In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. 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.
The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0. See Gentzen's consistency proof.
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 is the smallest ordinal (necessarily recursive, see next section) that the theory cannot prove is well founded—the supremum of all ordinals for which there exists a notation in Kleene's sense such that proves that is an ordinal notation. Equivalently, it is the supremum of all ordinals such that there exists a recursive relation on (the set of natural numbers) that well-orders it with ordinal and such that proves transfinite induction of arithmetical statements for .
The existence of a recursive ordinal that the theory fails to prove is well-ordered follows from the bounding theorem, as the set of natural numbers that 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) recursive ordinal, that is, less than the Church–Kleene ordinal .
Theories with proof-theoretic ordinal ωEdit
- Q, Robinson arithmetic (although the definition of the proof-theoretic ordinal for such weak theories has to be tweaked).
- PA–, the first-order theory of the nonnegative part of a discretely ordered ring.
Theories with proof-theoretic ordinal ω2Edit
- RFA, rudimentary function arithmetic.
- IΔ0, arithmetic with induction on Δ0-predicates without any axiom asserting that exponentiation is total.
Theories with proof-theoretic ordinal ω3Edit
- EFA, elementary function arithmetic.
- IΔ0 + exp, arithmetic with induction on Δ0-predicates augmented by an axiom asserting that exponentiation is total.
0, a second order form of EFA sometimes used in reverse mathematics.
0, a second order form of EFA sometimes used in reverse mathematics.
Friedman's grand conjecture suggests that much "ordinary" mathematics can be proved in weak systems having this as their proof-theoretic ordinal.
Theories with proof-theoretic ordinal ωn (for n = 2, 3, ... ω)Edit
- IΔ0 or EFA augmented by an axiom ensuring that each element of the n-th level of the Grzegorczyk hierarchy is total.
Theories with proof-theoretic ordinal ωωEdit
- RCA0, recursive comprehension.
- WKL0, weak König's lemma.
- PRA, primitive recursive arithmetic.
- IΣ1, arithmetic with induction on Σ1-predicates.
Theories with proof-theoretic ordinal ε0Edit
Theories with proof-theoretic ordinal the Feferman–Schütte ordinal Γ0Edit
- ATR0, arithmetical transfinite recursion.
- Martin-Löf type theory with arbitrarily many finite level universes.
This ordinal is sometimes considered to be the upper limit for "predicative" theories.
Theories with proof-theoretic ordinal the Bachmann–Howard ordinalEdit
- ID1, the theory of inductive definitions.
- KP, Kripke–Platek set theory with the axiom of infinity.
- CZF, Aczel's constructive Zermelo–Fraenkel set theory.
- EON, a weak variant of the Feferman's explicit mathematics system T0.
The Kripke-Platek or CZF set theories are weak set theories without axioms for the full powerset given as set of all subsets. Instead, they tend to either have axioms of restricted separation and formation of new sets, or they grant existence of certain function spaces (exponentiation) instead of carving them out from bigger relations.
Theories with larger proof-theoretic ordinalsEdit
- , Π11 comprehension has a rather large proof-theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams", and which is bounded by ψ0(Ωω) in Buchholz's notation. It is also the ordinal of , the theory of finitely iterated inductive definitions. And also the ordinal of MLW, Martin-Löf type theory with indexed W-Types Setzer (2004).
- T0, Feferman's constructive system of explicit mathematics has a larger proof-theoretic ordinal, which is also the proof-theoretic ordinal of the KPi, Kripke–Platek set theory with iterated admissibles and .
- KPM, an extension of Kripke–Platek set theory based on a Mahlo cardinal, has a very large proof-theoretic ordinal ϑ, which was described by Rathjen (1990).
- MLM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof-theoretic ordinal ψΩ1(ΩM + ω).
Most theories capable of describing the power set of the natural numbers have proof-theoretic ordinals that are so large that no explicit combinatorial description has yet been given. This includes second-order arithmetic and set theories with powersets including ZF and ZFC (as of 2019[update]). The strength of intuitionistic ZF (IZF) equals that of ZF.
- Buchholz, W.; Feferman, S.; Pohlers, W.; Sieg, W. (1981), Iterated inductive definitions and sub-systems of analysis, Lecture Notes in Math., 897, Berlin: Springer-Verlag, doi:10.1007/BFb0091894, ISBN 978-3-540-11170-2
- Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics, 1407, Berlin: Springer-Verlag, doi:10.1007/978-3-540-46825-7, ISBN 3-540-51842-8, MR 1026933
- Pohlers, Wolfram (1998), "Set Theory and Second Order Number Theory", Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, 137, Amsterdam: Elsevier Science B. V., pp. 210–335, doi:10.1016/S0049-237X(98)80019-0, ISBN 0-444-89840-9, MR 1640328
- Rathjen, Michael (1990), "Ordinal notations based on a weakly Mahlo cardinal.", Arch. Math. Logic, 29 (4): 249–263, doi:10.1007/BF01651328, MR 1062729
- Rathjen, Michael (2006), "The art of ordinal analysis" (PDF), International Congress of Mathematicians, II, Zürich: Eur. Math. Soc., pp. 45–69, MR 2275588, archived from the original on 2009-12-22 CS1 maint: discouraged parameter (link) CS1 maint: bot: original URL status unknown (link)
- Rose, H.E. (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, Berlin-New York: Springer-Verlag, pp. xii+299, ISBN 3-540-07911-4, MR 0505313
- Setzer, Anton (2004), "Proof theory of Martin-Löf type theory. An Overview", Mathématiques et Sciences Humaines. Mathematics and Social Sciences (165): 59–99
- Takeuti, Gaisi (1987), Proof theory, Studies in Logic and the Foundations of Mathematics, 81 (Second ed.), Amsterdam: North-Holland Publishing Co., ISBN 0-444-87943-9, MR 0882549
- Krajicek, Jan (1995). Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press. pp. 18–20. ISBN 9780521452052. defines the rudimentary sets and rudimentary functions, and proves them equivalent to the Δ0-predicates on the naturals. An ordinal analysis of the system can be found in Rose, H. E. (1984). Subrecursion: functions and hierarchies. University of Michigan: Clarendon Press. ISBN 9780198531890.