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Mathematical logic and foundations Definition

Ordinal analysis

Builds on Proof-theoretic ordinalProof-theoretic ordinalPeano axiomsEpsilon noughtTransfinite induction

Definition

Let TT be a recursively axiomatized theory that contains elementary arithmetic. An ordinal analysis of TT is the determination of its proof-theoretic ordinal ∣T∣\abs{T}. It consists of three parts.

  1. A computable notation system: a computable well-order ≺\prec on a set of codes in ℕ\N, whose order type is an ordinal α\alpha.
  2. A lower bound: for each β<α\beta < \alpha, a proof in TT of transfinite induction along the initial segment of ≺\prec of order type β\beta.
  3. An upper bound: a proof that TT does not prove transfinite induction along ≺\prec itself. More precisely, each computable well-order that TT proves well-founded has order type below α\alpha.

Then ∣T∣=α\abs{T} = \alpha.

Remarks

Example: Gentzen's analysis of Peano arithmetic uses the Cantor normal forms below epsilon nought as notation system, and gives ∣PA∣=ε0\abs{\mathrm{PA}} = \varepsilon_0. The upper bound comes from a cut-elimination argument for PA, together with transfinite induction up to ε0\varepsilon_0. Non-example: a proof that TT is consistent, by itself, is not an ordinal analysis, because it gives no ordinal. The upper bound is usually proved in a weak base theory together with transfinite induction along ≺\prec. This also gives a proof of the consistency of TT in that base theory.

Reading path

To understand this page from the ground up, read these 30 pages in order. The foundations come first.

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Computable function, concept, no evidence levelComputable functionconceptCountable 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 levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, 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-orderdefinitionTransfinite induction, statement, no evidence levelTransfinite inductionstatementOrdinal number, definition, no evidence levelOrdinal numberdefinitionEpsilon nought, object, no evidence levelEpsilon noughtobjectProof-theoretic ordinal, definition, no evidence levelProof-theoretic ordin…definitionProof-theoretic ordinal, concept, no evidence levelProof-theoretic ordin…conceptOrdinal analysis, definition, no evidence levelOrdinal analysisdefinition
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. Computable function concept
  2. Countable set concept
  3. Element definition
  4. Ordinal number concept
  5. Peano axioms concept
  6. Proposition concept
  7. Set definition
  8. Implication definition
  9. Predicate definition
  10. Ordered pair definition
  11. Universal quantifier definition
  12. Cartesian product definition
  13. Subset definition
  14. Binary relation definition
  15. Antisymmetric relation definition
  16. Reflexive relation definition
  17. Transitive relation definition
  18. Preorder definition
  19. Partial order definition
  20. Least element definition
  21. Minimal element definition
  22. Total order definition
  23. Well-founded relation definition
  24. Well-order definition
  25. Transfinite induction statement
  26. Ordinal number definition
  27. Epsilon nought object
  28. Proof-theoretic ordinal definition
  29. Proof-theoretic ordinal concept
  30. Ordinal analysis definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Ordinal_analysis
About this page
Tag57PTMS2ADY
URIm2://57PTMS2ADY
Kinddefinition
WritingW0 stub
MSC03F15, 03F05
DefinesOrdinal analysis
UsesProof-theoretic ordinal, Proof-theoretic ordinal, Peano axioms, Epsilon nought, Transfinite induction
Changed2 November 2026

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