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Skolem's paradox

Builds on Zermelo–Fraenkel set theoryCountable setUncountable setCantor's theoremModel

Example

Assume that ZFC, written in first-order logic, is consistent. Its signature has one binary relation symbol ∈\in, so it is countable. By the Löwenheim–Skolem theorem, ZFC has a countable model MM. By Cantor's theorem, ZFC proves that the power set of ω\omega is uncountable. So MM contains an element pp that MM believes is uncountable: no element of MM is a bijection between ωM\omega^M and pp. But MM is countable, so the set of elements of pp in MM is countable when seen from outside. There is no contradiction. A bijection between ωM\omega^M and the members of pp exists outside MM, but it is not an element of MM.

Remarks

The example shows that "uncountable" is not an absolute notion: it depends on which functions the model contains. The same reasoning shows that no countable first-order theory with an infinite model can pin down the size of its models. Thoralf Skolem pointed out this situation in the 1920s.

Reading path

To understand this page from the ground up, read these 33 pages in order. The foundations come first.

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Countable set, concept, no evidence levelCountable setconceptElement, definition, no evidence levelElementdefinitionModel, concept, no evidence levelModelconceptProposition, concept, no evidence levelPropositionconceptUncountable set, concept, no evidence levelUncountable setconceptSet, 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 levelBiconditionaldefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal 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 logicconceptUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionPower set, definition, no evidence levelPower setdefinitionPredicate logic, concept, no evidence levelPredicate logicconceptZermelo–Fraenkel set theory, concept, no evidence levelZermelo–Fraenkel set …conceptFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionCardinality, definition, no evidence levelCardinalitydefinitionCantor's theorem, statement, no evidence levelCantor's theoremstatementSkolem's paradox, example, no evidence levelSkolem's paradoxexample
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. Countable set concept
  2. Element definition
  3. Model concept
  4. Proposition concept
  5. Uncountable set concept
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Negation definition
  11. Predicate definition
  12. Ordered pair definition
  13. Biconditional definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Axiom of extensionality statement
  18. Subset definition
  19. Propositional logic concept
  20. Uniqueness quantifier definition
  21. Binary relation definition
  22. Power set definition
  23. Predicate logic concept
  24. Zermelo–Fraenkel set theory concept
  25. Function definition
  26. Codomain definition
  27. Injective function definition
  28. Surjective function definition
  29. Bijective function definition
  30. Equinumerous sets definition
  31. Cardinality definition
  32. Cantor's theorem statement
  33. Skolem's paradox example

Further reading

  1. Web https://en.wikipedia.org/wiki/Skolem's_paradox
About this page
Tag68QZWDXAQ4
URIm2://68QZWDXAQ4
Kindexample
WritingW0 stub
MSC03C62, 03E30, 03A05
IllustratesLöwenheim–Skolem theorem
UsesZermelo–Fraenkel set theory, Countable set, Uncountable set, Cantor's theorem, Model
Same as
Changed2 November 2026

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