Ordinal analysis
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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.
History[edit]
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 ε_{0}. See Gentzen's consistency proof.
Definition[edit]
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 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 for which there exists a notation in Kleene's sense such that proves that is an ordinal notation. Equivalently, it is the supremum of all ordinals such that there exists a recursive relation on (the set of natural numbers) that well-orders it with ordinal and such that proves transfinite induction of arithmetical statements for .
Ordinal notations[edit]
Some theories, such as subsystems of second-order arithmetic, have no conceptualization of or way to make arguments about transfinite ordinals. For example, to formalize what it means for a subsystem of Z_{2} to "prove well-ordered", we instead construct an ordinal notation with order type . 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,^{[1]} despite having order type - including such a notation in the ordinal analysis of PA would result in the false equality .
Upper bound[edit]
For any theory that's both -axiomatizable and -sound, the existence of a recursive ordering that the theory fails to prove is well-ordered follows from the bounding theorem, and said provably well-founded ordinal notations are in fact well-founded by -soundness. Thus the proof-theoretic ordinal of a -sound theory that has a axiomatization will always be a (countable) recursive ordinal, that is, less than the Church–Kleene ordinal . ^{[2]}
Examples[edit]
Theories with proof-theoretic ordinal ω[edit]
- 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 ω^{2}[edit]
- RFA, rudimentary function arithmetic.^{[3]}
- IΔ_{0}, arithmetic with induction on Δ_{0}-predicates without any axiom asserting that exponentiation is total.
Theories with proof-theoretic ordinal ω^{3}[edit]
- EFA, elementary function arithmetic.
- IΔ_{0} + exp, arithmetic with induction on Δ_{0}-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 ω^{n} (for n = 2, 3, ... ω)[edit]
- IΔ_{0} or EFA augmented by an axiom ensuring that each element of the n-th level of the Grzegorczyk hierarchy is total.
Theories with proof-theoretic ordinal ω^{ω}[edit]
- RCA_{0}, recursive comprehension.
- WKL_{0}, weak Kőnig's lemma.
- PRA, primitive recursive arithmetic.
- IΣ_{1}, arithmetic with induction on Σ_{1}-predicates.
Theories with proof-theoretic ordinal ε_{0}[edit]
- PA, Peano arithmetic (shown by Gentzen using cut elimination).
- ACA_{0}, arithmetical comprehension.
Theories with proof-theoretic ordinal the Feferman–Schütte ordinal Γ_{0}[edit]
- ATR_{0}, 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[edit]
- ID_{1}, 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 T_{0}.
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[edit]
What is the proof-theoretic ordinal of full second-order arithmetic?^{[4]}
- , Π_{1}^{1} comprehension has a rather large proof-theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams", and which is bounded by ψ_{0}(Ω_{ω}) in Buchholz's notation. It is also the ordinal of , 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.
- T_{0}, 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 .
- KPi, an extension of Kripke–Platek set theory based on a recursively inaccessible ordinal, has a very large proof-theoretic ordinal described in a 1983 paper of Jäger and Pohlers, where I is the smallest inaccessible.^{[5]} This ordinal is also the proof-theoretic ordinal of .
- 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).
- MLM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof-theoretic ordinal ψ_{Ω1}(Ω_{M + ω}).
- has a proof-theoretic ordinal equal to , where refers to the first weakly compact, using Rathjen's Psi function
- has a proof-theoretic ordinal equal to , where refers to the first -indescribable and , using Stegert's Psi function.
- has a proof-theoretic ordinal equal to where is a cardinal analogue of the least ordinal which is -stable for all and , using Stegert's Psi function.
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[edit]
Ordinal | First-order arithmetic | Second-order arithmetic | Kripke-Platek set theory | Type theory | Constructive set theory | Explicit mathematics |
---|---|---|---|---|---|---|
, | ||||||
, | ||||||
, | , | |||||
^{[1]} | , | |||||
, | , | |||||
, , | ||||||
, | ||||||
^{[2]} | ||||||
, | , | , | ||||
^{[3]} | , | |||||
^{[4]} | ||||||
, | ||||||
^{[5]} | ||||||
^{[6]} | ||||||
, | ||||||
^{[7]} | ||||||
^{[8]} | , | |||||
^{[9]} | ||||||
^{[10]} | ||||||
^{[11]} | ||||||
^{[12]} | ||||||
^{[13]} | ||||||
^{[14]} | ||||||
^{[15]} | ||||||
^{[6]} |
Key[edit]
This is a list of symbols used in this table:
- ψ represents Buchholz's psi unless stated otherwise.
- Ψ represents either Rathjen's or Stegert's Psi.
- φ represents Veblen's function.
- ω represents the first transfinite ordinal.
- ε_{α} represents the epsilon numbers.
- Γ_{α} represents the gamma numbers (Γ_{0} is the Feferman–Schütte ordinal)
- Ω_{α} represent the uncountable ordinals (Ω_{1}, abbreviated Ω, is ω_{1}).
This is a list of the abbreviations used in this table:
- First-order arithmetic
- is Robinson arithmetic
- is the first-order theory of the nonnegative part of a discretely ordered ring.
- is rudimentary function arithmetic.
- is arithmetic with induction restricted to Δ_{0}-predicates without any axiom asserting that exponentiation is total.
- is elementary function arithmetic.
- is arithmetic with induction restricted to Δ_{0}-predicates augmented by an axiom asserting that exponentiation is total.
- is elementary function arithmetic augmented by an axiom ensuring that each element of the n-th level of the Grzegorczyk hierarchy is total.
- is augmented by an axiom ensuring that each element of the n-th level of the Grzegorczyk hierarchy is total.
- is primitive recursive arithmetic.
- is arithmetic with induction restricted to Σ_{1}-predicates.
- is Peano arithmetic.
- is but with induction only for positive formulas.
- extends PA by ν iterated fixed points of monotone operators.
- is not exactly a first-order arithmetic system, but captures what one can get by predicative reasoning based on the natural numbers.
- is an automorphism on .
- extends PA by ν iterated least fixed points of monotone operators.
- is not exactly a first-order arithmetic system, but captures what one can get by predicative reasoning based on ν-times iterated generalized inductive definitions.
- is an automorphism on .
- is a weakened version of based on W-types.
- Second-order arithmetic
- is a second order form of sometimes used in reverse mathematics.
- is a second order form of sometimes used in reverse mathematics.
- is recursive comprehension.
- is weak Kőnig's lemma.
- is arithmetical comprehension.
- is plus the full second-order induction scheme.
- is arithmetical transfinite recursion.
- is plus the full second-order induction scheme.
- is plus the assertion "every true -sentence with parameters holds in a (countable coded) -model of ".
- Kripke-Platek set theory
- is Kripke-Platek set theory with the axiom of infinity.
- is Kripke-Platek set theory, whose universe is an admissible set containing .
- is a weakened version of based on W-types.
- asserts that the universe is a limit of admissible sets.
- is a weakened version of based on W-types.
- asserts that the universe is inaccessible sets.
- asserts that the universe is hyperinaccessible: an inaccessible set and a limit of inaccessible sets.
- asserts that the universe is a Mahlo set.
- is augmented by a certain first-order reflection scheme.
- is KPi augmented by the axiom .
- is KPI augmented by the assertion "at least one recursively Mahlo ordinal exists".
A superscript zero indicates that -induction is removed (making the theory significantly weaker).
- Type theory
- is the Herbelin-Patey Calculus of Primitive Recursive Constructions.
- is type theory without W-types and with universes.
- is type theory without W-types and with finitely many universes.
- is type theory with a next universe operator.
- is type theory without W-types and with a superuniverse.
- is an automorphism on type theory without W-types.
- is type theory with one universe and Aczel's type of iterative sets.
- is type theory with indexed W-Types.
- is type theory with W-types and one universe.
- is type theory with W-types and finitely many universes.
- is an automorphism on type theory with W-types.
- is type theory with a Mahlo universe.
- Constructive set theory
- is Aczel's constructive set theory.
- is plus the regular extension axiom.
- is plus the full-second order induction scheme.
- is with a Mahlo universe.
- Explicit mathematics
- is basic explicit mathematics plus elementary comprehension
- is plus join rule
- is plus join axioms
- is a weak variant of the Feferman's .
- is , where is inductive generation.
- is , where is the full second-order induction scheme.
See also[edit]
- Equiconsistency
- Large cardinal property
- Feferman–Schütte ordinal
- Bachmann–Howard ordinal
- Complexity class
- Gentzen's consistency proof
Notes[edit]
- 1.^ For
- 2.^ The Veblen function with countably infinitely iterated least fixed points.
- 3.^ Can also be commonly written as in Madore's ψ.
- 4.^ Uses Madore's ψ rather than Buchholz's ψ.
- 5.^ Can also be commonly written as in Madore's ψ.
- 6.^ represents the first recursively weakly compact ordinal. Uses Arai's ψ rather than Buchholz's ψ.
- 7.^ Also the proof-theoretic ordinal of , as the amount of weakening given by the W-types is not enough.
- 8.^ represents the first inaccessible cardinal. Uses Jäger's ψ rather than Buchholz's ψ.
- 9.^ represents the limit of the -inaccessible cardinals. Uses (presumably) Jäger's ψ.
- 10.^ represents the limit of the -inaccessible cardinals. Uses (presumably) Jäger's ψ.
- 11.^ represents the first Mahlo cardinal. Uses Rathjen's ψ rather than Buchholz's ψ.
- 12.^ represents the first weakly compact cardinal. Uses Rathjen's Ψ rather than Buchholz's ψ.
- 13.^ represents the first -indescribable cardinal. Uses Stegert's Ψ rather than Buchholz's ψ.
- 14.^ is the smallest such that ' is -indescribable') and ' is -indescribable '). Uses Stegert's Ψ rather than Buchholz's ψ.
- 15.^ represents the first Mahlo cardinal. Uses (presumably) Rathjen's ψ.
Citations[edit]
- ^ Rathjen, The Realm of Ordinal Analysis (p.3). Accessed 2021 September 29.
- ^ M. Rathjen, The Realm of Ordinal Analysis (theorem 2.21). Accessed 3 October 2022.
- ^ 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 Δ_{0}-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.
- ^ M. Rathjen, Proof Theory: From Arithmetic to Set Theory (p.28). Accessed 14 August 2022.
- ^ D. Madore, A Zoo of Ordinals (2017, p.2). Accessed 12 August 2022.
- ^ Arai, Toshiyasu (2022-01-10). "An ordinal analysis of $\Pi_{1}$-Collection". arXiv:2112.09871 [math.LO].
References[edit]
- 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, doi:10.1007/978-3-540-46825-7, ISBN 3-540-51842-8, MR 1026933
- Pohlers, Wolfram (1998), "Set Theory and Second Order Number Theory", Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, vol. 137, Amsterdam: Elsevier Science B. V., pp. 210–335, doi:10.1016/S0049-237X(98)80019-0, ISBN 0-444-89840-9, MR 1640328
- Rathjen, Michael (1990), "Ordinal notations based on a weakly Mahlo cardinal.", Arch. Math. Logic, 29 (4): 249–263, doi:10.1007/BF01651328, MR 1062729, S2CID 14125063
- Rathjen, Michael (2006), "The art of ordinal analysis" (PDF), International Congress of Mathematicians, vol. II, Zürich: Eur. Math. Soc., pp. 45–69, MR 2275588, archived from the original on 2009-12-22
{{citation}}
: CS1 maint: bot: original URL status unknown (link) - Rose, H.E. (1984), Subrecursion. Functions and Hierarchies, Oxford logic guides, vol. 9, Oxford, New York: Clarendon Press, Oxford University Press
- Schütte, Kurt (1977), Proof theory, Grundlehren der Mathematischen Wissenschaften, vol. 225, Berlin-New York: Springer-Verlag, pp. xii+299, ISBN 3-540-07911-4, MR 0505313
- Setzer, Anton (2004), "Proof theory of Martin-Löf type theory. An Overview", Mathématiques et Sciences Humaines. Mathematics and Social Sciences (165): 59–99
- 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, MR 0882549