Math 2.0

Mathematical logic and foundations Set theory: axioms and basics Concept

Zermelo–Fraenkel set theory

Builds on SetPredicate logicAxiom of extensionality

Summary

Zermelo–Fraenkel set theory, ZF, is an axiomatic theory of sets in first-order logic with one binary relation ∈\in. Its axioms are extensionality, regularity, separation, pairing, union, replacement, infinity and power set. Separation and replacement are axiom schemas. ZF with the axiom of choice is ZFC, the standard foundation of most of mathematics.

Remarks

The axiom of extensionality says that a set is determined by its elements. Separation forms only subsets {x∈A∣P(x)}\{x \in A \mid P(x)\} of a given set AA. This avoids Russell's paradox. The continuum hypothesis is independent of ZFC, if ZFC is consistent.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionAxiom of extensionality, statement, no evidence levelAxiom of extensionali…statementPropositional logic, concept, no evidence levelPropositional logicconceptPredicate logic, concept, no evidence levelPredicate logicconceptZermelo–Fraenkel set theory, concept, no evidence levelZermelo–Fraenkel set …concept
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  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Negation definition
  8. Predicate definition
  9. Biconditional definition
  10. Universal quantifier definition
  11. Axiom of extensionality statement
  12. Propositional logic concept
  13. Predicate logic concept
  14. Zermelo–Fraenkel set theory concept

Where this leads

30 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory, lead section
About this page
Tag0HE5WDQ9FJ
URIm2://0HE5WDQ9FJ
Aliaseszf, zfc, zermelo-fraenkel
Kindconcept
WritingW1 stated
MSC03E30
UsesSet, Predicate logic, Axiom of extensionality
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