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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 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 ?
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.
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 ordinals
, 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 .
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.
Rathjen, Michael (2006), "The art of ordinal analysis"(PDF), International Congress of Mathematicians, II, Zürich: Eur. Math. Soc., pp. 45-69, MR2275588, archived from the original on 2009-12-22CS1 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, ISBN3-540-07911-4, MR0505313