(en.wikipedia.org)
On the other hand, among axiomatic set theories, ZFC is comparatively weak. Unlike New Foundations, ZFC does not admit the existence of a universal set. Hence the universe of sets under ZFC is not closed under the elementary operations of the algebra of sets. Unlike von Neumann?Bernays?Godel set theory (NBG) and Morse?Kelley set theory (MK), ZFC does not admit the existence of proper classes.
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