Ordinal analysis

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.

History

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.

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 ${\displaystyle T}$  is the smallest ordinal (necessarily recursive, see next section) that the theory cannot prove is well founded—the supremum of all ordinals ${\displaystyle \alpha }$  for which there exists a notation ${\displaystyle o}$  in Kleene's sense such that ${\displaystyle T}$  proves that ${\displaystyle o}$  is an ordinal notation. Equivalently, it is the supremum of all ordinals ${\displaystyle \alpha }$  such that there exists a recursive relation ${\displaystyle R}$  on ${\displaystyle \omega }$  (the set of natural numbers) that well-orders it with ordinal ${\displaystyle \alpha }$  and such that ${\displaystyle T}$  proves transfinite induction of arithmetical statements for ${\displaystyle R}$ .

Upper bound

The existence of a recursive ordinal that the theory fails to prove is well-ordered follows from the ${\displaystyle \Sigma _{1}^{1}}$  bounding theorem, as the set of natural numbers that an effective theory proves to be ordinal notations is a ${\displaystyle \Sigma _{1}^{0}}$  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 ${\displaystyle \omega _{1}^{\mathrm {CK} }}$ .

Examples

Theories with proof-theoretic ordinal ω

• 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 ω2

• RFA, rudimentary function arithmetic.[1]
• 0, arithmetic with induction on Δ0-predicates without any axiom asserting that exponentiation is total.

Theories with proof-theoretic ordinal ω3

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)

• 0 or EFA augmented by an axiom ensuring that each element of the n-th level ${\displaystyle {\mathcal {E}}^{n}}$  of the Grzegorczyk hierarchy is total.

Theories with proof-theoretic ordinal the Feferman–Schütte ordinal Γ0

This ordinal is sometimes considered to be the upper limit for "predicative" theories.

Theories with proof-theoretic ordinal the Bachmann–Howard ordinal

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

• ${\displaystyle \Pi _{1}^{1}{\mbox{-}}{\mathsf {CA}}_{0}}$ , Π11 comprehension has a rather large proof-theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams", and which is bounded by ψ0ω) in Buchholz's notation. It is also the ordinal of ${\displaystyle ID_{<\omega }}$ , 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 ${\displaystyle \Sigma _{2}^{1}{\mbox{-}}{\mathsf {AC}}+{\mathsf {BI}}}$ .
• 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 ψΩ1M + 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). The strength of intuitionistic ZF (IZF) equals that of ZF.