Math 2.0

Mathematical logic and foundations Proof theory Concept

Proof-theoretic ordinal

Builds on Ordinal numberPeano axiomsTransfinite inductionEpsilon nought

Summary

The proof-theoretic ordinal of a formal theory measures how much transfinite induction the theory can prove. It is the supremum of the order types of the computable well-orderings that the theory proves to be well-founded. It is a countable ordinal, and it gives a scale on which theories can be compared by strength. For Peano arithmetic it is epsilon nought.

Reading path

To understand this page from the ground up, read these 27 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 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-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…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.
  • 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. Implication definition
  8. Predicate definition
  9. Ordered pair definition
  10. Universal quantifier definition
  11. Cartesian product definition
  12. Subset definition
  13. Binary relation definition
  14. Antisymmetric relation definition
  15. Reflexive relation definition
  16. Transitive relation definition
  17. Preorder definition
  18. Partial order definition
  19. Least element definition
  20. Minimal element definition
  21. Total order definition
  22. Well-founded relation definition
  23. Well-order definition
  24. Ordinal number definition
  25. Transfinite induction statement
  26. Epsilon nought object
  27. Proof-theoretic ordinal concept

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ordinal_analysis
About this page
TagC72RBH0CDR
URIm2://C72RBH0CDR
Kindconcept
WritingW0 stub
MSC03F15, 03F30
SpecializesOrdinal number
UsesOrdinal number, Peano axioms, Transfinite induction, Epsilon nought
Changed2 November 2026

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