Ordinal analysis

In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength.

Definition

All theories discussed are assumed to be recursive countable theories, and are assumed to be powerful enough to make statements about the natural numbers.

The proof theoretic ordinal of a theory is the smallest recursive ordinal that the theory cannot prove is well founded. More accurately, the proof theoretic ordinal of the theory $T$ can be defined the supremum of all ordinals $\alpha$ for which there exists a notation $o$ in Kleene's sense such that $T$ proves that $o$ is, indeed, an ordinal notation; or equivalently, as the supremum of all ordinals $\alpha$ such that there exists a recursive relation $R$ on $\omega$ (the set of natural numbers) which well-orders it with ordinal $\alpha$ and such that $T$ proves transfinite induction of arithmetical statements for $R$.

The proof theoretic ordinal of a theory is a countable ordinal less than the Church-Kleene ordinal. In practice, this 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 has a larger proof theoretic ordinal than another it can often prove the consistence of the second theory.

Examples

Theories with proof theoretic ordinal ω²

• ${\mathsf {RFA}}$, rudimentary function arithmetic.
• ${\mathsf {I}}\Delta _{0}$ (arithmetic with induction on $\Delta _{0}$ predicates) without any axiom asserting that exponentiation is total.

Theories with proof theoretic ordinal ω³

• ${\mathsf {EFA}}$, elementary function arithmetic.
• ${\mathsf {I}}\Delta _{0}$ (arithmetic with induction on $\Delta _{0}$ predicates) augmented by an axiom asserting that exponentiation is total.

Theories with proof theoretic ordinal ωⁿ

• ${\mathsf {I}}\Delta _{0}$ or ${\mathsf {EFA}}$ augmented by an axiom ensuring that each element of the nth level ${\mathcal {E}}^{n}$ of the Grzegorczyk hierarchy is total.

Theories with proof theoretic ordinal ω^ω

• ${\mathsf {RCA}}_{0}$, Recursive Comprehension.
• ${\mathsf {WKL}}_{0}$, Weak König's lemma.
• ${\mathsf {PRA}}$, primitive recursive arithmetic.
• ${\mathsf {I}}\Sigma _{1}$ (arithmetic with induction on $\Sigma _{1}$ predicates).

Theories with proof theoretic ordinal ε₀

• Gentzen showed using cut elimination that the proof theoretic ordinal of Peano arithmetic is ε₀.
• ${\mathsf {ACA}}_{0}$, Arithmetical comprehension.

Theories with proof theoretic ordinal the Feferman-Schütte ordinal Γ₀

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

• ${\mathsf {ATR}}_{0}$, Arithmetical Transfinite Recursion has proof theoretic ordinal the Feferman-Schütte ordinal Γ₀.

Theories with proof theoretic ordinal the Bachmann-Howard ordinal

• Kripke-Platek set theory

Theories with larger proof theoretic ordinals

• $\Pi _{1}^{1}{\mbox{-}}{\mathsf {CA}}_{0}$, Π₁¹ comprehension has a rather large proof theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams", and which is bounded by ψ₀(Ω_ω) in Buchholz's notation.
• KPM, an extension of Kripke-Platek set theory, has a very large proof theoretic ordinal, which was described by Rathjen (1990).

Most theories capable of describing the power set of the natural numbers have proof theoretic ordinals that are (as of 2008) so large that no explicit combinatorial description has yet been given. This includes second order arithmetic and set theories with powersets. (Kripke-Platek set theory mentioned above is a weak set theory without power sets.)

References

• Buchholz, W.; Feferman, S.; Pohlers, W.; Sieg, W. (1981), Iterated inductive definitions and sub-systems of analysis, Lecture Notes in Math., vol. 897, Berlin: Springer-Verlag, doi:10.1007/BFb0091894, ISBN 978-3-540-11170-2
• Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics, vol. 1407, Berlin: Springer-Verlag, ISBN 3-540-51842-8, MR1026933
• Rathjen, Michael (1990), "Ordinal notations based on a weakly Mahlo cardinal.", Arch. Math. Logic, 29 (4): 249–263, MR1062729
• Rathjen, Michael (2006), "The art of ordinal analysis", International Congress of Mathematicians (PDF), vol. II, Zürich,: Eur. Math. Soc., pp. 45–69, MR2275588{{citation}}: CS1 maint: extra punctuation (link)
• Schütte, Kurt (1977), Proof theory, Grundlehren der Mathematischen Wissenschaften, vol. 225, Berlin-New York: Springer-Verlag, pp. xii+299, ISBN 3-540-07911-4, MR0505313
• Takeuti, Gaisi (1987), Proof theory, Studies in Logic and the Foundations of Mathematics, vol. 81 (Second ed.), Amsterdam: North-Holland Publishing Co., ISBN 0-444-87943-9, MR0882549