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.

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

The existence of any recursive ordinal that the theory fails to prove is well ordered follows from the \Sigma^1_1 bounding theorem, as the set of natural numbers that an effective theory proves to be ordinal notations is a \Sigma^0_1 set (see Hyperarithmetical theory). Thus the proof theoretic ordinal of a theory will always be a countable ordinal less than the Church-Kleene ordinal \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.

Examples

Theories with proof theoretic ordinal ω2

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

Theories with proof theoretic ordinal ωω

Theories with proof theoretic ordinal ε0

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

Theories with larger proof theoretic ordinals

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 (as of 2008) been given. This includes second order arithmetic and set theories with powersets. (The CZF and Kripke-Platek set theories mentioned above are weak set theories without powersets.)

See also

References

  1. Krajicek, Jan (1995). Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press. pp. 18–20. ISBN 9780521452052. defines the rudimentary sets and rudimentary functions, and proves them equivalent to the Δ0-predicates on the naturals. An ordinal analysis of the system can be found in Rose, H. E. (1984). Subrecursion: functions and hierarchies. University of Michigan: Clarendon Press. ISBN 9780198531890.
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