Math 2.0

Mathematical logic and foundations Set theory: axioms and basics Concept

Axiom schema of replacement

Builds on Zermelo–Fraenkel set theoryFunctionSetOrdinal number

Summary

The axiom schema of replacement is a list of axioms of Zermelo–Fraenkel set theory, one for each formula. It says that the image of a set under a definable function is again a set. Fraenkel and Skolem proposed it in 1922 as an addition to Zermelo's axioms. It is needed to build large sets such as ℵω\aleph_\omega and to define functions on ordinals by transfinite recursion.

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To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionFunction, concept, no evidence levelFunctionconceptOrdinal number, concept, no evidence levelOrdinal numberconceptProposition, 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 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 schema of replacement, concept, no evidence levelAxiom schema of repla…concept
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.
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  1. Element definition
  2. Function concept
  3. Ordinal number concept
  4. Proposition concept
  5. Set concept
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Negation definition
  11. Predicate definition
  12. Biconditional definition
  13. Universal quantifier definition
  14. Axiom of extensionality statement
  15. Propositional logic concept
  16. Predicate logic concept
  17. Zermelo–Fraenkel set theory concept
  18. Axiom schema of replacement concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Axiom_schema_of_replacement
About this page
TagD96K8S8R61
URIm2://D96K8S8R61
Kindconcept
WritingW0 stub
MSC03E30
UsesZermelo–Fraenkel set theory, Function, Set, Ordinal number
Same as
Changed2 November 2026

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