Math 2.0

Mathematical logic and foundations Proof theory Concept

Ordinal analysis

Builds on Proof theoryProof-theoretic ordinalPeano axioms

Summary

Ordinal analysis is the part of proof theory that assigns to a formal theory its proof-theoretic ordinal. It measures the strength of a theory by the transfinite inductions that the theory can prove. It started with Gentzen's consistency proof for Peano arithmetic, which gives the ordinal ε0\varepsilon_0. An ordinal analysis usually also yields a consistency proof and a description of the provably total functions of the theory.

Reading path

To understand this page from the ground up, read these 37 pages in order. The foundations come first.

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Countable set, concept, no evidence levelCountable setconceptElement, definition, no evidence levelElementdefinitionOrdinal number, concept, no evidence levelOrdinal numberconceptPeano axioms, concept, no evidence levelPeano axiomsconceptProposition, 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 levelBiconditionaldefinitionModus ponens, concept, no evidence levelModus ponensconceptUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionMathematical proof, concept, no evidence levelMathematical proofconceptPropositional logic, concept, no evidence levelPropositional logicconceptBinary relation, definition, no evidence levelBinary relationdefinitionPredicate logic, concept, no evidence levelPredicate logicconceptAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionProof theory, concept, no evidence levelProof theoryconceptPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionLeast element, definition, no evidence levelLeast elementdefinitionMinimal element, definition, no evidence levelMinimal elementdefinitionTotal order, definition, no evidence levelTotal orderdefinitionWell-founded relation, definition, no evidence levelWell-founded relationdefinitionWell-order, definition, no evidence levelWell-orderdefinitionOrdinal number, definition, no evidence levelOrdinal numberdefinitionTransfinite induction, statement, no evidence levelTransfinite inductionstatementEpsilon nought, object, no evidence levelEpsilon noughtobjectProof-theoretic ordinal, concept, no evidence levelProof-theoretic ordin…conceptOrdinal analysis, concept, no evidence levelOrdinal analysisconcept
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. Ordinal number concept
  4. Peano axioms concept
  5. Proposition 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. Modus ponens concept
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Mathematical proof concept
  19. Propositional logic concept
  20. Binary relation definition
  21. Predicate logic concept
  22. Antisymmetric relation definition
  23. Reflexive relation definition
  24. Transitive relation definition
  25. Proof theory concept
  26. Preorder definition
  27. Partial order definition
  28. Least element definition
  29. Minimal element definition
  30. Total order definition
  31. Well-founded relation definition
  32. Well-order definition
  33. Ordinal number definition
  34. Transfinite induction statement
  35. Epsilon nought object
  36. Proof-theoretic ordinal concept
  37. Ordinal analysis concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Ordinal_analysis
About this page
TagC5E5TG8PGP
URIm2://C5E5TG8PGP
Kindconcept
WritingW0 stub
MSC03F15, 03F05
UsesProof theory, Proof-theoretic ordinal, Peano axioms
Same as
Changed2 November 2026

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