Math 2.0

Mathematical logic and foundations Definition

Set theory

Builds on SetElementPredicate logicZermelo–Fraenkel set theoryAxiom of extensionalityContinuum hypothesis

Definition

Set theory is the branch of mathematical logic whose objects of study are sets and the membership relation ∈\in between an element and a set. An axiomatic set theory is a theory in first-order logic with axioms about ∈\in. Its language has the binary relation symbol ∈\in, and possibly further symbols. The standard axiomatic set theory is ZFC, Zermelo–Fraenkel set theory with the axiom of choice.

Remarks

Example: ZFC is an axiomatic set theory. Its language has the single relation symbol ∈\in, and its axioms, such as the axiom of extensionality, speak only about ∈\in and equality. Non-example: Peano arithmetic is not a set theory in this sense, because its language has 00, successor, addition and multiplication, and its objects are natural numbers. Other axiomatic set theories include Zermelo set theory, von Neumann–Bernays–Gödel set theory, Morse–Kelley set theory and Quine's New Foundations. Questions such as the continuum hypothesis belong to set theory and are independent of ZFC, if ZFC is consistent.

Reading path

To understand this page from the ground up, read these 44 pages in order. The foundations come first.

  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. Ordered pair definition
  10. Biconditional definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Axiom of extensionality statement
  15. Subset definition
  16. Propositional logic concept
  17. Uniqueness quantifier definition
  18. Binary relation definition
  19. Power set definition
  20. Predicate logic concept
  21. Antisymmetric relation definition
  22. Reflexive relation definition
  23. Transitive relation definition
  24. Zermelo–Fraenkel set theory concept
  25. Function definition
  26. Preorder definition
  27. Codomain definition
  28. Injective function definition
  29. Partial order definition
  30. Surjective function definition
  31. Least element definition
  32. Total order definition
  33. Bijective function definition
  34. Well-order definition
  35. Countably infinite set definition
  36. Equinumerous sets definition
  37. Ordinal number definition
  38. Cardinality definition
  39. Cardinal number definition
  40. Order of cardinals definition
  41. Aleph-null object
  42. Cardinality of the continuum object
  43. Continuum hypothesis statement
  44. Set theory definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Set_theory
About this page
TagJRBRNZT680
URIm2://JRBRNZT680
Kinddefinition
WritingW0 stub
MSC03E30, 03E99
DefinesSet theory
UsesSet, Element, Predicate logic, Zermelo–Fraenkel set theory, Axiom of extensionality, Continuum hypothesis
Changed2 November 2026

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