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 \Delta^1_2 functions of the 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 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 \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.

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 T of Z2 to "prove \alpha well-ordered", we instead construct an ordinal notation with order type \alpha. T can now work with various transfinite induction principles along, 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 that is well-founded iff PA is consistent, p. 3 despite having order type \omega - including such a notation in the ordinal analysis of PA would result in the false equality.

Upper bound

Since an ordinal notation must be recursive, the proof-theoretic ordinal of any theory is less than or equal to the Church–Kleene ordinal. In particular, the proof-theoretic ordinal of an inconsistent theory is equal to, because an inconsistent theory trivially proves that all ordinal notations are well-founded. For any theory that's both \Sigma^1_1-axiomatizable and \Pi^1_1-sound, the existence of a recursive ordering that the theory fails to prove is well-ordered follows from the \Sigma^1_1 bounding theorem, and said provably well-founded ordinal notations are in fact well-founded by \Pi^1_1-soundness. Thus the proof-theoretic ordinal of a \Pi^1_1-sound theory that has a \Sigma^1_1 axiomatization will always be a (countable) recursive ordinal, that is, strictly less than. Theorem 2.21

Examples

Theories with proof-theoretic ordinal ω

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 (for n = 2, 3, ... ω)

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

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

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, full second-order arithmetic and set theories with powersets including ZF and ZFC. The strength of intuitionistic ZF (IZF) equals that of ZF.

Table of ordinal analyses

Key

This is a list of symbols used in this table: This is a list of the abbreviations used in this table: In general, a subscript 0 means that the induction scheme is restricted to a single set induction axiom. A superscript zero indicates that \in-induction is removed (making the theory significantly weaker).

Citations