Math 2.0

Mathematical logic and foundations Definition

Class

Builds on Zermelo–Fraenkel set theorySetPredicate logicRussell's paradoxOrdinal number

Definition

Work in Zermelo–Fraenkel set theory. Let φ(x,p1,…,pn)\varphi(x, p_1, \dots, p_n) be a formula of first-order logic in the language {∈}\set{\in}, and let p1,…,pnp_1, \dots, p_n be sets, the parameters. The class defined by φ\varphi with these parameters is the collection C={x:φ(x,p1,…,pn)},C = \set{x : \varphi(x, p_1, \dots, p_n)}, and x∈Cx \in C is an abbreviation for φ(x,p1,…,pn)\varphi(x, p_1, \dots, p_n). A class CC is a set if there is a set aa with ∀x (x∈a↔x∈C)\forall x\, (x \in a \leftrightarrow x \in C). Otherwise CC is a proper class.

In class theories such as von Neumann–Bernays–Gödel set theory, classes are instead objects of the theory. There, a set is a class that is an element of some class.

Remarks

Example: the class {x:x⊆ω}\set{x : x \subseteq \omega} is a set, namely the power set of ω\omega. Non-example: the class V={x:x=x}V = \set{x : x = x} of all sets is not a set, so it is a proper class. If VV were a set aa, then separation would give the set {x∈a:x∉x}\set{x \in a : x \notin x}, and Russell's paradox would follow. The class of all ordinals is also a proper class, by the Burali-Forti argument. Every set aa is a class, defined by the formula x∈px \in p with parameter p=ap = a.

Reading path

To understand this page from the ground up, read these 36 pages in order. The foundations come first.

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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 levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptModus ponens, concept, no evidence levelModus ponensconceptUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionAxiom of extensionality, statement, no evidence levelAxiom of extensionali…statementSubset, definition, no evidence levelSubsetdefinitionPropositional logic, concept, no evidence levelPropositional logicconceptMathematical proof, concept, no evidence levelMathematical proofconceptBinary relation, definition, no evidence levelBinary relationdefinitionTautology, definition, no evidence levelTautologydefinitionPredicate logic, concept, no evidence levelPredicate logicconceptAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionZermelo–Fraenkel set theory, concept, no evidence levelZermelo–Fraenkel set …conceptLogical equivalence, definition, no evidence levelLogical equivalencedefinitionPreorder, definition, no evidence levelPreorderdefinitionDouble negation law, statement, no evidence levelDouble negation lawstatementPartial order, definition, no evidence levelPartial orderdefinitionProof by contradiction, concept, no evidence levelProof by contradictionconceptLeast element, definition, no evidence levelLeast elementdefinitionTotal order, definition, no evidence levelTotal orderdefinitionRussell's paradox, statement, no evidence levelRussell's paradoxstatementWell-order, definition, no evidence levelWell-orderdefinitionOrdinal number, definition, no evidence levelOrdinal numberdefinitionClass, definition, no evidence levelClassdefinition
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 definition
  4. Conjunction definition
  5. Disjunction definition
  6. Implication definition
  7. Negation definition
  8. Predicate definition
  9. Ordered pair definition
  10. Biconditional definition
  11. Truth table concept
  12. Modus ponens concept
  13. Universal quantifier definition
  14. Cartesian product definition
  15. Axiom of extensionality statement
  16. Subset definition
  17. Propositional logic concept
  18. Mathematical proof concept
  19. Binary relation definition
  20. Tautology definition
  21. Predicate logic concept
  22. Antisymmetric relation definition
  23. Reflexive relation definition
  24. Transitive relation definition
  25. Zermelo–Fraenkel set theory concept
  26. Logical equivalence definition
  27. Preorder definition
  28. Double negation law statement
  29. Partial order definition
  30. Proof by contradiction concept
  31. Least element definition
  32. Total order definition
  33. Russell's paradox statement
  34. Well-order definition
  35. Ordinal number definition
  36. Class definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Class_(set_theory)
About this page
TagW473FH7Z0P
URIm2://W473FH7Z0P
Kinddefinition
WritingW0 stub
MSC03E30, 03E70
DefinesClass
UsesZermelo–Fraenkel set theory, Set, Predicate logic, Russell's paradox, Ordinal number
Changed2 November 2026

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