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Burali-Forti paradox

Builds on Ordinal numberWell-orderSetZermelo–Fraenkel set theory

Counterexample

Use von Neumann ordinals: an ordinal is a transitive set that is well-ordered by the membership relation ∈\in. Suppose that the collection Ω\Omega of all ordinals is a set. Every element of an ordinal is an ordinal, so Ω\Omega is transitive. The ordinals are well-ordered by ∈\in, so Ω\Omega is a transitive set well-ordered by ∈\in. Hence Ω\Omega is an ordinal, and so Ω∈Ω\Omega \in \Omega. This contradicts the fact that no ordinal is an element of itself. So the collection of all ordinals is not a set.

Remarks

In naive set theory every property defines a set, so the argument there is a paradox. In Zermelo–Fraenkel set theory it is a theorem: the ordinals form a proper class, not a set. In the original form, the order type of all ordinals would be an ordinal greater than every ordinal, including itself. Cesare Burali-Forti published the argument in 1897.

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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 levelElementdefinitionOrdinal number, concept, no evidence levelOrdinal numberconceptProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptWell-order, concept, no evidence levelWell-orderconceptSet, 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 …conceptBurali-Forti paradox, counterexample, no evidence levelBurali-Forti paradoxcounterexample
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. Ordinal number concept
  3. Proposition concept
  4. Set concept
  5. Well-order 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. Burali-Forti paradox counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Burali-Forti_paradox
About this page
TagCCNCGN3P59
URIm2://CCNCGN3P59
Kindcounterexample
WritingW0 stub
MSC03E10, 03A05
UsesOrdinal number, Well-order, Set, Zermelo–Fraenkel set theory
Same as
Changed2 November 2026

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