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In [[proof theory]], '''ordinal analysis''' assigns [[ordinal number|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 [[epsilon zero|&epsilon;<sub>0</sub>]].

{{Short description|Mathematical technique used in proof theory}}
In [[proof theory]], '''ordinal analysis''' assigns [[ordinal number|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 [[equiconsistency|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 <math>\Delta^1_2</math> functions of the theory.<ref>M. Rathjen, "[https://www1.maths.leeds.ac.uk/~rathjen/BEYOND.pdf Admissible Proof Theory and Beyond]". In ''Studies in Logic and the Foundations of Mathematics vol. 134'' (1995), pp.123--147.</ref>

== 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 [[epsilon numbers (mathematics)|ε<sub>0</sub>]]. See [[Gentzen's consistency proof]].


==Definition==
==Definition==
Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations.
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 smallest recursive ordinal that the theory cannot prove is well founded &mdash; the supremum of all ordinals <math>\alpha</math> for which there exists a [[Kleene's O|notation <math>o</math> in Kleene's sense]] such that <math>T</math> proves that <math>o</math> is an [[ordinal notation]]. Equivalently, it is the supremum of all ordinals <math>\alpha</math> such that there exists a [[Computable function|recursive relation]] <math>R</math> on <math>\omega</math> (the set of natural numbers) which [[well-order]]s it with ordinal <math>\alpha</math> and such that <math>T</math> proves [[transfinite induction]] of arithmetical statements for <math>R</math>.


The '''proof-theoretic ordinal''' of such a theory <math>T</math> is the supremum of the [[order type]]s of all [[ordinal notation]]s (necessarily [[recursive ordinal|recursive]], see next section) that the theory can prove are [[Well-founded relation|well founded]]&mdash;the supremum of all ordinals <math>\alpha</math> for which there exists a [[Kleene's O|notation <math>o</math> in Kleene's sense]] such that <math>T</math> proves that <math>o</math> is an ordinal notation. Equivalently, it is the supremum of all ordinals <math>\alpha</math> such that there exists a [[Computable function|recursive relation]] <math>R</math> on <math>\omega</math> (the set of natural numbers) that [[well-order]]s it with ordinal <math>\alpha</math> and such that <math>T</math> proves [[transfinite induction]] of arithmetical statements for <math>R</math>.
The existence of any recursive ordinal which the theory fails to prove is well ordered follows from the <math>\Sigma^1_1</math> bounding theorem, as the set of natural numbers which an effective theory proves to be ordinal notations is a <math>\Sigma^0_1</math> set (see [[Hyperarithmetical theory]]). Thus the proof theoretic ordinal of a theory will always be a countable ordinal less than the [[Church-Kleene ordinal]] <math>\omega_1^{\mathrm{CK}}</math>.
===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 <math>T</math> of Z<sub>2</sub> to "prove <math>\alpha</math> well-ordered", we instead construct an [[ordinal notation]] <math>(A,\tilde <)</math> with order type <math>\alpha</math>. <math>T</math> can now work with various transfinite induction principles along <math>(A,\tilde <)</math>, 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 <math>(\mathbb N,<_T)</math> that is well-founded iff PA is consistent,<ref name="Realm">Rathjen, [https://web.archive.org/web/20231207051454/http://www1.maths.leeds.ac.uk/~rathjen/realm.pdf The Realm of Ordinal Analysis]. Accessed 2021 September 29.</ref><sup>p. 3</sup> despite having order type <math>\omega</math> - including such a notation in the ordinal analysis of PA would result in the false equality <math>\mathsf{PTO(PA)}=\omega</math>.
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 [[equiconsistency|equiconsistent]], and if one theory has a larger proof theoretic ordinal than another it can often prove the consistency of the second theory.

==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]] <math>\omega_1^{\mathrm{CK}}</math>. In particular, the proof-theoretic ordinal of an [[inconsistent]] theory is equal to <math>\omega_1^{\mathrm{CK}}</math>, because an inconsistent theory trivially proves that all ordinal notations are well-founded.

For any theory that's both <math>\Sigma^1_1</math>-axiomatizable and <math>\Pi^1_1</math>-sound, the existence of a recursive ordering that the theory fails to prove is well-ordered follows from the <math>\Sigma^1_1</math> bounding theorem, and said provably well-founded ordinal notations are in fact well-founded by <math>\Pi^1_1</math>-soundness. Thus the proof-theoretic ordinal of a <math>\Pi^1_1</math>-sound theory that has a <math>\Sigma^1_1</math> axiomatization will always be a (countable) [[recursive ordinal]], that is, strictly less than <math>\omega_1^{\mathrm{CK}}</math>. <ref name="Realm" /><sup>Theorem 2.21</sup>


==Examples==
==Examples==


===Theories with proof theoretic ordinal ω<sup>2</sup>===
===Theories with proof-theoretic ordinal ω===
*Q, [[Robinson arithmetic]] (although the definition of the proof-theoretic ordinal for such weak theories has to be tweaked){{Citation needed|date=November 2022}}.
*<math>\mathsf{RFA}</math>, [[rudimentary function arithmetic]].
*PA<sup>&ndash;</sup>, the first-order theory of the nonnegative part of a discretely ordered ring.
*<math>\mathsf{I}\Delta_0</math> (arithmetic with induction on <math>\Delta_0</math> predicates) without any axiom asserting that exponentiation is total.

===Theories with proof-theoretic ordinal ω<sup>2</sup>===
*RFA, [[rudimentary function]] arithmetic.<ref name=Krajicek>{{cite book|last=Krajicek|first=Jan|title=Bounded Arithmetic, Propositional Logic and Complexity Theory|year=1995|publisher=Cambridge University Press|isbn=9780521452052|pages=[https://archive.org/details/boundedarithmeti0000kraj/page/18 18–20]|url=https://archive.org/details/boundedarithmeti0000kraj/page/18}} defines the rudimentary sets and rudimentary functions, and proves them equivalent to the Δ<sub>0</sub>-predicates on the naturals. <!--I think that --> An ordinal analysis of the system can be found in {{cite book|last=Rose|first=H. E.|title=Subrecursion: functions and hierarchies|year=1984|publisher=Clarendon Press|location=University of Michigan|isbn= 9780198531890}}</ref>
*IΔ<sub>0</sub>, arithmetic with induction on Δ<sub>0</sub>-predicates without any axiom asserting that exponentiation is total.

===Theories with proof-theoretic ordinal ω<sup>3</sup>===
*EFA, [[elementary function arithmetic]].
*IΔ<sub>0</sub> + exp, arithmetic with induction on Δ<sub>0</sub>-predicates augmented by an axiom asserting that exponentiation is total.
*RCA{{su|p=*|b=0}}, a second order form of EFA sometimes used in [[reverse mathematics]].
*WKL{{su|p=*|b=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<sup>''n''</sup> (for ''n'' = 2, 3, ... ω)===
*IΔ<sub>0</sub> or EFA augmented by an axiom ensuring that each element of the ''n''-th level <math>\mathcal{E}^n</math> of the [[Grzegorczyk hierarchy]] is total.

===Theories with proof-theoretic ordinal ω<sup>ω</sup>===
*RCA<sub>0</sub>, [[Reverse mathematics#The base system RCA0|recursive comprehension]].
*WKL<sub>0</sub>, [[Reverse mathematics#Weak Kőnig's lemma WKL0|weak Kőnig's lemma]].
*PRA, [[primitive recursive arithmetic]].
*IΣ<sub>1</sub>, arithmetic with induction on Σ<sub>1</sub>-predicates.

===Theories with proof-theoretic ordinal ε<sub>0</sub>===
*PA, [[Peano arithmetic]] ([[Gentzen's consistency proof|shown]] by [[Gentzen]] using [[cut elimination]]).
*ACA<sub>0</sub>, [[Reverse mathematics#Arithmetical comprehension ACA0|arithmetical comprehension]].

===Theories with proof-theoretic ordinal the [[Feferman–Schütte_ordinal|Feferman–Schütte ordinal &Gamma;<sub>0</sub>]]===
*ATR<sub>0</sub>, [[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]]===
* ID<sub>1</sub>, the first [[Buchholz's ID hierarchy|theory of inductive definitions]].
* KP, [[Kripke–Platek set theory]] with the [[axiom of infinity]].
* CZF, Aczel's [[CZF|constructive Zermelo–Fraenkel set theory]].
* EON, a weak variant of the [[Solomon Feferman|Feferman]]'s explicit mathematics system T<sub>0</sub>.

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===
{{unsolved|mathematics|What is the proof-theoretic ordinal of full second-order arithmetic?<ref name="RathjenFromArithmetic">M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf Proof Theory: From Arithmetic to Set Theory] (p.28). Accessed 14 August 2022.</ref>}}

*<math>\Pi^1_1\mbox{-}\mathsf{CA}_0</math>, [[second order arithmetic|&Pi;<sub>1</sub><sup>1</sup> comprehension]] has a rather large proof-theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams",<ref>{{citation|chapter-url=https://web.archive.org/web/20240114000508/https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf|chapter=The art of ordinal analysis|first=Michael|last=Rathjen|mr=2275588|title=International Congress of Mathematicians|volume=II|pages=45–69|publisher=Eur. Math. Soc.|place=Zürich|year=2006|url-status=bot: unknown|archiveurl=https://web.archive.org/web/20091222002129/http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf|archivedate=2009-12-22}}</ref><sup>p. 13</sup> and which is bounded by [[Buchholz's ordinal|&psi;<sub>0</sub>(&Omega;<sub>ω</sub>)]] in [[Buchholz's notation]]. It is also the ordinal of <math>ID_{<\omega}</math>, the theory of finitely iterated inductive definitions. And also the ordinal of MLW, Martin-Löf type theory with indexed W-Types {{harvtxt|Setzer|2004}}.
*ID<sub>ω</sub>, the [[Buchholz's ID hierarchy|theory of ω-iterated inductive definitions]]. Its proof-theoretic ordinal is equal to the [[Takeuti–Feferman–Buchholz ordinal|Takeuti-Feferman-Buchholz ordinal]].
*T<sub>0</sub>, 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 <math>\Sigma^1_2\mbox{-}\mathsf{AC} + \mathsf{BI}</math>.
*KPi, an extension of [[Kripke–Platek set theory]] based on a [[Admissible ordinal|recursively inaccessible ordinal]], has a very large proof-theoretic ordinal <math>\psi(\varepsilon_{I + 1})</math> described in a 1983 paper of Jäger and Pohlers, where I is the smallest inaccessible.<ref>D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017, p.2). Accessed 12 August 2022.</ref> This ordinal is also the proof-theoretic ordinal of <math>\Delta^1_2\mbox{-}\mathsf{CA} + \mathsf{BI}</math>.
*KPM, an extension of [[Kripke–Platek set theory]] based on a [[Admissible ordinal|recursively Mahlo ordinal]], has a very large proof-theoretic ordinal θ, which was described by {{harvtxt|Rathjen|1990}}.
*TTM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof-theoretic ordinal <math>\psi_{\Omega_1}(\Omega_{M+\omega})</math>.
*<math>\mathsf{KP} + \Pi_3 - Ref</math> has a proof-theoretic ordinal equal to <math>\Psi(\varepsilon_{K + 1})</math>, where <math>K</math> refers to the first weakly compact, due to (Rathjen 1993)
*<math>\mathsf{KP} + \Pi_\omega - Ref</math> has a proof-theoretic ordinal equal to <math>\Psi^{\varepsilon_{\Xi + 1}}_X</math>, where <math>\Xi</math> refers to the first <math>\Pi^2_0</math>-indescribable and <math>\mathbb{X} = (\omega^+; P_0; \epsilon, \epsilon, 0)</math>, due to (Stegert 2010).
*<math>\mathsf{Stability}</math> has a proof-theoretic ordinal equal to <math>\Psi^{\varepsilon_{\Upsilon+1}}_{\mathbb{X}}</math> where <math>\Upsilon</math> is a cardinal analogue of the least ordinal <math>\alpha</math> which is <math>\alpha+\beta</math>-stable for all <math>\beta < \alpha</math> and <math>\mathbb{X} = (\omega^+; P_0; \epsilon, \epsilon, 0)</math>, due to (Stegert 2010).

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 <math>\Pi^1_2 - CA_0</math>, full [[second-order arithmetic]] (<math>\Pi^1_\infty - CA_0</math>) and set theories with powersets including [[Zermelo–Fraenkel set theory |ZF]] and ZFC. The strength of [[Intuitionistic logic|intuitionistic]] ZF (IZF) equals that of ZF.

== Table of ordinal analyses ==
{| class="wikitable"
|+Table of proof-theoretic ordinals
!Ordinal
!First-order arithmetic
!Second-order arithmetic
!Kripke-Platek set theory
!Type theory
!Constructive set theory
!Explicit mathematics
|-
|<math>\omega</math>
|<math>\mathsf{Q}</math>, <math>\mathsf{PA}^-</math>
|
|
|
|
|
|-
|<math>\omega^2</math>
|<math>\mathsf{RFA}</math>, <math>\mathsf{I\Delta}_0</math>
|
|
|
|
|
|-
|<math>\omega^3</math>
|<math>\mathsf{EFA}</math>, <math>\mathsf{I\Delta}_0^{\mathsf{+}}</math>
|<math>\mathsf{RCA}_0^*</math>, <math>\mathsf{WKL}_0^*</math>
|
|
|
|
|-
|<math>\omega^n</math>{{ref|a}}
|<math>\mathsf{EFA}^{\mathsf{n}}</math>, <math>\mathsf{I\Delta}_0^{\mathsf{n+}}</math>
|
|
|
|
|
|-
|<math>\omega^\omega</math>
|<math>\mathsf{PRA}</math>, <math>\mathsf{I\Sigma}_1</math><ref name="AvigadSommer97">J. Avigad, R. Sommer, "[https://www.andrew.cmu.edu/user/avigad/Papers/alpha1.pdf A Model-Theoretic Approach to Ordinal Analysis]" (1997).</ref><sup>p. 13</sup>
|<math>\mathsf{RCA}_0</math><ref name="AvigadSommer97" /><sup>p. 13</sup>, <math>\mathsf{WKL}_0</math><ref name="AvigadSommer97" /><sup>p. 13</sup>
|
|<math>\mathsf{CPRC}</math>
|
|
|-
|<math>\omega^{\omega^{\omega^\omega}}</math>
|<math>\mathsf I\Sigma_3</math><ref>M. Rathjen, W. Carnielli, "[https://web.archive.org/web/20220708135503/http://www1.maths.leeds.ac.uk/~rathjen/HydraeSubArith.pdf Hydrae and subsystems of arithmetic]" (1991)</ref><ref name="AvigadSommer97" /><sup>p. 13</sup>
|<math>\mathsf{RCA}_0+(\Pi^0_2)^-\mathsf{-IND}</math><ref>{{cite arXiv|eprint=1411.4481 |author1=Jeroen Van der Meeren |last2=Rathjen |first2=Michael |last3=Weiermann |first3=Andreas |title=An order-theoretic characterization of the Howard-Bachmann-hierarchy |date=2014 |class=math.LO }}</ref>{{rp|40}}
|
|
|
|
|-
|<math>\varepsilon_0</math>
|<math>\mathsf{PA}</math><ref name="AvigadSommer97" /><sup>p. 13</sup>
|<math>\mathsf{ACA}_0</math><ref name="AvigadSommer97" /><sup>p. 13</sup>, <math>\mathsf{\Delta}_1^1\mathsf{-CA}_0</math>, <math>\mathsf{\Sigma}_1^1\mathsf{-AC}_0</math><ref name="AvigadSommer97" /><sup>p. 13</sup>, <math>\text{R-}\widehat{\mathbf{E}\boldsymbol{\Omega}}</math><ref name="JagerStrahm">G. Jäger, T. Strahm, "[https://home.inf.unibe.ch/thomas.strahm/download/pdf/secord.pdf Second order theories with ordinals and elementary comprehension]".</ref><sup>p. 8</sup>
|<math>\mathrm{KPu}^r</math><ref name="Jäger84" /><sup>p. 869</sup>
|
|
|<math>\mathsf{EM}_0</math>
|-
|<math>\varepsilon_\omega</math>
|
|<math>\mathsf{ACA}_0+\mathsf{iRT}</math>,<ref>B. Afshari, M. Rathjen, "Ordinal Analysis and the Infinite Ramsey Theorem" (2012)</ref> <math>\mathsf{RCA}_0+\forall Y\forall n\exists X(\textrm{TJ}(n,X,Y))</math><ref name="MarconeMontalbán2010" />{{rp|8}}
|
|
|
|
|-
|<math>\varepsilon_{\varepsilon_0}</math>
|
|<math>\mathsf{ACA}</math><ref>S. Feferman, "Theories of finite type related to mathematical practice". In ''Handbook of Mathematical Logic'', Studies in Logic and the Foundations of Mathematics vol. 90 (1977), ed. J. Barwise, pub. North Holland.</ref><sup>p. 959</sup>
|
|
|
|
|-
|<math>\zeta_0</math>
|
|<math>\mathsf{ACA}_0+\forall X\exists Y(\textrm{TJ}(\omega,X,Y))</math>,<ref name="Heissenbüttel2001">M. Heissenbüttel, "Theories of ordinal strength <math>\varphi20</math> and <math>\varphi2\varepsilon_0</math>" (2001)</ref><ref name="MarconeMontalbán2010">{{cite journal|arxiv=0910.5442 |doi=10.2178/jsl/1305810765 |title=The Veblen functions for computability theorists |date=2011 |last1=Marcone |first1=Alberto |last2=Montalbán |first2=Antonio |journal=The Journal of Symbolic Logic |volume=76 |issue=2 |pages=575–602 |s2cid=675632 }}</ref> <math>\mathsf p_1(\mathsf{ACA}_0)</math>,<ref name="Probst2017" />{{rp|7}} <math>\mathsf{RFN}_0</math><ref name="Heissenbüttel2001" /><sup>p. 17</sup>, <math>\mathsf{ACA}_0+(\mathsf{BR})</math><ref name="Heissenbüttel2001" /><sup>p. 5</sup>
|
|
|
|
|-
|<math>\varphi(2,\varepsilon_0)</math>
|
|<math>\mathsf{RFN}</math>, <math>\mathsf{ACA}+\forall X\exists Y(\textrm{TJ}(\omega,X,Y))</math><ref name="Heissenbüttel2001" /><sup>p. 52</sup>
|
|
|
|
|-
|<math>\varphi(\omega, 0)</math>
|<math>\mathsf{ID}_1\#</math>
|<math>\mathsf{\Delta}_1^1\mathsf{-CR}</math>, <math>\Sigma^1_1\mathsf{-DC}_0</math><ref>A. Cantini, "On the relation between choice and comprehension principles in second order arithmetic", Journal of Symbolic Logic vol. 51 (1986), pp. 360--373.</ref>
|
|
|
|<math>\mathsf{EM}_0 \mathsf{+JR}</math>
|-
|<math>\varphi(\varepsilon_0, 0)</math>
|<math>\widehat{\mathsf{ID}}_1</math>, <math>\mathsf{KFL}</math><ref name="FNF20" /><sup>p. 17</sup>, <math>\mathsf{KF}</math><ref name="FNF20" /><sup>p. 17</sup>
|<math>\mathsf{\Delta}_1^1\mathsf{-CA}</math><ref name="Simpson85">S. G. Simpson, "Friedman's Research on Subsystems of Second Order Arithmetic". In ''Harvey Friedman's Research on the Foundations of Mathematics'', Studies in Logic and the Foundations of Mathematics vol. 117 (1985), ed. L. Harrington, M. Morley, A. Šcedrov, S. G. Simpson, pub. North-Holland.</ref><sup>p. 140</sup>, <math>\mathsf{\Sigma}_1^1\mathsf{-AC}</math><ref name="Simpson85" /><sup>p. 140</sup>, <math>\mathsf{\Sigma}_1^1\mathsf{-DC}</math><ref name="Simpson85" /><sup>p. 140</sup>, <math>\text{W-}\widehat{\mathbf{E}\boldsymbol{\Omega}}</math><ref name="JagerStrahm" /><sup>p. 8</sup>
|<math>\mathrm{KPu}^r+(\mathrm{IND}_N)</math><ref name="Jäger84" /><sup>p. 870</sup>
|<math>\mathsf{ML}_1</math>
|
|<math>\mathsf{EM}_0 \mathsf{+J}</math>
|-
|<math>\varphi(\varepsilon_{\varepsilon_0}, 0)</math>
|
|<math>\widehat{\mathbf{E}\boldsymbol{\Omega}}</math><ref name="JagerStrahm" /><sup>p. 27</sup>, <math>\widehat{\mathbf{EID}}_{\boldsymbol{1}}</math><ref name="JagerStrahm" /><sup>p. 27</sup>
|
|
|
|
|-
|<math>\varphi(\varphi(\omega,0), 0)</math>
|<math>\mathrm{PRS}\omega</math><ref>J. Avigad, "[https://www.andrew.cmu.edu/user/avigad/Papers/admissible.pdf An ordinal analysis of admissible set theory using recursion on ordinal notations]". Journal of Mathematical Logic vol. 2, no. 1, pp.91--112 (2002).</ref><sup>p.9</sup>
|
|
|
|
|
|-
|<math>\varphi(\mathsf{<}\Omega, 0)</math>{{ref|b}}
|<math>\mathsf{Aut(ID\#)}</math>
|
|
|
|
|
|-
|<math>\Gamma_0</math>
|<math>\widehat{\mathsf{ID}}_{<\omega}</math>,<ref>S. Feferman, "[https://www.sciencedirect.com/science/article/abs/pii/S0049237X08713648 Iterated inductive fixed-point theories: application fo Hancock's conjecture]". In ''Patras Logic Symposion'', Studies in Logic and the Foundations of Mathematics vol. 109 (1982).</ref> <math>\mathsf{U(PA)}</math>, <math>\mathbf{KFL}^*</math><ref name="FNF20">{{cite arXiv|eprint=2007.07188 |last1=Fischer |first1=Martin |last2=Nicolai |first2=Carlo |author3=Pablo Dopico Fernandez |title=Nonclassical truth with classical strength. A proof-theoretic analysis of compositional truth over HYPE |date=2020 |class=math.LO }}</ref><sup>p. 22</sup>, <math>\mathbf{KF}^*</math><ref name="FNF20" /><sup>p. 22</sup>, <math>\mathcal U(\mathrm{NFA})</math><ref>S. Feferman, T. Strahm, "[https://math.stanford.edu/~feferman/papers/unfolding.pdf The unfolding of non-finitist arithmetic]", Annals of Pure and Applied Logic vol. 104, no.1--3 (2000), pp.75--96.</ref>
|<math>\mathsf{ATR}_0</math>, <math>\mathsf{\Delta}_1^1\mathsf{-CA+BR}</math>, <math>\Delta^1_1\mathrm{-CA}_0+\mathrm{(SUB)}</math>,<ref name="FJ83">S. Feferman, G. Jäger, "Choice principles, the bar rule and autonomously iterated comprehension schemes in analysis", Journal of Symbolic Logic vol. 48, no. (1983), pp.63--70.</ref> <math>\mathrm{FP}_0</math><ref name="BJS16">U. Buchholtz, G. Jäger, T. Strahm, "[https://ulrikbuchholtz.dk/bjs.pdf Theories of proof-theoretic strength <math>\psi(\Gamma_{\Omega+1})</math>]". In ''Concepts of Proof in Mathematics, Philosophy, and Computer Science'' (2016), ed. D. Probst, P. Schuster. DOI 10.1515/9781501502620-007.</ref><sup>p. 26</sup>
|<math>\mathsf{KPi}^0</math><ref name="Jäger84">G. Jäger, "[https://www.jstor.org/stable/2274140 The Strength of Admissibility Without Foundation]". Journal of Symbolic Logic vol. 49, no. 3 (1984).</ref><sup>p. 878</sup>, <math>\mathsf{KPu}^0+(\mathrm{BR})</math><ref name="Jäger84" /><sup>p. 878</sup>
|<math>\mathsf{ML}_{<\omega}</math>, <math>\mathsf{MLU}</math>
|
|
<!--|- Doesn't seem to have a source
|<math>\Gamma_{\omega}</math>
|
|
|<math>\mathsf{KPI}^0 \mathsf{ + \Sigma_1-I}_\omega</math>
|
|
|-->
|-
|<math>\Gamma_{\omega^\omega}</math>
|
|
|<math>\mathsf{KPI}^0 + (\mathsf{ \Sigma_1-I}_\omega)</math><ref>T. Strahm, "[https://home.inf.unibe.ch/thomas.strahm/download/pdf/prag.pdf Autonomous fixed point progressions and fixed point transfinite recursion]" (2000). In ''Logic Colloquium '98'', ed. S. R. Buss, P. Hájek, and P. Pudlák . DOI [https://doi.org/10.1017/9781316756140.031 10.1017/9781316756140.031]</ref><sup>p.13<!--, stated without proof--></sup>
|
|
|
|-
|<math>\Gamma_{\varepsilon_0}</math>
|<math>\widehat{\mathsf{ID}}_{\omega}</math>
|<math>\mathsf{ATR}</math><ref>G. Jäger, T. Strahm, "Fixed point theories and dependent choice". Archive for Mathematical Logic vol. 39 (2000), pp.493--508.</ref>
|<math>\mathsf{KPI}^0 \mathsf{ + F-I}_\omega</math>
|
|
|
|-
|<math>\varphi(1, \omega, 0)</math>
|<math>\widehat{\mathsf{ID}}_{<\omega^\omega}</math>
|<math>\mathsf{ATR}_0+(\mathsf{\Sigma}_1^1\mathsf{-DC})</math><ref name="Probst2017" />{{rp|7}}
|<math>\mathsf{KPi}^0 \mathsf{ + \Sigma_1-I}_\omega</math>
|
|
|
|-
|<math>\varphi(1, \varepsilon_0, 0)</math>
|<math>\widehat{\mathsf{ID}}_{<\varepsilon_0}</math>
|<math>\mathsf{ATR}+(\mathsf{\Sigma}_1^1\mathsf{-DC})</math><ref name="Probst2017" />{{rp|7}}
|<math>\mathsf{KPi}^0 \mathsf{ + F-I}_\omega</math>
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|<math>\varphi(1, \Gamma_0, 0)</math>
|<math>\widehat{\mathsf{ID}}_{<\Gamma_0}</math>
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|<math>\mathsf{MLS}</math>
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|<math>\varphi(2, 0, 0)</math>
|<math>\mathsf{Aut(\widehat{ID})}</math>, <math>\mathsf{FTR}_0</math><ref name="Strahm2000">T. Strahm, "Autonomous fixed point progressions and fixed point transfinite recursion" (2000)</ref>
|<math>Ax_{\Sigma^1_1\mathsf{-AC}}\mathsf{TR}_0</math><ref name="Ruede2002">C. Rüede, "[https://core.ac.uk/download/pdf/85211647.pdf Transfinite dependent choice and ω-model reflection]". Journal of Symbolic Logic vol. 67, no. 3 (2002).</ref><sup>p.1167</sup>, <math>Ax_{\mathsf{ATR}+\Sigma^1_1\mathsf{-DC}}\mathsf{RFN}_0</math><ref name="Ruede2002" /><sup>p.1167</sup>
|<math>\mathsf{KPh}^0</math>
|<math>\mathsf{Aut(ML)}</math>
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|<math>\varphi(2, 0, \varepsilon_0)</math>
|<math>\mathsf{FTR}</math><ref name="Strahm2000" />
|<math>Ax_{\Sigma^1_1\mathsf{-AC}}\mathsf{TR}</math><ref name="Ruede2002" /><sup>p.1167</sup>, <math>Ax_{\mathsf{ATR}+\Sigma^1_1\mathsf{-DC}}\mathsf{RFN}</math><ref name="Ruede2002" /><sup>p.1167</sup>
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|<math>\varphi(2, \varepsilon_0, 0)</math>
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|<math>\mathsf{KPh}_0+(\mathsf{F-I}_\omega)</math><ref name="Strahm2000" />{{rp|11}}
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|<math>\varphi(\omega, 0, 0)</math>
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|<math>(\Pi^1_2\mathsf{-RFN})^{\Sigma^1_1\mathsf{-DC}}_0</math><ref name="Ruede2003">C. Rüede, "[https://www.sciencedirect.com/science/article/pii/S016800720300006X The proof-theoretic analysis of Σ<sup>1</sup><sub>1</sub> transfinite dependent choice]". Annals of Pure and Applied Logic vol. 122 (2003).</ref><sup>p.233</sup>, <math>\Sigma^1_1\mathsf{-TDC}_0</math><ref name="Ruede2003" /><sup>p.233</sup>
|<math>\mathsf{KPm}^0</math><ref name="StrahmMahlo2002">T. Strahm, "[https://doc.rero.ch/record/293367/files/S0022481200009981.pdf Wellordering Proofs for Metapredicative Mahlo]". Journal of Symbolic Logic vol. 67, no. 1 (2002)</ref><sup>p.276</sup>
|<math>\mathsf{EMA}</math><ref name="StrahmMahlo2002" /><sup>p.276</sup>
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|<math>\varphi(\varepsilon_0, 0, 0)</math>
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|<math>(\Pi^1_2\mathsf{-RFN})^{\Sigma^1_1\mathsf{-DC}}</math><ref name="Ruede2003" /><sup>p.233</sup>, <math>\Sigma^1_1\mathsf{-TDC}</math><ref name=Probst2017>D. Probst, "A modular ordinal analysis of metapredicative subsystems of second-order arithmetic" (2017)</ref>
|<math>\mathsf{KPm}^0+(\mathcal{L}^*\mathsf{-I}_\mathsf{N})</math><ref name="StrahmMahlo2002" /><sup>p.277</sup><!--\mathcal{L} should be \mathscr{L}-->
|<math>\mathsf{EMA}+(\mathbb{L}\mathsf{-I}_\mathsf{N})</math><ref name="StrahmMahlo2002" /><sup>p.277</sup>
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|<math>\varphi(1, 0, 0, 0)</math>
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|<math>\mathsf p_1(\Sigma^1_1\mathsf{-TDC}_0)</math><ref name="Probst2017"/>{{rp|7}}
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|-
|<math>\psi_{\Omega_1}(\Omega^{\Omega^\omega})</math>
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|<math>\mathsf{RCA}_0^*+\Pi^1_1\mathsf{-CA}^-</math>,<ref>F. Ranzi, T. Strahm, "A flexible type system for the small Veblen ordinal" (2019). Archive for Mathematical Logic 58: 711–751.</ref> <math>\mathsf p_3(\mathsf{ACA}_0)</math><ref name="Probst2017" />{{rp|7}}
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|<math>\vartheta(\Omega^\Omega)</math>
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|<math>\mathsf p_1(\mathsf p_3(\mathsf{ACA}_0))</math><ref name="Probst2017" />{{rp|7}}
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|<math>\psi_0(\varepsilon_{\Omega+1})</math>{{ref|c}}
|<math>\mathsf{ID}_1</math>
|<math>\text{W-}\widetilde{\mathbf{E}\boldsymbol{\Omega}}</math><ref name="JagerStrahm" /><sup>p. 8</sup>
|<math>\mathsf{KP}</math>,<ref name="Realm" /> <math>\mathsf{KP\omega}</math>, <math>\mathrm{KPu}</math><ref name="Jäger84" /><sup>p. 869</sup>
|<math>\mathsf{ML}_1\mathsf{V}</math>
|<math>\mathsf{CZF}</math>
|<math>\mathsf{EON}</math>
|-
|<math>\psi(\varepsilon_{\Omega+\varepsilon_0})</math><!--psi from Pohlers's ''Proof Theory: An Introduction'', sections 23--24-->
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|<math>\widetilde{\mathbf{E}\boldsymbol{\Omega}}</math><ref name="JagerStrahm" /><sup>p. 31</sup>, <math>\widetilde{\mathbf{EID}}_{\boldsymbol{1}}</math><ref name="JagerStrahm" /><sup>p. 31</sup>, <math>\mathbf{ACA}+(\Pi^1_1\text{-CA})^-</math><ref name="JagerStrahm" /><sup>p. 31</sup>
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|<math>\psi(\varepsilon_{\Omega+\Omega})</math>
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|<math>(\mathsf{ID}^2_1)_0+\mathsf{BR}</math><!--Not sure of notation for BR = bar rule - don't have access to original source--><ref>K. Fujimoto, "Notes on some second-order systems of iterated inductive definitions and <math>\Pi^1_1</math>-comprehensions and relevant subsystems of set theory". Annals of Pure and Applied Logic, vol. 166 (2015), pp. 409--463.</ref>
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|<math>\psi(\varepsilon_{\varepsilon_{\Omega+1}})</math><!--psi from Pohlers's ''Proof Theory: An Introduction'', sections 23--24-->
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|<math>\mathbf{E}\boldsymbol{\Omega}</math><ref name="JagerStrahm" /><sup>p. 33</sup>, <math>\mathbf{EID}_{\boldsymbol{1}}</math><ref name="JagerStrahm" /><sup>p. 33</sup>, <math>\mathbf{ACA}+(\Pi^1_1\text{-CA})^-+(\mathrm{BI}_\mathrm{PR})^-</math><ref name="JagerStrahm" /><sup>p. 33</sup>
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|<math>\psi_0(\Gamma_{\Omega+1})</math>{{ref|d}}
|<math>\mathsf{U(ID}_1\mathsf{)}</math>, <math>\widehat{\mathsf{ID}}^\bullet_{<\omega}</math><ref name="BJS16" /><sup>p. 26</sup>, <math>\Sigma^1_1\mathsf{-DC}^\bullet_0+(\mathsf{SUB}^\bullet)</math><ref name="BJS16" /><sup>p. 26</sup>, <math>\mathsf{ATR}^\bullet_0</math><ref name="BJS16" /><sup>p. 26</sup>, <math>\Sigma^1_1\mathsf{-AC}^\bullet_0+(\mathsf{SUB}^\bullet)</math><ref name="BJS16" /><sup>p. 26</sup>, <math>\mathcal U(\mathsf{ID}_1)</math><ref name="BJS16" /><sup>p. 26</sup>
|<math>\mathsf{FP}^\bullet_0</math><ref name="BJS16" /><sup>p. 26</sup>, <math>\mathsf{ATR}^\bullet_0</math><ref name="BJS16" /><sup>p. 26</sup>
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|<math>\psi_0(\varphi(\mathsf{<}\Omega, 0, \Omega+1))</math>
|<math>\mathsf{Aut(U(ID))}</math>
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|<math>\psi_0(\Omega_\omega)</math>
|<math>\mathsf{ID}_{<\omega}</math><ref name="RathjenFromArithmetic" /><sup>p. 28</sup>
|<math>\mathsf{\Pi}_1^1\mathsf{-CA}_0</math><ref name="RathjenFromArithmetic" /><sup>p. 28</sup>, <math>\mathsf{\Delta}_2^1\mathsf{-CA}_0</math>
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|<math>\mathsf{MLW}</math>
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|<math>\psi_0(\Omega_\omega\omega^\omega)</math>
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|<math>\Pi^1_1\mathsf{-CA}_0+\Pi^1_2\mathsf{-IND}</math><ref name="RathjenKrombholz2019">{{Cite arXiv |eprint=1907.00412 |last1=Krombholz |first1=Martin |last2=Rathjen |first2=Michael |title=Upper bounds on the graph minor theorem |date=2019 |class=math.LO }}</ref>
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|<math>\psi_0(\Omega_\omega\varepsilon_0)</math>
|<math>\mathsf{W-ID}_{\omega}</math>
|<math>\mathsf{\Pi}_1^1\mathsf{-CA}</math><ref>W. Buchholz, S. Feferman, W. Pohlers, W. Sieg, ''Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies''</ref><sup>p. 14</sup>
|<math>\mathsf{W-KPI}</math>
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|<math>\psi_0(\Omega_\omega\Omega)</math>
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|<math>\Pi^1_1\mathsf{-CA+BR}</math><ref>W. Buchholz, ''Proof Theory of Impredicative Subsystems of Analysis (Studies in Proof Theory, Monographs, Vol 2'' (1988)</ref>
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|<math>\psi_0(\Omega_\omega^\omega)</math>
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|<math>\Pi^1_1\mathsf{-CA}_0+\Pi^1_2\mathsf{-BI}</math><ref name="RathjenKrombholz2019" />
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|<math>\psi_0(\Omega_\omega^{\omega^\omega})</math>
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|<math>\Pi^1_1\mathsf{-CA}_0+\Pi^1_2\mathsf{-BI}+\Pi^1_3\mathsf{-IND}</math><ref name="RathjenKrombholz2019" />
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|<math>\psi_0(\varepsilon_{\Omega_\omega+1})</math>{{ref|e}}
|<math>\mathsf{ID}_{\omega}</math>
|<math>\mathsf{\Pi}_1^1\mathsf{-CA+BI}</math>
|<math>\mathsf{KPI}</math>
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|<math>\psi_0(\Omega_{\omega^\omega})</math>
|<math>\mathsf{ID}_{<\omega^\omega}</math>
|<math>\mathsf{\Delta}_2^1\mathsf{-CR}</math><ref name="RathjenFromArithmetic" /><sup>p. 28</sup>
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|<math>\psi_0(\Omega_{\varepsilon_0})</math>
|<math>\mathsf{ID}_{<\varepsilon_0}</math>
|<math>\mathsf{\Delta}_2^1\mathsf{-CA}</math><ref name="RathjenFromArithmetic" /><sup>p. 28</sup>, <math>\mathsf{\Sigma}_2^1\mathsf{-AC}</math>
|<math>\mathsf{W-KPi}</math>
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|<math>\psi_0(\Omega_\Omega)</math>
|<math>\mathsf{Aut(ID)}</math>{{ref|g}}
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|<math>\psi_{\Omega_1}(\varepsilon_{\Omega_\Omega+1})</math>
|<math>\mathsf{ID}_{\prec^*}</math>, <math>\mathsf{BID}^{2*}</math>, <math>\mathsf{ID}^{2*}+\mathsf{BI}</math><ref name="RathjenInvestigations">M. Rathjen, "[https://www1.maths.leeds.ac.uk/~rathjen/Dissertation_Ontos.pdf Investigations of Subsystems of Second Order Arithmetic and Set Theory in Strength between <math>\Pi^1_1\mathsf{-CA}</math> and <math>\Delta^1_2\mathsf{-CA+BI}</math>: Part I]". Accessed 21 September 2023.</ref>
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|<math>\mathsf{KPl}^*</math>, <math>\mathsf{KPl}^r_\Omega</math>
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|<math>\psi_0(\Phi_1(0))</math>
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|<math>\Pi^1_1\mathsf{-TR}_0</math>, <math>\Pi^1_1\mathsf{-TR}_0+\Delta^1_2\mathsf{-CA}_0</math>, <math>\Delta^1_2\mathsf{-CA+BI(impl-}\Sigma^1_2)</math>,<math>\Delta^1_2\mathsf{-CA+BR(impl-}\Sigma^1_2)</math>,<math>\mathbf{AUT-ID}^{pos}_0</math>, <math>\mathbf{AUT-ID}^{mon}_0</math><ref name="RathjenInvestigations" />{{rp|72}}
|<math>\mathsf{KPi}^w+\mathsf{FOUNDR}(\mathsf{impl-})\Sigma)</math>,<ref name="RathjenInvestigations" />{{rp|72}} <math>\mathsf{KPi}^w+\mathsf{FOUND}(\mathsf{impl-})\Sigma)</math>,<ref name="RathjenInvestigations" />{{rp|72}}
<math>\mathbf{AUT-KPl}^r</math>, <math>\mathbf{AUT-KPl}^r+\mathbf{KPi}^r</math><ref name="RathjenInvestigations" />{{rp|72}}
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|<math>\psi_0(\Phi_1(0)\varepsilon_0)</math>
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|<math>\Pi^1_1\mathsf{-TR}</math>, <math>\mathbf{AUT-ID}^{pos}</math>, <math>\mathbf{AUT-ID}^{mon}</math><ref name="RathjenInvestigations" />{{rp|72}}
|<math>\mathbf{AUT-KPl}^w</math><ref name="RathjenInvestigations" />{{rp|72}}
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|<math>\psi_0(\varepsilon_{\Phi_1(0)+1})</math>
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|<math>\Pi^1_1\mathsf{-TR}+(\mathsf{BI})</math>, <math>\mathbf{AUT-ID}^{pos}_2</math>, <math>\mathbf{AUT-ID}^{mon}_2</math><ref name="RathjenInvestigations" />{{rp|72}}
|<math>\mathbf{AUT-KPl}</math><ref name="RathjenInvestigations" />{{rp|72}}
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|<math>\psi_0(\Phi_1(\varepsilon_0))</math>
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|<math>\Pi^1_1\mathsf{-TR}+\Delta^1_2\mathsf{-CA}</math>, <math>\Pi^1_1\mathsf{-TR}+\Sigma^1_2\mathsf{-AC}</math><ref name="RathjenInvestigations" />{{rp|72}}
|<math>\mathbf{AUT-KPl}^w+\mathbf{KPi}^w</math><ref name="RathjenInvestigations" />{{rp|72}}
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|<math>\psi_0(\Phi_\omega(0))</math>
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|<math>\Delta^1_2\mathsf{-TR}_0</math>, <math>\Sigma^1_2\mathsf{-TRDC}_0</math>, <math>\Delta^1_2\mathsf{-CA}_0+(\Sigma^1_2\mathsf{-BI})</math><ref name="RathjenInvestigations" />{{rp|72}}
|<math>\mathbf{KPi}^r+(\Sigma\mathsf{-FOUND})</math>, <math>\mathbf{KPi}^r+(\Sigma\mathsf{-REC})</math><ref name="RathjenInvestigations" />{{rp|72}}
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|<math>\psi_0(\Phi_{\varepsilon_0}(0))</math>
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|<math>\Delta^1_2\mathsf{-TR}</math>, <math>\Sigma^1_2\mathsf{-TRDC}</math>, <math>\Delta^1_2\mathsf{-CA}+(\Sigma^1_2\mathsf{-BI})</math><ref name="RathjenInvestigations" />{{rp|72}}
|<math>\mathbf{KPi}^w+(\Sigma\mathsf{-FOUND})</math>, <math>\mathbf{KPi}^w+(\Sigma\mathsf{-REC})</math><ref name="RathjenInvestigations" />{{rp|72}}
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|<math>\psi(\varepsilon_{I+1})</math>{{ref|h}}
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|<math>\mathsf{\Delta}_2^1\mathsf{-CA+BI}</math><ref name="RathjenFromArithmetic" /><sup>p. 28</sup>, <math>\mathsf{\Sigma}_2^1\mathsf{-AC+BI}</math>
|<math>\mathsf{KPi}</math>
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|<math>\mathsf{CZF+REA}</math>
|<math>\mathsf{T}_0</math>
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|<math>\psi(\Omega_{I+\omega})</math>
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|<math>\mathsf{ML}_1\mathsf{W}</math><ref>M. Rathjen, "[https://www1.maths.leeds.ac.uk/~rathjen/typeOHIO.pdf The Strength of Some Martin-Löf Type Theories]"</ref>{{rp|38}}
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|<math>\psi(\Omega_L)</math>{{ref|i}}
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|<math>\mathsf{KPh}</math>
|<math>\mathsf{ML}_{<\omega}\mathsf{W}</math>
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|<math>\psi(\Omega_{L^*})</math>{{ref|j}}
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|<math>\mathsf{Aut(MLW)}</math>
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|<math>\psi_\Omega(\chi_{\varepsilon_{M+1}}(0))</math>{{ref|k}}
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|<math>\mathsf{\Delta}_2^1\mathsf{-CA+BI+(M)}</math><ref>See conservativity result in {{cite|author=Rathjen|title=The Recursively Mahlo Property in Second Order Arithmetic|url=https://doi.org/10.1002/malq.19960420106|journal=Math. Log. Quart.|volume=42|year=1996}} giving same ordinal as <math>\mathsf{KPM}</math></ref>
|<math>\mathsf{KPM}</math>
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|<math>\mathsf{CZFM}</math>
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|<math>\psi(\Omega_{M+\omega})</math>{{ref|o}}
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|<math>\mathsf{KPM}^+</math><ref name="Setzer96">A. Setzer, "[https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=71a5c4eb40059aa6fda9e8096c7952ce65fa5ac6 A Model for a type theory with Mahlo universe]" (1996).</ref>
|<math>\mathsf{TTM}</math><ref name="Setzer96" />
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|<math>\Psi^0_\Omega(\varepsilon_{K+1})</math>{{ref|l}}
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|<math>\mathsf{KP + \Pi}_3 - \mathsf{Ref}</math><ref>M. Rathjen, "[https://www.sciencedirect.com/science/article/pii/0168007294900744 Proof Theory of Reflection]". Annals of Pure and Applied Logic vol. 68, iss. 2 (1994), pp.181--224.</ref>
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|<math>\Psi^{\varepsilon_{\Xi+1}}_{(\omega^+; P_0, \epsilon, \epsilon, 0)}</math>{{ref|m}}
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|<math>\mathsf{KP + \Pi}_\omega - \mathsf{Ref}</math><ref name="Stegert10">Stegert, Jan-Carl, "[https://d-nb.info/1017849250/34 Ordinal Proof Theory of Kripke-Platek Set Theory Augmented by Strong Reflection Principles]" (2010).</ref>
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|<math>\Psi^{\varepsilon_{\Upsilon+1}}_{(\omega^+; P_0, \epsilon, \epsilon, 0)}</math>{{ref|n}}
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|<math>\mathsf{Stability}</math><ref name="Stegert10" />
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|<math>\psi_{\omega_1^{CK}}(\varepsilon_{\mathbb S^++1})</math><ref name="Arai23A">{{Cite arXiv|last=Arai|first=Toshiyasu|date=2023-04-01|title=Lectures on Ordinal Analysis|class=math.LO |eprint=2304.00246}}</ref>
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|<math>\mathsf{KP}\omega+\Pi_1^1-\mathsf{Ref}</math>,<ref name="Arai23A" /> <math>\mathsf{KP}\omega+(M\prec_{\Sigma_1}V)</math><ref name="Arai23B">{{Cite arXiv|last=Arai|first=Toshiyasu|date=2023-04-07|title=Well-foundedness proof for <math>\Pi_1^1</math>-reflection|class=math.LO |eprint=2304.03851}}</ref>
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|<math>\psi_{\omega_1^{CK}}(\varepsilon_{\mathbb I+1})</math><ref name="Arai23A" />
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|<math>\Sigma_3^1\mathsf{-DC+BI}</math>, <math>\Sigma_3^1\mathsf{-AC+BI}</math>
|<math>\mathsf{KP}\omega+\Pi_1-\mathsf{Collection}+(V=L)</math>
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|<math>\psi_{\omega_1^{CK}}(\varepsilon_{\mathbb I_N+1})</math><ref name="Arai24">{{Cite arXiv|last=Arai|first=Toshiyasu|date=2024-02-12|title=An ordinal analysis of <math>\Pi_N</math>-Collection|class=math.LO |eprint=2311.12459}}</ref>
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|<math>\Sigma_{N+2}^1\mathsf{-DC+BI}</math>, <math>\Sigma_{N+2}^1\mathsf{-AC+BI}</math>
|<math>\mathsf{KP}\omega+\Pi_N-\mathsf{Collection}+(V=L)</math>
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|?
|<math>\mathsf{PA}+\bigcup\limits_{N<\omega}\mathsf{TI}[\Pi_0^{1-},\psi_{\omega_1^{CK}}(\varepsilon_{\mathbb I_N+1})]</math><ref name="Arai24" />
|<math>\mathbf{Z}_2</math>, <math>\Pi^1_\infty - \mathsf{CA}</math>
|<math>\mathsf{KP}+\Pi_\omega^{\text{set}}-\mathsf{Separation}</math>
|<math>\lambda 2</math><ref>Valentin Blot. "[https://inria.hal.science/hal-03698879/document A direct computational interpretation of second-order arithmetic via update recursion]" (2022).</ref>
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=== Key ===
This is a list of symbols used in this table:

* ψ represents various [[Ordinal collapsing function|ordinal collapsing functions]] as defined in their respective citations.
* Ψ represents either Rathjen's or Stegert's Psi.
* φ represents Veblen's function.
* ω represents the first transfinite ordinal.
* ε<sub>α</sub> represents the [[Epsilon numbers (mathematics)|epsilon numbers]].
* Γ<sub>α</sub> represents the gamma numbers (Γ<sub>0</sub> is the [[Feferman–Schütte ordinal]])
* Ω<sub>α</sub> represent the uncountable ordinals (Ω<sub>1</sub>, abbreviated Ω, is [[First uncountable ordinal|ω<sub>1</sub>]]). Countability is considered necessary for an ordinal to be regarded as proof theoretic.
* <math>\mathbb S</math> is an ordinal term denoting a stable ordinal, and <math>\mathbb S^+</math> the least admissible ordinal above <math>\mathbb S</math>.
* <math>\mathbb I_N</math> is an ordinal term denoting an ordinal such that <math>L_{\mathbb{I}_N}\models\mathsf{KP}\omega+\Pi_N-\mathsf{Collection}+(V=L)</math>; N is a variable that defines a series of ordinal analyses of the results of <math>\Pi_N-\mathsf{Collection}</math> forall <math>1 \leq N < \omega</math>. when N=1, <math>\psi_{\omega_1^{CK}}(\varepsilon_{\mathbb I_1+1})=\psi_{\omega_1^{CK}}(\varepsilon_{\mathbb I+1})</math>

This is a list of the abbreviations used in this table:


* First-order arithmetic
===Theories with proof theoretic ordinal ω<sup>3</sup>===
*<math>\mathsf{EFA}</math>, [[ELEMENTARY|elementary function arithmetic]].
** <math>\mathsf{Q}</math> is [[Robinson arithmetic]]
*<math>\mathsf{I}\Delta_0</math> (arithmetic with induction on <math>\Delta_0</math> predicates) augmented by an axiom asserting that exponentiation is total.
** <math>\mathsf{PA}^-</math> is the first-order theory of the nonnegative part of a discretely ordered ring.
** <math>\mathsf{RFA}</math> is [[Jensen hierarchy|rudimentary function]] arithmetic.
** <math>\mathsf{I\Delta}_0</math> is arithmetic with induction restricted to Δ<sub>0</sub>-predicates without any axiom asserting that exponentiation is total.
** <math>\mathsf{EFA}</math> is [[elementary function arithmetic]].
** <math>\mathsf{I\Delta}_0^{\mathsf{+}}</math> is arithmetic with induction restricted to Δ<sub>0</sub>-predicates augmented by an axiom asserting that exponentiation is total.
** <math>\mathsf{EFA}^{\mathsf{n}}</math> is elementary function arithmetic augmented by an axiom ensuring that each element of the ''n''-th level <math>\mathcal{E}^n</math> of the [[Grzegorczyk hierarchy]] is total.
** <math>\mathsf{I\Delta}_0^{\mathsf{n+}}</math> is <math>\mathsf{I\Delta}_0^{\mathsf{+}}</math> augmented by an axiom ensuring that each element of the ''n''-th level <math>\mathcal{E}^n</math> of the [[Grzegorczyk hierarchy]] is total.
** <math>\mathsf{PRA}</math> is [[primitive recursive arithmetic]].
** <math>\mathsf{I\Sigma}_1</math> is arithmetic with induction restricted to Σ<sub>1</sub>-predicates.
** <math>\mathsf{PA}</math> is [[Peano axioms|Peano arithmetic]].
** <math>\mathsf{ID}_\nu\#</math> is <math>\widehat{\mathsf{ID}}_\nu</math> but with induction only for positive formulas.
** <math>\widehat{\mathsf{ID}}_\nu</math> extends PA by ν iterated fixed points of monotone operators.
** <math>\mathsf{U(PA)}</math> is not exactly a first-order arithmetic system, but captures what one can get by predicative reasoning based on the natural numbers.
** <math>\mathsf{Aut(\widehat{ID})}</math> is autonomously iterated <math>\widehat{\mathsf{ID}}_\nu</math> (in other words, once an ordinal is defined, it can be used to index a new series of definitions.)
** <math>\mathsf{ID}_\nu</math> extends PA by ν iterated '''least''' fixed points of monotone operators.
** <math>\mathsf{U(ID}_\nu\mathsf{)}</math> 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.
** <math>\mathsf{Aut(U(ID))}</math> is autonomously iterated <math>\mathsf{U(ID}_\nu\mathsf{)}</math>.
** <math>\mathsf{W-ID}_{\nu}</math> is a weakened version of <math>\mathsf{ID}_{\nu}</math> based on W-types.
** <math>\mathsf{TI}[\Pi_0^{1-},\alpha]</math> is a transfinite induction of length α no more than <math>\Pi_0^1</math>-formulas. It happens to be the representation of the ordinal notation when used in first-order arithmetic.


* Second-order arithmetic
===Theories with proof theoretic ordinal ω<sup>''n''</sup>===
In general, a subscript 0 means that the induction scheme is restricted to a single set induction axiom.
*<math>\mathsf{I}\Delta_0</math> or <math>\mathsf{EFA}</math> augmented by an axiom ensuring that each element of the ''n''th level <math>\mathcal{E}^n</math> of the [[Grzegorczyk hierarchy]] is total.
** <math>\mathsf{RCA}_0^*</math> is a second order form of <math>\mathsf{EFA}</math> sometimes used in [[reverse mathematics]].
** <math>\mathsf{WKL}_0^*</math> is a second order form of <math>\mathsf{EFA}</math> sometimes used in reverse mathematics.
** <math>\mathsf{RCA}_0</math> is [[second-order arithmetic#Recursive comprehension|recursive comprehension]].
** <math>\mathsf{WKL}_0</math> is [[Reverse mathematics#Weak Kőnig's lemma WKL0|weak Kőnig's lemma]].
** <math>\mathsf{ACA}_0</math> is [[Second-order arithmetic#Arithmetical comprehension|arithmetical comprehension]].
** <math>\mathsf{ACA}</math> is <math>\mathsf{ACA}_0</math> plus the full second-order induction scheme.
** <math>\mathsf{ATR}_0</math> is [[Reverse mathematics#Arithmetical transfinite recursion ATR0|arithmetical transfinite recursion]].
** <math>\mathsf{ATR}</math> is <math>\mathsf{ATR}_0</math> plus the full second-order induction scheme.
** <math>\mathsf{\Delta}_2^1\mathsf{-CA+BI+(M)}</math> is <math>\mathsf{\Delta}_2^1\mathsf{-CA+BI}</math> plus the assertion ''"every true <math>\mathsf{\Pi}^1_3</math>-sentence with parameters holds in a (countable coded) <math>\beta</math>-model of <math>\mathsf{\Delta}_2^1\mathsf{-CA}</math>".''


* Kripke-Platek set theory
===Theories with proof theoretic ordinal ω<sup>ω</sup>===
*<math>\mathsf{RCA}_0</math>, [[Recursive Comprehension]].
** <math>\mathsf{KP}</math> is [[Kripke–Platek set theory|Kripke-Platek set theory]] with the axiom of infinity.
*<math>\mathsf{WKL}_0</math>, [[Weak König's lemma]].
** <math>\mathsf{KP\omega}</math> is Kripke-Platek set theory, whose universe is an admissible set containing <math>\omega</math>.
*<math>\mathsf{PRA}</math>, [[primitive recursive arithmetic]].
** <math>\mathsf{W-KPI}</math> is a weakened version of <math>\mathsf{KPI}</math> based on W-types.
*<math>\mathsf{I}\Sigma_1</math> (arithmetic with induction on <math>\Sigma_1</math> predicates).
** <math>\mathsf{KPI}</math> asserts that the universe is a limit of admissible sets.
** <math>\mathsf{W-KPi}</math> is a weakened version of <math>\mathsf{KPi}</math> based on W-types.
** <math>\mathsf{KPi}</math> asserts that the universe is inaccessible sets.
** <math>\mathsf{KPh}</math> asserts that the universe is hyperinaccessible: an inaccessible set and a limit of inaccessible sets.
** <math>\mathsf{KPM}</math> asserts that the universe is a Mahlo set.
** <math>\mathsf{KP + \Pi}_\mathsf{n} - \mathsf{Ref}</math> is <math>\mathsf{KP}</math> augmented by a certain first-order reflection scheme.
** <math>\mathsf{Stability}</math> is KPi augmented by the axiom <math>\forall \alpha \exists \kappa \geq \alpha (L_\kappa \preceq_1 L_{\kappa + \alpha})</math>.
** <math>\mathsf{KPM}^+</math> is KPI augmented by the assertion ''"at least one recursively Mahlo ordinal exists".''
** <math>\mathsf{KP}\omega+(M\prec_{\Sigma_1}V)</math> is <math>\mathsf{KP}\omega</math> with an axiom stating that 'there exists a non-empty and transitive set M such that <math>M\prec_{\Sigma_1}V</math>'.


A superscript zero indicates that <math>\in</math>-induction is removed (making the theory significantly weaker).
===Theories with proof theoretic ordinal &epsilon;<sub>0</sub>===
*[[Gentzen's consistency proof|Gentzen showed]] using [[cut elimination]] that the proof theoretic ordinal of [[Peano arithmetic]] is [[epsilon zero|&epsilon;<sub>0</sub>]].
*<math>\mathsf{ACA}_0</math>, [[Arithmetical comprehension]].


* Type theory
===Theories with proof theoretic ordinal the [[Feferman-Schütte ordinal]] &Gamma;<sub>0</sub>===
** <math>\mathsf{CPRC}</math> is the Herbelin-Patey Calculus of Primitive Recursive Constructions.
This ordinal is sometimes considered to be the upper limit for "predicative" theories.
** <math>\mathsf{ML}_\mathsf{n}</math> is type theory without W-types and with <math>n</math> universes.
** <math>\mathsf{ML}_{<\omega}</math> is type theory without W-types and with finitely many universes.
** <math>\mathsf{MLU}</math> is type theory with a next universe operator.
** <math>\mathsf{MLS}</math> is type theory without W-types and with a superuniverse.
**<math>\mathsf{Aut(ML)}</math> is an automorphism on type theory without W-types.
** <math>\mathsf{ML}_1\mathsf{V}</math> is type theory with one universe and Aczel's type of iterative sets.
** <math>\mathsf{MLW}</math> is type theory with indexed W-Types.
** <math>\mathsf{ML}_1\mathsf{W}</math> is type theory with W-types and one universe.
** <math>\mathsf{ML}_{<\omega}\mathsf{W}</math> is type theory with W-types and finitely many universes.
**<math>\mathsf{Aut(MLW)}</math> is an automorphism on type theory with W-types.
** <math>\mathsf{TTM}</math> is type theory with a Mahlo universe.
** <math>\lambda 2</math> is [[System F]], also polymorphic lambda calculus or second-order lambda calculus.


* Constructive set theory
*<math>\mathsf{ATR}_0</math>, [[Arithmetical Transfinite Recursion]].
** <math>\mathsf{CZF}</math> is Aczel's constructive set theory.
*[[Martin-Löf type theory]].
** <math>\mathsf{CZF+REA}</math> is <math>\mathsf{CZF}</math> plus the regular extension axiom.
** <math>\mathsf{CZF+REA+FZ}_2</math> is <math>\mathsf{CZF+REA}</math> plus the full-second order induction scheme.
** <math>\mathsf{CZFM}</math> is <math>\mathsf{CZF}</math> with a Mahlo universe.


* Explicit mathematics
===Theories with proof theoretic ordinal the [[Bachmann-Howard ordinal]]===
* <math>\mathsf{ID}_1</math>, the theory of [[inductive definition]]s.
** <math>\mathsf{EM}_0</math> is basic explicit mathematics plus elementary comprehension
** <math>\mathsf{EM}_0 \mathsf{+JR}</math> is <math>\mathsf{EM}_0</math> plus join rule
* [[Kripke-Platek set theory]] with the [[axiom of infinity]].
** <math>\mathsf{EM}_0 \mathsf{+J}</math> is <math>\mathsf{EM}_0</math> plus join axioms
* The system CZF of [[constructive set theory]].
** <math>\mathsf{EON}</math> is a weak variant of the [[Solomon Feferman|Feferman]]'s <math>\mathsf{T}_0</math>.
** <math>\mathsf{T}_0</math> is <math>\mathsf{EM}_0 \mathsf{+J+IG}</math>, where <math>\mathsf{IG}</math> is inductive generation.
** <math>\mathsf{T}</math> is <math>\mathsf{EM}_0 \mathsf{+J+IG+FZ}_2</math>, where <math>\mathsf{FZ}_2</math> is the full second-order induction scheme.


== See also ==
===Theories with larger proof theoretic ordinals===
*[[Equiconsistency]]
*<math>\Pi^1_1\mbox{-}\mathsf{CA}_0</math>, [[second order arithmetic|&Pi;<sub>1</sub><sup>1</sup> comprehension]] has a rather large proof theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams", and which is bounded by &psi;<sub>0</sub>(&Omega;<sub>&omega;</sub>) in [[Buchholz's notation]]. It is also the ordinal of <math>ID_{<\omega}</math>, the theory of finitely iterated inductive definitions.
*[[Large cardinal property]]
*KPM, an extension of [[Kripke-Platek set theory]], has a very large proof theoretic ordinal, which was described by {{harvtxt|Rathjen|1990}}.
*[[Feferman–Schütte ordinal]]
*[[Bachmann–Howard ordinal]]
*[[Complexity class]]
*[[Gentzen's consistency proof]]


== Notes ==
Most theories capable of describing the power set of the natural numbers have proof theoretic ordinals
:1.{{note|a}}For <math>1 < n \leq \omega</math>
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.)
:2.{{note|b}}The Veblen function <math>\varphi</math> with countably infinitely iterated least fixed points.{{clarification needed|date=September 2023}}
:3.{{note|c}}Can also be commonly written as <math>\psi(\varepsilon_{\Omega+1})</math> in Madore's ψ.
:4.{{note|d}}Uses Madore's ψ rather than Buchholz's ψ.
:5.{{note|e}}Can also be commonly written as <math>\psi(\varepsilon_{\Omega_\omega+1})</math> in Madore's ψ.
:6.{{note|f}}<math>K</math> represents the first recursively weakly compact ordinal. Uses Arai's ψ rather than Buchholz's ψ.
:7.{{note|g}}Also the proof-theoretic ordinal of <math>\mathsf{Aut(W-ID)}</math>, as the amount of weakening given by the W-types is not enough.
:8.{{note|h}}<math>I</math> represents the first inaccessible cardinal. Uses Jäger's ψ rather than Buchholz's ψ.
:9.{{note|i}}<math>L</math> represents the limit of the <math>\omega</math>-inaccessible cardinals. Uses (presumably) Jäger's ψ.
:10.{{note|j}}<math>L^*</math>represents the limit of the <math>\Omega</math>-inaccessible cardinals. Uses (presumably) Jäger's ψ.
:11.{{note|k}}<math>M</math> represents the first Mahlo cardinal. Uses Rathjen's ψ rather than Buchholz's ψ.
:12.{{note|l}}<math>K</math> represents the first weakly compact cardinal. Uses Rathjen's Ψ rather than Buchholz's ψ.
:13.{{note|m}}<math>\Xi</math> represents the first <math>\Pi^2_0</math>-indescribable cardinal. Uses Stegert's Ψ rather than Buchholz's ψ.
:14.{{note|n}}<math>Y</math> is the smallest <math>\alpha</math> such that <math>\forall \theta < Y \exists \kappa < Y (</math>'<math>\kappa</math> is <math>\theta</math>-indescribable') and <math>\forall \theta < Y \forall \kappa < Y (</math>'<math>\kappa</math> is <math>\theta</math>-indescribable <math>\rightarrow \theta < \kappa</math>'). Uses Stegert's Ψ rather than Buchholz's ψ.
:15.{{note|o}}<math>M</math> represents the first Mahlo cardinal. Uses (presumably) Rathjen's ψ.


==See also==
==Citations==
{{reflist}}
*[[equiconsistency]]
*[[large cardinal property]]


==References==
==References==
{{refbegin}}
*{{citation|last=Buchholz|first= W.|last2= Feferman|first2= S.|last3= Pohlers|first3= W.|last4= Sieg|first4= W. |title=Iterated inductive definitions and sub-systems of analysis|series= Lecture Notes in Math.|volume= 897|publisher= Springer-Verlag|place= Berlin |year=1981|doi=10.1007/BFb0091894
*{{citation|last1= Buchholz|first1= W.|last2= Feferman|first2= S.|last3= Pohlers|first3= W.|last4= Sieg|first4= W.|title= Iterated inductive definitions and sub-systems of analysis|series= Lecture Notes in Math.|volume= 897|publisher= Springer-Verlag|place= Berlin|year= 1981|doi= 10.1007/BFb0091894|isbn= 978-3-540-11170-2|url-access= registration|url= https://archive.org/details/iteratedinductiv0000unse}}
|isbn =978-3-540-11170-2}}
*{{citation|last= Pohlers|first= Wolfram|title= Proof Theory|mr= 1026933|series= Lecture Notes in Mathematics|volume= 1407|publisher= Springer-Verlag|place= Berlin|year= 1989|isbn= 3-540-51842-8|doi= 10.1007/978-3-540-46825-7|url-access= registration|url= https://archive.org/details/prooftheoryintro0000pohl}}
*{{citation|last=Pohlers|first=Wolfram |title=Proof theory|id={{MR|1026933}}
*{{citation|last=Pohlers|first=Wolfram | title=Handbook of Proof Theory|chapter=Set Theory and Second Order Number Theory| pages=210–335| series= Studies in Logic and the Foundations of Mathematics|volume= 137|publisher= Elsevier Science B. V.|place= Amsterdam|year= 1998|mr=1640328|isbn= 0-444-89840-9| doi=10.1016/S0049-237X(98)80019-0}}
|series= Lecture Notes in Mathematics|volume= 1407|publisher= Springer-Verlag|place= Berlin|year= 1989|isbn= 3-540-51842-8 }}
*{{citation|mr=1062729
*{{citation|last=Pohlers|first=Wolfram | title=Handbook of Proof Theory|chapter=Set Theory and Second Order Number Theory
|book = Handbook of Proof Theory| pages=210–335| series= Studies in Logic and the Foundations of Mathematics|volume= 137|publisher= Elsevier Science B. V.|place= Amsterdam|year= 1998|id={{MR|1640328}}|isbn= 0-444-89840-9 }}
*{{citation|id={{MR|1062729}}
|last=Rathjen|first= Michael
|last=Rathjen|first= Michael
|title=Ordinal notations based on a weakly Mahlo cardinal.
|title=Ordinal notations based on a weakly Mahlo cardinal.
|journal=Arch. Math. Logic|volume= 29 |year=1990|issue= 4|pages=249–263|doi=10.1007/BF01651328}}
|journal=Arch. Math. Logic|volume= 29 |year=1990|issue= 4|pages=249–263|doi=10.1007/BF01651328|s2cid=14125063 }}
*{{citation|url=http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf|chapter=The art of ordinal analysis
*{{citation|chapter-url=http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf|chapter=The art of ordinal analysis|first=Michael|last=Rathjen|mr=2275588|title=International Congress of Mathematicians|volume=II|pages=45–69|publisher=Eur. Math. Soc.|place=Zürich|year=2006|url-status=bot: unknown|archiveurl=https://web.archive.org/web/20091222002129/http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf|archivedate=2009-12-22}}
|first=Michael|last= Rathjen
*{{citation| first=H.E. | last= Rose
|id={{MR|2275588}}
|title=International Congress of Mathematicians|volume= II|pages= 45–69|publisher= Eur. Math. Soc.|place= Zürich, |year=2006}}
*{{citation|first=H. | middle=E. | last= Rose
|title=Subrecursion. Functions and Hierarchies |series= Oxford logic guides |volume = 9
|title=Subrecursion. Functions and Hierarchies |series= Oxford logic guides |volume = 9
|publisher= Clarendon Press, Oxford University Press|place= Oxford, New York |year=1984}}
|publisher= Clarendon Press, Oxford University Press|place= Oxford, New York |year=1984}}
*{{citation|id={{MR|0505313}}|last= Schütte|first= Kurt |title=Proof theory|series= Grundlehren der Mathematischen Wissenschaften|volume= 225|publisher= Springer-Verlag|place= Berlin-New York|year= 1977|pages= xii+299 | isbn= 3-540-07911-4}}
*{{citation|mr=0505313|last= Schütte|first= Kurt |title=Proof theory|series= Grundlehren der Mathematischen Wissenschaften|volume= 225|publisher= Springer-Verlag|place= Berlin-New York|year= 1977|pages= xii+299 | isbn= 3-540-07911-4}}
*{{citation|id={{MR|0882549}}|last= Takeuti|first= Gaisi |title=Proof theory|edition= Second |series= Studies in Logic and the Foundations of Mathematics|volume= 81|publisher= North-Holland Publishing Co.|place= Amsterdam|year=1987| isbn= 0-444-87943-9}}
*{{citation|url=https://journals.openedition.org/msh/pdf/2959|last=Setzer|first=Anton|title=Proof theory of Martin-Löf type theory. An Overview|journal= Mathématiques et Sciences Humaines. Mathematics and Social Sciences|issue=165|year=2004|pages=59–99}}
*{{citation|mr=0882549|last= Takeuti|first= Gaisi |title=Proof theory|edition= Second |series= Studies in Logic and the Foundations of Mathematics|volume= 81|publisher= North-Holland Publishing Co.|place= Amsterdam|year=1987| isbn= 0-444-87943-9}}
*{{citation|last= Rathjen|first= Michael|title= Proof theory of Reflection|year= 1994|journal= Annals of Pure and Applied Logic|volume= 68|issue= 2|pages= 181–224|doi= 10.1016/0168-0072(94)90074-4|url= https://dx.doi.org/10.1016/0168-0072%2894%2990074-4}}
*{{citation|last= Stegert|first= Jan-Carl|title= Ordinal Proof Theory of Kripke-Platek Set Theory Augmented by Strong Reflection Principles|year= 2010|url= https://d-nb.info/1017849250/34}}
{{refend}}


[[Category:Proof theory]]
[[Category:Proof theory]]

Latest revision as of 07:44, 25 October 2024

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 functions of the theory.[1]

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 ε0. 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 or way to make arguments about transfinite ordinals. For example, to formalize what it means for a subsystem of Z2 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,[2]p. 3 despite having order type - including such a notation in the ordinal analysis of PA would result in the false equality .

Upper bound

[edit]

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 -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, strictly less than . [2]Theorem 2.21

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 ω2

[edit]
  • RFA, rudimentary function arithmetic.[3]
  • 0, arithmetic with induction on Δ0-predicates without any axiom asserting that exponentiation is total.

Theories with proof-theoretic ordinal ω3

[edit]

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)

[edit]
  • 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 ωω

[edit]

Theories with proof-theoretic ordinal ε0

[edit]

Theories with proof-theoretic ordinal the Feferman–Schütte ordinal Γ0

[edit]

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

Theories with proof-theoretic ordinal the Bachmann–Howard ordinal

[edit]

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]
Unsolved problem in mathematics:
What is the proof-theoretic ordinal of full second-order arithmetic?[4]
  • , Π11 comprehension has a rather large proof-theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams",[5]p. 13 and which is bounded by ψ0ω) 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).
  • IDω, the theory of ω-iterated inductive definitions. Its proof-theoretic ordinal is equal to the Takeuti-Feferman-Buchholz ordinal.
  • T0, 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.[6] 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).
  • TTM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof-theoretic ordinal .
  • has a proof-theoretic ordinal equal to , where refers to the first weakly compact, due to (Rathjen 1993)
  • has a proof-theoretic ordinal equal to , where refers to the first -indescribable and , due to (Stegert 2010).
  • has a proof-theoretic ordinal equal to where is a cardinal analogue of the least ordinal which is -stable for all and , due to (Stegert 2010).

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]
Table of proof-theoretic ordinals
Ordinal First-order arithmetic Second-order arithmetic Kripke-Platek set theory Type theory Constructive set theory Explicit mathematics
,
,
, ,
[1] ,
, [7]p. 13 [7]p. 13, [7]p. 13
[8][7]p. 13 [9]: 40 
[7]p. 13 [7]p. 13, , [7]p. 13, [10]p. 8 [11]p. 869
,[12] [13]: 8 
[14]p. 959
,[15][13] ,[16]: 7  [15]p. 17, [15]p. 5
, [15]p. 52
, [17]
, [18]p. 17, [18]p. 17 [19]p. 140, [19]p. 140, [19]p. 140, [10]p. 8 [11]p. 870
[10]p. 27, [10]p. 27
[20]p.9
[2]
,[21] , [18]p. 22, [18]p. 22, [22] , , ,[23] [24]p. 26 [11]p. 878, [11]p. 878 ,
[25]p.13
[26]
[16]: 7 
[16]: 7 
, [27] [28]p.1167, [28]p.1167
[27] [28]p.1167, [28]p.1167
[27]: 11 
[29]p.233, [29]p.233 [30]p.276 [30]p.276
[29]p.233, [16] [30]p.277 [30]p.277
[16]: 7 
,[31] [16]: 7 
[16]: 7 
[3] [10]p. 8 ,[2] , [11]p. 869
[10]p. 31, [10]p. 31, [10]p. 31
[32]
[10]p. 33, [10]p. 33, [10]p. 33
[4] , [24]p. 26, [24]p. 26, [24]p. 26, [24]p. 26, [24]p. 26 [24]p. 26, [24]p. 26
[4]p. 28 [4]p. 28,
[33]
[34]p. 14
[35]
[33]
[33]
[5]
[4]p. 28
[4]p. 28,
[6]
, , [36] ,
, , ,,, [36]: 72  ,[36]: 72  ,[36]: 72 

, [36]: 72 

, , [36]: 72  [36]: 72 
, , [36]: 72  [36]: 72 
, [36]: 72  [36]: 72 
, , [36]: 72  , [36]: 72 
, , [36]: 72  , [36]: 72 
[7] [4]p. 28,
[37]: 38 
[8]
[9]
[10] [38]
[11] [39] [39]
[12] [40]
[13] [41]
[14] [41]
[42] ,[42] [43]
[42] ,
[44] ,
? [44] , [45]

Key

[edit]

This is a list of symbols used in this table:

  • ψ represents various ordinal collapsing functions as defined in their respective citations.
  • Ψ represents either Rathjen's or Stegert's Psi.
  • φ represents Veblen's function.
  • ω represents the first transfinite ordinal.
  • εα represents the epsilon numbers.
  • Γα represents the gamma numbers (Γ0 is the Feferman–Schütte ordinal)
  • Ωα represent the uncountable ordinals (Ω1, abbreviated Ω, is ω1). Countability is considered necessary for an ordinal to be regarded as proof theoretic.
  • is an ordinal term denoting a stable ordinal, and the least admissible ordinal above .
  • is an ordinal term denoting an ordinal such that ; N is a variable that defines a series of ordinal analyses of the results of forall . when N=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 Δ0-predicates without any axiom asserting that exponentiation is total.
    • is elementary function arithmetic.
    • is arithmetic with induction restricted to Δ0-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 Σ1-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 autonomously iterated (in other words, once an ordinal is defined, it can be used to index a new series of definitions.)
    • 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 autonomously iterated .
    • is a weakened version of based on W-types.
    • is a transfinite induction of length α no more than -formulas. It happens to be the representation of the ordinal notation when used in first-order arithmetic.
  • Second-order arithmetic

In general, a subscript 0 means that the induction scheme is restricted to a single set induction axiom.

    • 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".
    • is with an axiom stating that 'there exists a non-empty and transitive set M such that '.

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.
    • is System F, also polymorphic lambda calculus or second-order lambda calculus.
  • 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]

Notes

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1.^ For
2.^ The Veblen function with countably infinitely iterated least fixed points.[clarification needed]
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

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  1. ^ M. Rathjen, "Admissible Proof Theory and Beyond". In Studies in Logic and the Foundations of Mathematics vol. 134 (1995), pp.123--147.
  2. ^ a b c Rathjen, The Realm of Ordinal Analysis. Accessed 2021 September 29.
  3. ^ 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.
  4. ^ a b c d e f M. Rathjen, Proof Theory: From Arithmetic to Set Theory (p.28). Accessed 14 August 2022.
  5. ^ 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)
  6. ^ D. Madore, A Zoo of Ordinals (2017, p.2). Accessed 12 August 2022.
  7. ^ a b c d e f g J. Avigad, R. Sommer, "A Model-Theoretic Approach to Ordinal Analysis" (1997).
  8. ^ M. Rathjen, W. Carnielli, "Hydrae and subsystems of arithmetic" (1991)
  9. ^ Jeroen Van der Meeren; Rathjen, Michael; Weiermann, Andreas (2014). "An order-theoretic characterization of the Howard-Bachmann-hierarchy". arXiv:1411.4481 [math.LO].
  10. ^ a b c d e f g h i j k G. Jäger, T. Strahm, "Second order theories with ordinals and elementary comprehension".
  11. ^ a b c d e G. Jäger, "The Strength of Admissibility Without Foundation". Journal of Symbolic Logic vol. 49, no. 3 (1984).
  12. ^ B. Afshari, M. Rathjen, "Ordinal Analysis and the Infinite Ramsey Theorem" (2012)
  13. ^ a b Marcone, Alberto; Montalbán, Antonio (2011). "The Veblen functions for computability theorists". The Journal of Symbolic Logic. 76 (2): 575–602. arXiv:0910.5442. doi:10.2178/jsl/1305810765. S2CID 675632.
  14. ^ S. Feferman, "Theories of finite type related to mathematical practice". In Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics vol. 90 (1977), ed. J. Barwise, pub. North Holland.
  15. ^ a b c d M. Heissenbüttel, "Theories of ordinal strength and " (2001)
  16. ^ a b c d e f g D. Probst, "A modular ordinal analysis of metapredicative subsystems of second-order arithmetic" (2017)
  17. ^ A. Cantini, "On the relation between choice and comprehension principles in second order arithmetic", Journal of Symbolic Logic vol. 51 (1986), pp. 360--373.
  18. ^ a b c d Fischer, Martin; Nicolai, Carlo; Pablo Dopico Fernandez (2020). "Nonclassical truth with classical strength. A proof-theoretic analysis of compositional truth over HYPE". arXiv:2007.07188 [math.LO].
  19. ^ a b c S. G. Simpson, "Friedman's Research on Subsystems of Second Order Arithmetic". In Harvey Friedman's Research on the Foundations of Mathematics, Studies in Logic and the Foundations of Mathematics vol. 117 (1985), ed. L. Harrington, M. Morley, A. Šcedrov, S. G. Simpson, pub. North-Holland.
  20. ^ J. Avigad, "An ordinal analysis of admissible set theory using recursion on ordinal notations". Journal of Mathematical Logic vol. 2, no. 1, pp.91--112 (2002).
  21. ^ S. Feferman, "Iterated inductive fixed-point theories: application fo Hancock's conjecture". In Patras Logic Symposion, Studies in Logic and the Foundations of Mathematics vol. 109 (1982).
  22. ^ S. Feferman, T. Strahm, "The unfolding of non-finitist arithmetic", Annals of Pure and Applied Logic vol. 104, no.1--3 (2000), pp.75--96.
  23. ^ S. Feferman, G. Jäger, "Choice principles, the bar rule and autonomously iterated comprehension schemes in analysis", Journal of Symbolic Logic vol. 48, no. (1983), pp.63--70.
  24. ^ a b c d e f g h U. Buchholtz, G. Jäger, T. Strahm, "Theories of proof-theoretic strength ". In Concepts of Proof in Mathematics, Philosophy, and Computer Science (2016), ed. D. Probst, P. Schuster. DOI 10.1515/9781501502620-007.
  25. ^ T. Strahm, "Autonomous fixed point progressions and fixed point transfinite recursion" (2000). In Logic Colloquium '98, ed. S. R. Buss, P. Hájek, and P. Pudlák . DOI 10.1017/9781316756140.031
  26. ^ G. Jäger, T. Strahm, "Fixed point theories and dependent choice". Archive for Mathematical Logic vol. 39 (2000), pp.493--508.
  27. ^ a b c T. Strahm, "Autonomous fixed point progressions and fixed point transfinite recursion" (2000)
  28. ^ a b c d C. Rüede, "Transfinite dependent choice and ω-model reflection". Journal of Symbolic Logic vol. 67, no. 3 (2002).
  29. ^ a b c C. Rüede, "The proof-theoretic analysis of Σ11 transfinite dependent choice". Annals of Pure and Applied Logic vol. 122 (2003).
  30. ^ a b c d T. Strahm, "Wellordering Proofs for Metapredicative Mahlo". Journal of Symbolic Logic vol. 67, no. 1 (2002)
  31. ^ F. Ranzi, T. Strahm, "A flexible type system for the small Veblen ordinal" (2019). Archive for Mathematical Logic 58: 711–751.
  32. ^ K. Fujimoto, "Notes on some second-order systems of iterated inductive definitions and -comprehensions and relevant subsystems of set theory". Annals of Pure and Applied Logic, vol. 166 (2015), pp. 409--463.
  33. ^ a b c Krombholz, Martin; Rathjen, Michael (2019). "Upper bounds on the graph minor theorem". arXiv:1907.00412 [math.LO].
  34. ^ W. Buchholz, S. Feferman, W. Pohlers, W. Sieg, Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies
  35. ^ W. Buchholz, Proof Theory of Impredicative Subsystems of Analysis (Studies in Proof Theory, Monographs, Vol 2 (1988)
  36. ^ a b c d e f g h i j k l m n o M. Rathjen, "Investigations of Subsystems of Second Order Arithmetic and Set Theory in Strength between and : Part I". Accessed 21 September 2023.
  37. ^ M. Rathjen, "The Strength of Some Martin-Löf Type Theories"
  38. ^ See conservativity result in Rathjen (1996), "The Recursively Mahlo Property in Second Order Arithmetic", Math. Log. Quart., 42 giving same ordinal as
  39. ^ a b A. Setzer, "A Model for a type theory with Mahlo universe" (1996).
  40. ^ M. Rathjen, "Proof Theory of Reflection". Annals of Pure and Applied Logic vol. 68, iss. 2 (1994), pp.181--224.
  41. ^ a b Stegert, Jan-Carl, "Ordinal Proof Theory of Kripke-Platek Set Theory Augmented by Strong Reflection Principles" (2010).
  42. ^ a b c Arai, Toshiyasu (2023-04-01). "Lectures on Ordinal Analysis". arXiv:2304.00246 [math.LO].
  43. ^ Arai, Toshiyasu (2023-04-07). "Well-foundedness proof for -reflection". arXiv:2304.03851 [math.LO].
  44. ^ a b Arai, Toshiyasu (2024-02-12). "An ordinal analysis of -Collection". arXiv:2311.12459 [math.LO].
  45. ^ Valentin Blot. "A direct computational interpretation of second-order arithmetic via update recursion" (2022).

References

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