 Ordinal analysis

In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. The field was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof theoretic ordinal of Peano arithmetic is ε_{0}.
Definition
Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations. The proof theoretic ordinal of such a theory T is the smallest recursive ordinal that the theory cannot prove is well founded — the supremum of all ordinals α for which there exists a notation o in Kleene's sense such that T proves that o is an ordinal notation. Equivalently, it is the supremum of all ordinals α such that there exists a recursive relation R on ω (the set of natural numbers) which wellorders it with ordinal α and such that T proves transfinite induction of arithmetical statements for R.
The existence of any recursive ordinal which the theory fails to prove is well ordered follows from the bounding theorem, as the set of natural numbers which an effective theory proves to be ordinal notations is a set (see Hyperarithmetical theory). Thus the proof theoretic ordinal of a theory will always be a countable ordinal less than the ChurchKleene ordinal .
In practice, the proof theoretic ordinal of a theory is a good measure of the strength of a theory. If theories have the same proof theoretic ordinal they are often equiconsistent, and if one theory has a larger proof theoretic ordinal than another it can often prove the consistency of the second theory.
Examples
Theories with proof theoretic ordinal ω^{2}
 RFA, rudimentary function arithmetic.
 IΔ_{0}, arithmetic with induction on Δ_{0}predicates without any axiom asserting that exponentiation is total.
Theories with proof theoretic ordinal ω^{3}
Friedman's grand conjecture suggests that much "ordinary" mathematics can be proved in weak systems having this as their prooftheoretic ordinal.
 EFA, elementary function arithmetic.
 IΔ_{0} + exp, arithmetic with induction on Δ_{0}predicates augmented by an axiom asserting that exponentiation is total.
 RCA*
0, a second order form of EFA sometimes used in reverse mathematics.  WKL*
0, a second order form of EFA sometimes used in reverse mathematics.
Theories with proof theoretic ordinal ω^{n}
 IΔ_{0} or EFA augmented by an axiom ensuring that each element of the nth level of the Grzegorczyk hierarchy is total.
Theories with proof theoretic ordinal ω^{ω}
 RCA_{0}, recursive comprehension.
 WKL_{0}, weak König's lemma.
 PRA, primitive recursive arithmetic.
 IΣ_{1}, arithmetic with induction on Σ_{1}predicates.
Theories with proof theoretic ordinal ε_{0}
 PA, Peano arithmetic (shown by Gentzen using cut elimination).
 ACA_{0}, arithmetical comprehension.
Theories with proof theoretic ordinal the FefermanSchütte ordinal Γ_{0}
This ordinal is sometimes considered to be the upper limit for "predicative" theories.
 ATR_{0}, arithmetical transfinite recursion.
 MartinLöf type theory.
Theories with proof theoretic ordinal the BachmannHoward ordinal
 ID_{1}, the theory of inductive definitions.
 KP, KripkePlatek set theory with the axiom of infinity.
 CZF, Aczel's constructive ZermeloFraenkel set theory.
Theories with larger proof theoretic ordinals
 , Π_{1}^{1} comprehension has a rather large proof theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams", and which is bounded by ψ_{0}(Ω_{ω}) in Buchholz's notation. It is also the ordinal of ID _{< ω}, the theory of finitely iterated inductive definitions.
 KPM, an extension of KripkePlatek set theory based on a Mahlo cardinal, has a very large proof theoretic ordinal, which was described by Rathjen (1990).
Most theories capable of describing the power set of the natural numbers have proof theoretic ordinals that are (as of 2008^{[update]}) so large that no explicit combinatorial description has yet been given. This includes second order arithmetic and set theories with powersets. (KripkePlatek set theory mentioned above is a weak set theory without power sets.)
See also
References
 Buchholz, W.; Feferman, S.; Pohlers, W.; Sieg, W. (1981), Iterated inductive definitions and subsystems of analysis, Lecture Notes in Math., 897, Berlin: SpringerVerlag, doi:10.1007/BFb0091894, ISBN 9783540111702
 Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics, 1407, Berlin: SpringerVerlag, ISBN 3540518428, MR1026933
 Pohlers, Wolfram (1998), Set Theory and Second Order Number Theory, "Handbook of Proof Theory", Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics (Amsterdam: Elsevier Science B. V.) 137: pp. 210–335, ISBN 0444898409, MR1640328
 Rathjen, Michael (1990), "Ordinal notations based on a weakly Mahlo cardinal.", Arch. Math. Logic 29 (4): 249–263, doi:10.1007/BF01651328, MR1062729
 Rathjen, Michael (2006), "The art of ordinal analysis", International Congress of Mathematicians, II, Zürich,: Eur. Math. Soc., pp. 45–69, MR2275588, http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf
 Rose, H. (1984), Subrecursion. Functions and Hierarchies, Oxford logic guides, 9, Oxford, New York: Clarendon Press, Oxford University Press
 Schütte, Kurt (1977), Proof theory, Grundlehren der Mathematischen Wissenschaften, 225, BerlinNew York: SpringerVerlag, pp. xii+299, ISBN 3540079114, MR0505313
 Takeuti, Gaisi (1987), Proof theory, Studies in Logic and the Foundations of Mathematics, 81 (Second ed.), Amsterdam: NorthHolland Publishing Co., ISBN 0444879439, MR0882549
Categories: Proof theory
 Ordinal numbers
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