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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.

In addition to obtaining the proof-theoretic ordinal of a theory, in practice ordinal analysis usually also yields various other pieces of information about the theory being analyzed, for example characterizations of the classes of provably recursive, hyperarithmetical, or ${\displaystyle \Delta _{2}^{1}}$ functions of the theory.^[1]

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 supremum of the order types of all ordinal notations (necessarily recursive, see next section) that the theory can prove are 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 ${\displaystyle R}$.

Ordinal notations

Some theories, such as subsystems of second-order arithmetic, have no conceptualization or way to make arguments about transfinite ordinals. For example, to formalize what it means for a subsystem of Z₂ ${\displaystyle T}$ to "prove ${\displaystyle \alpha }$ well-ordered", we instead construct an ordinal notation ${\displaystyle (A,{\tilde {<}})}$ with order type ${\displaystyle \alpha }$. ${\displaystyle T}$ can now work with various transfinite induction principles along ${\displaystyle (A,{\tilde {<}})}$, which substitute for reasoning about set-theoretic ordinals.

However, some pathological notation systems exist that are unexpectedly difficult to work with. For example, Rathjen gives a primitive recursive notation system ${\displaystyle (\mathbb {N} ,<_{T})}$ that is well-founded iff PA is consistent,^{[2]p. 3} despite having order type ${\displaystyle \omega }$ - including such a notation in the ordinal analysis of PA would result in the false equality ${\displaystyle {\mathsf {PTO(PA)}}=\omega }$.

Upper bound

For any theory that's both ${\displaystyle \Sigma _{1}^{1}}$-axiomatizable and ${\displaystyle \Pi _{1}^{1}}$-sound, the existence of a recursive ordering that the theory fails to prove is well-ordered follows from the ${\displaystyle \Sigma _{1}^{1}}$ bounding theorem, and said provably well-founded ordinal notations are in fact well-founded by ${\displaystyle \Pi _{1}^{1}}$-soundness. Thus the proof-theoretic ordinal of a ${\displaystyle \Pi _{1}^{1}}$-sound theory that has a ${\displaystyle \Sigma _{1}^{1}}$ axiomatization will always be a (countable) recursive ordinal, that is, less than the Church–Kleene ordinal ${\displaystyle \omega _{1}^{\mathrm {CK} }}$. ^{[2]Theorem 2.21}

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)^{[citation needed]}.
• PA^–, the first-order theory of the nonnegative part of a discretely ordered ring.

Theories with proof-theoretic ordinal ω²

• RFA, rudimentary function arithmetic.^[3]
• IΔ₀, arithmetic with induction on Δ₀-predicates without any axiom asserting that exponentiation is total.

Theories with proof-theoretic ordinal ω³

• EFA, elementary function arithmetic.
• IΔ₀ + exp, arithmetic with induction on Δ₀-predicates augmented by an axiom asserting that exponentiation is total.
• RCA^*
  ₀, a second order form of EFA sometimes used in reverse mathematics.
• WKL^*
  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)

• IΔ₀ 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 ω^ω

• RCA₀, recursive comprehension.
• WKL₀, weak Kőnig's lemma.
• PRA, primitive recursive arithmetic.
• IΣ₁, arithmetic with induction on Σ₁-predicates.

Theories with proof-theoretic ordinal ε₀

• PA, Peano arithmetic (shown by Gentzen using cut elimination).
• ACA₀, arithmetical comprehension.

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

• 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 ordinal

• ID₁, the first 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 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",^{[5]p. 13} and which is bounded by ψ₀(Ω_ω) 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).
• ID_ω, the theory of ω-iterated inductive definitions. Its proof-theoretic ordinal is equal to the Takeuti-Feferman-Buchholz ordinal.
• 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 ${\displaystyle \Sigma _{2}^{1}{\mbox{-}}{\mathsf {AC}}+{\mathsf {BI}}}$.
• KPi, an extension of Kripke–Platek set theory based on a recursively inaccessible ordinal, has a very large proof-theoretic ordinal ${\displaystyle \psi (\varepsilon _{I+1})}$ described in a 1983 paper of Jäger and Pohlers, where I is the smallest inaccessible.^[6] This ordinal is also the proof-theoretic ordinal of ${\displaystyle \Delta _{2}^{1}{\mbox{-}}{\mathsf {CA}}+{\mathsf {BI}}}$.
• KPM, an extension of Kripke–Platek set theory based on a recursively Mahlo ordinal, has a very large proof-theoretic ordinal θ, which was described by Rathjen (1990).
• TTM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof-theoretic ordinal ψ_Ω1(Ω_{M + ω}).
• ${\displaystyle {\mathsf {KP}}+\Pi _{3}-Ref}$ has a proof-theoretic ordinal equal to ${\displaystyle \mathrm{\Psi <!--\; \Psi \; -->}}$Zdroj:https://en.wikipedia.org?pojem=Ordinal_analysis
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