Math 2.0

Mathematical logic and foundations Set theory: axioms and basics Concept

Naive set theory

Builds on Set theorySetRussell's paradoxZermelo–Fraenkel set theory

Summary

Naive set theory is set theory done in ordinary language, without a formal list of axioms. In its original form, every property defines a set, the set of all objects with that property. This principle leads to contradictions, the best known being Russell's paradox. The axiomatic theories, such as Zermelo–Fraenkel set theory, restrict how sets may be formed to avoid them. The name is also used for the informal set theory taught at the start of most courses, which is consistent with those axioms.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptSet, 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 levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptModus ponens, concept, no evidence levelModus ponensconceptUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionAxiom of extensionality, statement, no evidence levelAxiom of extensionali…statementPropositional logic, concept, no evidence levelPropositional logicconceptMathematical proof, concept, no evidence levelMathematical proofconceptTautology, definition, no evidence levelTautologydefinitionPredicate logic, concept, no evidence levelPredicate logicconceptLogical equivalence, definition, no evidence levelLogical equivalencedefinitionZermelo–Fraenkel set theory, concept, no evidence levelZermelo–Fraenkel set …conceptSet theory, concept, no evidence levelSet theoryconceptDouble negation law, statement, no evidence levelDouble negation lawstatementProof by contradiction, concept, no evidence levelProof by contradictionconceptRussell's paradox, statement, no evidence levelRussell's paradoxstatementNaive set theory, concept, no evidence levelNaive set theoryconcept
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. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Negation definition
  9. Predicate definition
  10. Biconditional definition
  11. Truth table concept
  12. Modus ponens concept
  13. Universal quantifier definition
  14. Axiom of extensionality statement
  15. Propositional logic concept
  16. Mathematical proof concept
  17. Tautology definition
  18. Predicate logic concept
  19. Logical equivalence definition
  20. Zermelo–Fraenkel set theory concept
  21. Set theory concept
  22. Double negation law statement
  23. Proof by contradiction concept
  24. Russell's paradox statement
  25. Naive set theory concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Naive_set_theory
About this page
TagXGAMQBSE1G
URIm2://XGAMQBSE1G
Kindconcept
WritingW0 stub
MSC03E30, 03A05
SpecializesSet theory
UsesSet theory, Set, Russell's paradox, Zermelo–Fraenkel set theory
Same as
Changed2 November 2026

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