# Ordinal analysis

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 α 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 α such that there exists a recursive relation R on ω (the set of natural numbers) which well-orders it with ordinal α and such that T proves transfinite induction of arithmetical statements for R.

The existence of any recursive ordinal which the theory fails to prove is well ordered follows from the $\Sigma^1_1$ bounding theorem, as the set of natural numbers which 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

• RFA, rudimentary function arithmetic.
• 0, arithmetic with induction on Δ0-predicates without any axiom asserting that exponentiation is total.

### 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.

• EFA, elementary function arithmetic.
• 0 + 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.

### Theories with proof theoretic ordinal ωn

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

### 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

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

• ATR0, arithmetical transfinite recursion.
• Martin-Löf type 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 ψ0ω) in Buchholz's notation. It is also the ordinal of ID < ω, the theory of finitely iterated inductive definitions.
• KPM, an extension of Kripke-Platek set theory based on a Mahlo cardinal, 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., 897, Berlin: Springer-Verlag, doi:10.1007/BFb0091894, ISBN 978-3-540-11170-2
• Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics, 1407, Berlin: Springer-Verlag, ISBN 3-540-51842-8, MR1026933
• Pohlers, Wolfram (1998), Set Theory and Second Order Number Theory, "Handbook of Proof Theory", Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics (Amsterdam: Elsevier Science B. V.) 137: pp. 210–335, ISBN 0-444-89840-9, MR1640328
• Rathjen, Michael (1990), "Ordinal notations based on a weakly Mahlo cardinal.", Arch. Math. Logic 29 (4): 249–263, doi:10.1007/BF01651328, MR1062729
• Rathjen, Michael (2006), "The art of ordinal analysis", International Congress of Mathematicians, II, Zürich,: Eur. Math. Soc., pp. 45–69, MR2275588
• Rose, H. (1984), Subrecursion. Functions and Hierarchies, Oxford logic guides, 9, Oxford, New York: Clarendon Press, Oxford University Press
• Schütte, Kurt (1977), Proof theory, Grundlehren der Mathematischen Wissenschaften, 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, 81 (Second ed.), Amsterdam: North-Holland Publishing Co., ISBN 0-444-87943-9, MR0882549

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Ordinal collapsing function — In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger …   Wikipedia

• Ordinal notation — In mathematical logic and set theory, an ordinal notation is a finite sequence of symbols from a finite alphabet which names an ordinal number according to some scheme which gives meaning to the language. There are many such schemes of ordinal… …   Wikipedia

• Ordinal utility — theory states that while the utility of a particular good or service cannot be measured using a numerical scale bearing economic meaning in and of itself, pairs of alternative bundles (combinations) of goods can be ordered such that one is… …   Wikipedia

• analysis of variance — n analysis of variation in an experimental outcome and esp. of a statistical variance in order to determine the contributions of given factors or variables to the variance * * * (ANOVA) a statistical method for analyzing the effects of each of… …   Medical dictionary

• Ordinal optimization — In mathematical optimization, ordinal optimization is the maximization of functions taking values in a partially ordered set ( poset ). Ordinal optimization has applications in the theory of queuing networks. Contents 1 Mathematical foundations 1 …   Wikipedia

• analysis of covariance — (ANCOVA) a statistical procedure used with one dependent variable and multiple independent variables of both categorical (ordinal, dichotomous, or nominal) and continuous types; it is a variation of analysis of variance that adjusts for… …   Medical dictionary

• Data analysis — Analysis of data is a process of inspecting, cleaning, transforming, and modeling data with the goal of highlighting useful information, suggesting conclusions, and supporting decision making. Data analysis has multiple facets and approaches,… …   Wikipedia

• Failure mode and effects analysis — A failure modes and effects analysis (FMEA) is a procedure for analysis of potential failure modes within a system for classification by severity or determination of the effect of failures on the system. It is widely used in manufacturing… …   Wikipedia

• multivariate analysis — Univariate analysis consists in describing and explaining the variation in a single variable. Bivariate analysis does the same for two variables taken together (covariation). Multivariate analysis (MVA) considers the simultaneous effects of many… …   Dictionary of sociology

• Regression analysis — In statistics, regression analysis is a collective name for techniques for the modeling and analysis of numerical data consisting of values of a dependent variable (response variable) and of one or more independent variables (explanatory… …   Wikipedia