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 ε0. 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 ε0. 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 {displaystyle T} is the smallest recursive ordinal that the theory cannot prove is well founded—the supremum of all ordinals α {displaystyle alpha } for which there exists a notation o {displaystyle o} in Kleene's sense such that T {displaystyle T} proves that o {displaystyle o} is an ordinal notation. Equivalently, it is the supremum of all ordinals α {displaystyle alpha } such that there exists a recursive relation R {displaystyle R} on ω {displaystyle omega } (the set of natural numbers) that well-orders it with ordinal α {displaystyle alpha } and such that T {displaystyle T} proves transfinite induction of arithmetical statements for R {displaystyle R} . The existence of any recursive ordinal that the theory fails to prove is well ordered follows from the Σ 1 1 {displaystyle Sigma _{1}^{1}} bounding theorem, as the set of natural numbers that an effective theory proves to be ordinal notations is a Σ 1 0 {displaystyle Sigma _{1}^{0}} set (see Hyperarithmetical theory). Thus the proof-theoretic ordinal of a theory will always be a countable ordinal less than the Church–Kleene ordinal ω 1 C K {displaystyle omega _{1}^{mathrm {CK} }} . 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. Friedman's grand conjecture suggests that much 'ordinary' mathematics can be proved in weak systems having this as their proof-theoretic ordinal.