In proof theory, ordinal analysis assigns 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.
Ordinal analysis concerns true, effective 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 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 that well-orders it with ordinal and such that proves transfinite induction of arithmetical statements for.
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 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 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.
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.
MLM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof-theoretic ordinal ψΩ1.
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. The strength of intuitionistic ZF equals that of ZF.
