View source for Ordinal analysis

{{more footnotes|date=September 2021}}

{{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==
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 <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>.
===Ordinal notations===
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<sub>2</sub> <math>T</math> 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>.

==Upper bound==
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, less than the [[Church–Kleene ordinal]] <math>\omega_1^{\mathrm{CK}}</math>. <ref name="Realm" /><sup>Theorem 2.21</sup>

==Examples==

===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}}.
*PA<sup>&ndash;</sup>, the first-order theory of the nonnegative part of a discretely ordered ring.

===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>, [[second-order arithmetic#Recursive comprehension|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>, [[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>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 [[Psi0(Omega omega)|&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}}.
*MLM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof-theoretic ordinal &psi;<sub>&Omega;<sub>1</sub></sub>(&Omega;<sub>M + ω</sub>).
*<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>
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|
|
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|-
|<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>
|<math>\mathsf{RCA}_0</math>, <math>\mathsf{WKL}_0</math>
|
|<math>\mathsf{CPRC}</math>
|
|
|-
|<math>\omega^{\omega^{\omega^\omega}}</math>
|
|<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>
|<math>\mathsf{ACA}_0</math>, <math>\mathsf{\Delta}_1^1\mathsf{-CA}_0</math>, <math>\mathsf{\Sigma}_1^1\mathsf{-AC}_0</math>, <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>\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}}
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|-
|<math>\varepsilon_{\varepsilon_0}</math>
|
|<math>\mathsf{ACA}</math>
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|
|
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|-
|<math>\zeta_0</math>
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|<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}}
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|-
|<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" />
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|-
|<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>
|
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|
|<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>, <math>\mathsf{\Sigma}_1^1\mathsf{-AC}</math>, <math>\text{W-}\widehat{\mathbf{E}\boldsymbol{\Omega}}</math><ref name="JagerStrahm" /><sup>p. 8</sup>
|
|<math>\mathsf{ML}_1</math>
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|<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>
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|-
|<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>
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|-
|<math>\varphi(\mathsf{<}\Omega, 0)</math>{{ref|b}}
|<math>\mathsf{Aut(ID\#)}</math>
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|-
|<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}}</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>
|<math>\mathsf{ML}_{<\omega}</math>, <math>\mathsf{MLU}</math>
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|
<!--|- Doesn't seem to have a source
|<math>\Gamma_{\omega}</math>
|
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|<math>\mathsf{KPI}^0 \mathsf{ + \Sigma_1-I}_\omega</math>
|
|
|-->
|-
|<math>\Gamma_{\omega^\omega}</math>
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|<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>
|
|
|<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>\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>
|<math>\mathsf{\Pi}_1^1\mathsf{-CA}_0</math>, <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>
|<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>
|<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>
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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>, <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>, <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>, <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>
|-
|<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>
|<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>
|<math>\mathsf{MLM}</math>
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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>
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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>
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|-
|<math>\psi_{\omega_1^{CK}}(\varepsilon_{\mathbb I+1})</math><ref>{{Cite arXiv|last=Arai|first=Toshiyasu|date=2022-01-10|title=An ordinal analysis of $\Pi_{1}$-Collection|class=math.LO |eprint=2112.09871}}</ref>
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|<math>\Sigma_3^1\mathsf{-DC+BI}</math>
|<math>\mathsf{KP+\Pi}_1\mathsf{-collection}</math>
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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.

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

* First-order arithmetic
** <math>\mathsf{Q}</math> is [[Robinson arithmetic]]
** <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 an [[automorphism]] on <math>\widehat{\mathsf{ID}}_\nu</math>.
** <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 an automorphism on <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.
* Second-order arithmetic
** <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
** <math>\mathsf{KP}</math> is [[Kripke–Platek set theory|Kripke-Platek set theory]] with the axiom of infinity.
** <math>\mathsf{KP\omega}</math> is Kripke-Platek set theory, whose universe is an admissible set containing <math>\omega</math>.
** <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 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".'' 

A superscript zero indicates that <math>\in</math>-induction is removed (making the theory significantly weaker).

* Type theory
** <math>\mathsf{CPRC}</math> is the Herbelin-Patey Calculus of Primitive Recursive Constructions.
** <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{MLM}</math> is type theory with a Mahlo universe.
* Constructive set theory
** <math>\mathsf{CZF}</math> is Aczel's constructive set 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
** <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
** <math>\mathsf{EM}_0 \mathsf{+J}</math> is <math>\mathsf{EM}_0</math> plus join axioms
** <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 ==
*[[Equiconsistency]]
*[[Large cardinal property]]
*[[Feferman–Schütte ordinal]]
*[[Bachmann–Howard ordinal]]
*[[Complexity class]]
*[[Gentzen's consistency proof]]

== Notes ==
:1.{{note|a}}For <math>1 < n \leq \omega</math>
: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.

==Citations==
{{reflist}}

==References==
{{refbegin}}
*{{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}}
*{{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=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}}
*{{citation|mr=1062729
|last=Rathjen|first= Michael
|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|s2cid=14125063 }}
*{{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}}
*{{citation| first=H.E. | last= Rose
|title=Subrecursion. Functions and Hierarchies |series= Oxford logic guides |volume = 9
|publisher=  Clarendon Press, Oxford University Press|place= Oxford, New York |year=1984}}
*{{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|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:Ordinal numbers]]