Math 2.0

Mathematical logic and foundations Set theory: axioms and basics Concept

Axiom of regularity

Builds on Zermelo–Fraenkel set theorySetWell-founded relation

Summary

The axiom of regularity, also called the axiom of foundation, is an axiom of Zermelo–Fraenkel set theory. It says that every nonempty set has an element that shares no element with it. In other words, the membership relation ∈\in is well-founded. It rules out sets such as x={x}x = \set{x}, and it implies that every set appears at some stage of the cumulative hierarchy VαV_\alpha.

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To understand this page from the ground up, read these 17 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptWell-founded relation, concept, no evidence levelWell-founded relationconceptSet, 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 …conceptAxiom of regularity, concept, no evidence levelAxiom of regularityconcept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set concept
  4. Well-founded relation concept
  5. Set definition
  6. Conjunction definition
  7. Disjunction definition
  8. Implication definition
  9. Negation definition
  10. Predicate definition
  11. Biconditional definition
  12. Universal quantifier definition
  13. Axiom of extensionality statement
  14. Propositional logic concept
  15. Predicate logic concept
  16. Zermelo–Fraenkel set theory concept
  17. Axiom of regularity concept

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Axiom_of_regularity
About this page
TagNFQ9FVCP31
URIm2://NFQ9FVCP31
Kindconcept
WritingW0 stub
MSC03E30
UsesZermelo–Fraenkel set theory, Set, Well-founded relation
Same as
Changed2 November 2026

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