Ordinal analysis explained

In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.

In addition to obtaining the proof-theoretic ordinal of a theory, in practice ordinal analysis usually also yields various other pieces of information about the theory being analyzed, for example characterizations of the classes of provably recursive, hyperarithmetical, or

\Delta ^{1} _{2} functions of the theory.^[1]

History

The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε₀. See Gentzen's consistency proof.

Definition

Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations.

The proof-theoretic ordinal of such a theory

T is the supremum of the order types of all ordinal notations (necessarily recursive, see next section) that the theory can prove are well founded - the supremum of all ordinals \alpha for which there exists a notation o in Kleene's sense such that T proves that o is an ordinal notation. Equivalently, it is the supremum of all ordinals \alpha such that there exists a recursive relation R on \omega (the set of natural numbers) that well-orders it with ordinal \alpha and such that T proves transfinite induction of arithmetical statements for R .

Ordinal notations

Some theories, such as subsystems of second-order arithmetic, have no conceptualization or way to make arguments about transfinite ordinals. For example, to formalize what it means for a subsystem of Z₂

T to "prove \alpha well-ordered", we instead construct an ordinal notation (A,\tilde <) with order type \alpha . T can now work with various transfinite induction principles along (A,\tilde <) , 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

( N,< _{T)} that is well-founded iff PA is consistent,^{[2] p. 3} despite having order type \omega - including such a notation in the ordinal analysis of PA would result in the false equality PTO(PA) = \omega .

Upper bound

\omega ^{CK} _{1} . In particular, the proof-theoretic ordinal of an inconsistent theory is equal to \omega ^{CK} _{1} , because an inconsistent theory trivially proves that all ordinal notations are well-founded.

For any theory that's both

\Sigma ^{1} _{1} -axiomatizable and \Pi ^{1} _{1} -sound, the existence of a recursive ordering that the theory fails to prove is well-ordered follows from the \Sigma ^{1} _{1} bounding theorem, and said provably well-founded ordinal notations are in fact well-founded by \Pi ^{1} _{1} -soundness. Thus the proof-theoretic ordinal of a \Pi ^{1} _{1} -sound theory that has a \Sigma ^{1} _{1} axiomatization will always be a (countable) recursive ordinal, that is, strictly less than \omega ^{CK} _{1} . ^{Theorem 2.21}

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).
• PA ^- , the first-order theory of the nonnegative part of a discretely ordered ring.

Theories with proof-theoretic ordinal ω²

• RFA, rudimentary function arithmetic.^[3]
• IΔ₀, arithmetic with induction on Δ₀-predicates without any axiom asserting that exponentiation is total.

Theories with proof-theoretic ordinal ω³

• EFA, elementary function arithmetic.
• IΔ₀ + exp, arithmetic with induction on Δ₀-predicates augmented by an axiom asserting that exponentiation is total.
• RCA, a second order form of EFA sometimes used in reverse mathematics.
• WKL, 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 ωⁿ (for n = 2, 3, ... ω)

• IΔ₀ or EFA augmented by an axiom ensuring that each element of the n-th level

l{E} ^{n} of the Grzegorczyk hierarchy is total.

Theories with proof-theoretic ordinal ω^ω

• RCA₀, recursive comprehension.
• WKL₀, weak Kőnig's lemma.
• PRA, primitive recursive arithmetic.
• IΣ₁, arithmetic with induction on Σ₁-predicates.

Theories with proof-theoretic ordinal ε₀

• PA, Peano arithmetic (shown by Gentzen using cut elimination).
• ACA₀, arithmetical comprehension.

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

• ATR₀, 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

• ID₁, the first theory of inductive definitions.
• KP, Kripke–Platek set theory with the axiom of infinity.
• CZF, Aczel's constructive Zermelo–Fraenkel set theory.
• EON, a weak variant of the Feferman's explicit mathematics system T₀.

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

\Pi ^{1} _{1 - CA} _{0} , Π₁¹ comprehension has a rather large proof-theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams",^{p. 13} and which is bounded by ψ₀(Ω_ω) in Buchholz's notation. It is also the ordinal of ID _{<\omega} , the theory of finitely iterated inductive definitions. And also the ordinal of MLW, Martin-Löf type theory with indexed W-Types .

• ID_ω, 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

\Sigma ^{1} _{2 - AC} + BI .

• KPi, an extension of Kripke–Platek set theory based on a recursively inaccessible ordinal, has a very large proof-theoretic ordinal

\psi(\varepsilon _{I} ) described in a 1983 paper of Jäger and Pohlers, where I is the smallest inaccessible.^[4] This ordinal is also the proof-theoretic ordinal of \Delta ^{1} _{2 - CA} + BI .

• 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 .
• TTM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof-theoretic ordinal ψ_Ω1(Ω_{M + ω}).

KP + \Pi _{3} - Ref has a proof-theoretic ordinal equal to \Psi(\varepsilon _{K} ) , where K refers to the first weakly compact, due to (Rathjen 1993)KP + \Pi _{\omega} - Ref has a proof-theoretic ordinal equal to \Psi ^{\varepsilon _{\Xi}} _{X} , where \Xi refers to the first \Pi ^{2} _{0} -indescribable and X = (\omega ^{+;} P _{0;} \epsilon, \epsilon, 0 ) , due to (Stegert 2010).Stability has a proof-theoretic ordinal equal to \Psi ^{\varepsilon _{\Upsilon+1}} _{X} where \Upsilon is a cardinal analogue of the least ordinal \alpha which is \alpha + \beta -stable for all \beta < \alpha and X = (\omega ^{+;} P _{0;} \epsilon, \epsilon, 0 ) , 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

\Pi ^{1} _{2} - CA _{0} , full second-order arithmetic (\Pi ^{1} _{inf ty} - CA _{0} ) and set theories with powersets including ZF and ZFC. The strength of intuitionistic ZF (IZF) equals that of ZF.

Table of ordinal analyses

Table of proof-theoretic ordinals!Ordinal!First-order arithmetic!Second-order arithmetic!Kripke-Platek set theory!Type theory!Constructive set theory!Explicit mathematics

\omega

Q , PA ^{-}

\omega ^{2}

RFA , I\Delta _{0}

\omega ^{3}

EFA , I\Delta ^{+} _{0}

RCA ^{*} _{0} , WKL ^{*} _{0}

\omega ^{n}

EFA ^{n} , I\Delta ^{n+} _{0}

\omega ^{\omega}

PRA , I\Sigma _{1} ^[5] ^p. 13

RCA _{0} ^p. 13, WKL _{0} ^p. 13

CPRC

\omega ^{\omega ^{\omega ^{\omega}}}

I\Sigma _{3} ^[6] ^p. 13

RCA ^{- -IND} _{2 )} ^[7]

\varepsilon _{0}

PA ^p. 13

ACA _{0} ^p. 13, \Delta ^{1 -CA} _{0} , \Sigma ^{1 -AC} _{0} ^p. 13, R- \widehat{ E \boldsymbol{\Omega}} ^[8] ^p. 8

KPu ^{r} ^p. 869

EM _{0}

\varepsilon _{\omega}

ACA _{0+iRT} ,^[9] RCA _{0+\forall} Y\forall n\exists X( rm{TJ}(n,X,Y))

\varepsilon _{\varepsilon _{0}}

ACA ^[10] ^p. 959

\zeta _{0}

ACA _{0+\forall} X\exists Y( rm{TJ}(\omega,X,Y)) ,^[11] ^[12] p _{1(ACA} _{0)} , RFN _{0} ^p. 17, ACA _{0+(BR)} ^p. 5

\varphi( 2 ,\varepsilon _{0)}

RFN , ACA + \forall X\exists Y( rm{TJ}(\omega,X,Y)) ^p. 52

\varphi(\omega, 0 )

ID _{1\#}

\Delta ^{1 -CR} _{1} , \Sigma ^{1} _{1 -DC} _{0} ^[13]

EM _{0} +JR

\varphi(\varepsilon _{0,} 0 )

\widehat{ ID }_1, KFL ^p. 17, KF ^p. 17

\Delta ^{1 -CA} _{1} ^[14] ^p. 140, \Sigma ^{1 -AC} _{1} ^p. 140, \Sigma ^{1 -DC} _{1} ^p. 140, W- \widehat{ E \boldsymbol{\Omega}} ^p. 8

KPu ^{r + ( IND} _{N)} ^p. 870

ML _{1}

EM _{0} +J

\varphi(\varepsilon _{\varepsilon _{0}} , 0 )

\widehat{ E \boldsymbol{\Omega}} ^p. 27, \widehat{ EID }_^p. 27

\varphi(\varphi(\omega, 0 ), 0 )

PRS \omega ^[15] ^p.9

\varphi( < \Omega, 0 )

Aut(ID\#)

\Gamma _{0}

\widehat{ ID }_,^[16] U(PA) , KFL ^{*} ^[17] ^p. 22, KF ^{*} ^p. 22, l U( NFA ) ^[18]

ATR _{0} , \Delta ^{1 -CA+BR} _{1} , \Delta ^{1} _{1 -CA} _{0+(SUB)} ,^[19] FP _{0} ^[20] ^p. 26

KPi ^{0} ^[21] ^p. 878, KPu ^{0+(BR)} ^p. 878

ML _{<\omega} , MLU

- Doesn't seem to have a source

\Gamma _{\omega}

KPI ^{0} +\Sigma _{1-I} _{\omega}

\Gamma _{\omega ^{\omega}}

KPI ^{0} + ( \Sigma _{1-I} _{\omega)} ^[22] ^p.13

\Gamma _{\varepsilon _{0}}

\widehat{ ID }_

ATR ^[23]

KPI ^{0} +F-I _{\omega}

\varphi( 1 , \omega, 0 )

\widehat{ ID }_

ATR _{0+(\Sigma} ^{1 -DC )} ^{1}

KPi ^{0} +\Sigma _{1-I} _{\omega}

\varphi( 1 , \varepsilon _{0,} 0 )

\widehat{ ID }_

ATR + ( \Sigma ^{1 -DC )} _{1}

KPi ^{0} +F-I _{\omega}

\varphi( 1 , \Gamma _{0,} 0 )

\widehat{ ID }_

MLS

\varphi( 2 , 0 , 0 )

Aut(\widehat{ID )} , FTR _{0} ^[24]

Ax _{\Sigma ^{1} _{1 -AC}} TR _{0} ^[25] ^p.1167, Ax _{ATR + \Sigma ^{1} _{1 -DC}} RFN _{0} ^p.1167

KPh ^{0}

Aut(ML)

\varphi( 2 , 0 , \varepsilon _{0)}

FTR

Ax _{\Sigma ^{1} _{1 -AC}} TR ^p.1167, Ax _{ATR + \Sigma ^{1} _{1 -DC}} RFN ^p.1167

\varphi( 2 , \varepsilon _{0,} 0 )

KPh _{0+(F-I} _{\omega)}

\varphi(\omega, 0 , 0 )

(\Pi ^{\Sigma ^{1} _{1 -DC}} _{0} ^[26] ^p.233, \Sigma ^{1} _{1 -TDC} _{0} ^p.233

KPm ^{0} ^[27] ^p.276

EMA ^p.276

\varphi(\varepsilon _{0,} 0 , 0 )

(\Pi ^{\Sigma ^{1} _{1 -DC}} _{2 -RFN )} ^p.233, \Sigma ^{1} _{1 -TDC} ^[28]

KPm ^{0+(l{L}} ^{* -I} ^{N )} ^p.277

EMA + ( L -I _{N)} ^p.277

\varphi( 1 , 0 , 0 , 0 )

p ^{1} _{1 -TDC} _{0)}

\psi _{\Omega _{1}} (\Omega ^{\Omega ^{\omega}} )

RCA ^{* + \Pi} _{0} ^{-} ^{1 -CA} ,^[29] p _{3(ACA} _{0)}

\vartheta(\Omega ^{\Omega)}

p _{1(p} _{3(ACA} _{0))}

\psi _{0(\varepsilon} _{\Omega+1} )

ID _{1}

W- \widetilde{ E \boldsymbol{\Omega}} ^p. 8

KP , KP\omega , KPu ^p. 869

ML _{1V}

CZF

EON

\psi(\varepsilon _{\Omega + \varepsilon _{0}} )

\widetilde{ E \boldsymbol{\Omega}} ^p. 31, \widetilde{ EID }_^p. 31, ACA + (\Pi ^{-} _{1 -CA )} ^p. 31

\psi(\varepsilon _{\Omega+\Omega} )

( ID ^{2} _{1 )} _{0+BR} ^[30]

\psi(\varepsilon _{\varepsilon _{\Omega+1}} )

E \boldsymbol{\Omega} ^p. 33, EID _{\boldsymbol{1} }^p. 33, ACA + (\Pi ^{-} _{PR )} ^p. 33

\psi _{0(\Gamma} _{\Omega+1} )

U(ID _{1)} , \widehat{ ID }^\bullet_^p. 26, \Sigma ^{\bullet)} _{0+ ( SUB} ^p. 26, ATR ^{\bullet} _{0} ^p. 26, \Sigma ^{\bullet)} _{0+ ( SUB} ^p. 26, l U( ID _{1)} ^p. 26

FP ^{\bullet} _{0} ^p. 26, ATR ^{\bullet} _{0} ^p. 26

\psi _{0(\varphi(<\Omega,} 0 , \Omega +1 ))

Aut(U(ID))

\psi _{0(\Omega} _{\omega)}

ID _{<\omega} ^p. 28

\Pi ^{1 -CA} _{0} ^p. 28, \Delta ^{1 -CA} _{0}

MLW

\psi _{0(\Omega} ^{\omega)} ^{\omega\omega}

\Pi ^{1} _{1 -CA} ^{1} ^{2 -IND} ^[31]

\psi _{0(\Omega} _{\omega\varepsilon} _{0)}

W-ID _{\omega}

\Pi ^{1 -CA} _{1} ^[32] ^p. 14

W-KPI

\psi _{0(\Omega} _{\omega\Omega)}

\Pi ^{1} _{1 -CA+BR} ^[33]

\psi _{0(\Omega} ^{\omega)} ^{\omega}

\Pi ^{1} _{1 -CA} ^{1} ^{2 -BI}

\psi _{0(\Omega} ^{\omega ^{\omega}} ^{\omega} )

\Pi ^{1} _{1 -CA} ^{1} ^{3 -IND}

\psi _{0(\varepsilon} ^{\Omega _{\omega+1}} )

ID _{\omega}

\Pi ^{1 -CA+BI} _{1}

KPI

\psi _{0(\Omega} ^{\omega ^{\omega}} )

ID _{<\omega ^{\omega}}

\Delta ^{1 -CR} _{2} ^p. 28

\psi _{0(\Omega} ^{\varepsilon _{0}} )

ID _{<\varepsilon _{0}}

\Delta ^{1 -CA} _{2} ^p. 28, \Sigma ^{1 -AC} _{2}

W-KPi

\psi _{0(\Omega} _{\Omega)}

Aut(ID)

\psi _{\Omega _{1}} (\varepsilon _{\Omega _{\Omega+1}} )

ID _{\prec ^{*}} , BID ^{2*} , ID ^{2*} + BI ^[34]

KPl ^{*} , KPl ^{r} _{\Omega}

\psi _{0(\Phi} _{1(0))}

\Pi ^{1} _{1 -TR} _{0} , \Pi ^{1} _{1 -TR} ^{1} ^{2 -CA} _{0} , \Delta ^{1} _{2 )} ,\Delta ^{1} _{2 )} , AUT-ID ^{pos} _{0} , AUT-ID ^{mon} _{0}

KPi ^{w+FOUNDR(impl-)\Sigma)} , KPi ^{w+FOUND(impl-)\Sigma)}

AUT-KPl ^{r} , AUT-KPl ^{r+KPi} ^{r}

\psi _{0(\Phi} _{1(0)\varepsilon} _{0)}

\Pi ^{1} _{1 -TR} , AUT-ID ^{pos} , AUT-ID ^{mon}

AUT-KPl ^{w}

\psi _{0(\varepsilon} ^{\Phi _{1(0)+1}} )

\Pi ^{1} _{1 -TR + ( BI )} , AUT-ID ^{pos} _{2} , AUT-ID ^{mon} _{2}

AUT-KPl

\psi _{0(\Phi} _{1(\varepsilon} _{0))}

\Pi ^{1} _{2 -CA} , \Pi ^{1} _{2 -AC} , AUT-KPl ^{w+KPi} ^{w}

\psi _{0(\Phi} _{\omega(0))}

\Delta ^{1} _{2 -TR} _{0} , \Sigma ^{1} _{2 -TRDC} _{0} , \Delta ^{1} _{2 -CA} ^{1} ^{2 -BI )}

KPi ^{r+(\Sigma-FOUND)} , KPi ^{r+(\Sigma-REC)}

\psi _{0(\Phi} ^{\varepsilon _{0}} ( 0 ))

\Delta ^{1} _{2 -TR} , \Sigma ^{1} _{2 -TRDC} , \Delta ^{1} _{2 -BI )}

KPi ^{w+(\Sigma-FOUND)} , KPi ^{w+(\Sigma-REC)}

\psi(\varepsilon _{I+1} )

\Delta ^{1 -CA+BI} _{2} ^p. 28, \Sigma ^{1 -AC+BI} _{2}

KPi

CZF+REA

T _{0}

\psi(\Omega _{I+\omega} )

ML _{1W} ^[35]

\psi(\Omega _{L)}

KPh

ML _{<\omega} W

\psi(\Omega _{L ^{*}} )

Aut(MLW)

\psi _{\Omega(\chi} ^{\varepsilon _{M+1}} ( 0 ))

\Delta ^{1 -CA+BI+(M)} _{2}

KPM

CZFM

\psi(\Omega _{M+\omega} )

KPM ^{+} ^[36]

TTM

\Psi ^{0} _{\Omega(\varepsilon} _{K+1} )

KP+\Pi _{3} - Ref ^[37]

\Psi ^{\varepsilon _{\Xi+1}} _{(\omega ^{+;} P _{0,} \epsilon, \epsilon, 0 )}

KP+\Pi _{\omega} - Ref ^[38]

\Psi ^{\varepsilon _{\Upsilon+1}} _{(\omega ^{+;} P _{0,} \epsilon, \epsilon, 0 )}

Stability

\psi _{\omega ^{CK} _{1}} (\varepsilon _{S ^{++1}} ) ^[39]

KP \omega + \Pi ^{1- Ref} _{1} , KP \omega + (M\prec _{\Sigma _{1}} V) ^[40]

\psi _{\omega ^{CK} _{1}} (\varepsilon _{I+1} )

\Sigma ^{1 -DC+BI} _{3} , \Sigma ^{1 -AC+BI} _{3}

KP \omega + \Pi _{1-Collection+(V=L)}

\psi _{\omega ^{CK} _{1}} (\varepsilon _{I _{N+1}} ) ^[41]

\Sigma ^{1 -DC+BI} _{N +2} , \Sigma ^{1 -AC+BI} _{N +2}

KP \omega + \Pi _{N-Collection+(V=L)}

?

PA + c up\limits _{N<\omega} TI [\Pi ^{1-} _{0} ,\psi _{\omega ^{CK} _{1}} (\varepsilon _{I _{N+1}} )]

Z _{2} , \Pi ^{1} _{inf ty} - CA

KP + \Pi ^{set} _{\omega} - Separation

λ 2 ^[42]

Key

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.
• ε_α represents the epsilon numbers.
• Γ_α represents the gamma numbers (Γ₀ is the Feferman–Schütte ordinal)
• Ω_α represent the uncountable ordinals (Ω₁, abbreviated Ω, is ω₁). Countability is considered necessary for an ordinal to be regarded as proof theoretic.

S is an ordinal term denoting a stable ordinal, and S ^{+} the least admissible ordinal above S .I _{N} is an ordinal term denoting an ordinal such that L _{I _{N}} \models KP \omega + \Pi _{N-Collection+(V=L)} . N is a variable that defines a series of ordinal analyses of the results of \Pi _{N-Collection} forall 1 \leq N < \omega . when N=1, \psi _{\omega ^{CK} _{1}} (\varepsilon _{I _{1+1}} ) = \psi _{\omega ^{CK} _{1}} (\varepsilon _{I+1} )

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

• First-order arithmetic

Q is Robinson arithmeticPA ^{-} is the first-order theory of the nonnegative part of a discretely ordered ring.RFA is rudimentary function arithmetic.I\Delta _{0} is arithmetic with induction restricted to Δ₀-predicates without any axiom asserting that exponentiation is total.EFA is elementary function arithmetic.I\Delta ^{+} _{0} is arithmetic with induction restricted to Δ₀-predicates augmented by an axiom asserting that exponentiation is total.EFA ^{n} is elementary function arithmetic augmented by an axiom ensuring that each element of the n-th level l{E} ^{n} of the Grzegorczyk hierarchy is total.I\Delta ^{n+} _{0} is I\Delta ^{+} _{0} augmented by an axiom ensuring that each element of the n-th level l{E} ^{n} of the Grzegorczyk hierarchy is total.PRA is primitive recursive arithmetic.I\Sigma _{1} is arithmetic with induction restricted to Σ₁-predicates.PA is Peano arithmetic.ID _{\nu\#} is \widehat{ ID }_\nu but with induction only for positive formulas.\widehat{ ID }_\nu extends PA by ν iterated fixed points of monotone operators.U(PA) is not exactly a first-order arithmetic system, but captures what one can get by predicative reasoning based on the natural numbers.Aut(\widehat{ID )} is an automorphism on \widehat{ ID }_\nu.ID _{\nu} extends PA by ν iterated least fixed points of monotone operators.U(ID _{\nu)} 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.Aut(U(ID)) is an automorphism on U(ID _{\nu)} .W-ID _{\nu} is a weakened version of ID _{\nu} based on W-types.TI [\Pi ^{1-} _{0} ,\alpha] is a transfinite induction of length α no more than \Pi ^{1} _{0} -formulas. It happens to be the representation of the ordinal notation when used in first-order arithmetic.

• Second-order arithmetic

RCA ^{*} _{0} is a second order form of EFA sometimes used in reverse mathematics.WKL ^{*} _{0} is a second order form of EFA sometimes used in reverse mathematics.RCA _{0} is recursive comprehension.WKL _{0} is weak Kőnig's lemma.ACA _{0} is arithmetical comprehension.ACA is ACA _{0} plus the full second-order induction scheme.ATR _{0} is arithmetical transfinite recursion.ATR is ATR _{0} plus the full second-order induction scheme.\Delta ^{1 -CA+BI+(M)} _{2} is \Delta ^{1 -CA+BI} _{2} plus the assertion "every true \Pi ^{1} _{3} -sentence with parameters holds in a (countable coded) \beta -model of \Delta ^{1 -CA} _{2} ".

• Kripke-Platek set theory

KP is Kripke-Platek set theory with the axiom of infinity.KP\omega is Kripke-Platek set theory, whose universe is an admissible set containing \omega .W-KPI is a weakened version of KPI based on W-types.KPI asserts that the universe is a limit of admissible sets.W-KPi is a weakened version of KPi based on W-types.KPi asserts that the universe is inaccessible sets.KPh asserts that the universe is hyperinaccessible: an inaccessible set and a limit of inaccessible sets.KPM asserts that the universe is a Mahlo set.KP+\Pi _{n} - Ref is KP augmented by a certain first-order reflection scheme.Stability is KPi augmented by the axiom \forall \alpha \exists \kappa \geq \alpha (L _{\kappa} \preceq _{1} L _{\kappa} ) .KPM ^{+} is KPI augmented by the assertion "at least one recursively Mahlo ordinal exists". KP \omega + (M\prec _{\Sigma _{1}} V) is KP \omega with an axiom stating that 'there exists an non-empty and transitive set M such that M\prec _{\Sigma _{1}} V '.

A superscript zero indicates that

\in -induction is removed (making the theory significantly weaker).

• Type theory

CPRC is the Herbelin-Patey Calculus of Primitive Recursive Constructions.ML _{n} is type theory without W-types and with n universes.ML _{<\omega} is type theory without W-types and with finitely many universes.MLU is type theory with a next universe operator.MLS is type theory without W-types and with a superuniverse.Aut(ML) is an automorphism on type theory without W-types.ML _{1V} is type theory with one universe and Aczel's type of iterative sets.MLW is type theory with indexed W-Types.ML _{1W} is type theory with W-types and one universe.ML _{<\omega} W is type theory with W-types and finitely many universes.Aut(MLW) is an automorphism on type theory with W-types.TTM is type theory with a Mahlo universe.λ 2 is System F, also polymorphic lambda calculus or second-order lambda calculus.

• Constructive set theory

CZF is Aczel's constructive set theory.CZF+REA is CZF plus the regular extension axiom.CZF+REA+FZ _{2} is CZF+REA plus the full-second order induction scheme.CZFM is CZF with a Mahlo universe.

• Explicit mathematics

EM _{0} is basic explicit mathematics plus elementary comprehensionEM _{0} +JR is EM _{0} plus join ruleEM _{0} +J is EM _{0} plus join axiomsEON is a weak variant of the Feferman's T _{0} .T _{0} is EM _{0} +J+IG , where IG is inductive generation.T is EM _{0} +J+IG+FZ _{2} , where FZ _{2} 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.For

1 < n \leq \omega

2.The Veblen function

\varphi with countably infinitely iterated least fixed points.

3.Can also be commonly written as

\psi(\varepsilon _{\Omega+1} ) in Madore's ψ.

4.Uses Madore's ψ rather than Buchholz's ψ.

5.Can also be commonly written as

\psi(\varepsilon _{\Omega _{\omega+1}} ) in Madore's ψ.

6.

K represents the first recursively weakly compact ordinal. Uses Arai's ψ rather than Buchholz's ψ.

7.Also the proof-theoretic ordinal of

Aut(W-ID) , as the amount of weakening given by the W-types is not enough.

8.

I represents the first inaccessible cardinal. Uses Jäger's ψ rather than Buchholz's ψ.

9.

L represents the limit of the \omega -inaccessible cardinals. Uses (presumably) Jäger's ψ.

10.

L ^{*} represents the limit of the \Omega -inaccessible cardinals. Uses (presumably) Jäger's ψ.

11.

M represents the first Mahlo cardinal. Uses Rathjen's ψ rather than Buchholz's ψ.

12.

K represents the first weakly compact cardinal. Uses Rathjen's Ψ rather than Buchholz's ψ.

13.

\Xi represents the first \Pi ^{2} _{0} -indescribable cardinal. Uses Stegert's Ψ rather than Buchholz's ψ.

14.

Y is the smallest \alpha such that \forall \theta < Y \exists \kappa < Y ( '\kappa is \theta -indescribable') and \forall \theta < Y \forall \kappa < Y ( '\kappa is \theta -indescribable → \theta < \kappa '). Uses Stegert's Ψ rather than Buchholz's ψ.

15.

M represents the first Mahlo cardinal. Uses (presumably) Rathjen's ψ.

Notes and References

1. M. Rathjen, "Admissible Proof Theory and Beyond". In Studies in Logic and the Foundations of Mathematics vol. 134 (1995), pp.123--147.
2. Rathjen, The Realm of Ordinal Analysis. Accessed 2021 September 29.
3. Book: Krajicek, Jan. Bounded Arithmetic, Propositional Logic and Complexity Theory. 1995. Cambridge University Press. 9780521452052. 18–20. 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 Book: Rose, H. E.. Subrecursion: functions and hierarchies. 1984. Clarendon Press. University of Michigan. 9780198531890.
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