Ordinal analysis

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 ${\mathrm{\Delta }}_2^1$ functions of the theory.

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 (Z₂), have no conceptualization or way to make arguments about transfinite ordinals. For example, to formalize what it means for a subsystem $T$ of Z₂ to "prove $\alpha$ well-ordered", we instead construct an ordinal notation $(A,\widetilde{<})$ with order type $\alpha$. $T$ can now work with various transfinite induction principles along $(A,\widetilde{<})$, 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 $(\mathbb{N},{<}_T)$ that is well-founded if and only if PA is consistent,^p. 3 despite having order type $\omega$. Including such a notation in the ordinal analysis of PA would result in the false equality $𝖯𝖳𝖮\mathsf{(}𝖯𝖠\mathsf{)}=\omega$.

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 ${\omega }_1^{\mathrm{C}\mathrm{K}}$. In particular, the proof-theoretic ordinal of an inconsistent theory is equal to ${\omega }_1^{\mathrm{C}\mathrm{K}}$, because an inconsistent theory trivially proves that all ordinal notations are well-founded.

For any theory that's both ${\mathrm{\Sigma }}_1^1$-axiomatizable and ${\mathrm{\Pi }}_1^1$-sound, the existence of a recursive ordering that the theory fails to prove is well-ordered follows from the ${\mathrm{\Sigma }}_1^1$ bounding theorem, and said provably well-founded ordinal notations are in fact well-founded by ${\mathrm{\Pi }}_1^1$-soundness. Thus the proof-theoretic ordinal of a ${\mathrm{\Pi }}_1^1$-sound theory that has a ${\mathrm{\Sigma }}_1^1$ axiomatization will always be a (countable) recursive ordinal, that is, strictly less than ${\omega }_1^{\mathrm{C}\mathrm{K}}$.^{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)Template:Citation needed.
• PA^–, the first-order theory of the nonnegative part of a discretely ordered ring.

Theories with proof-theoretic ordinal ω²

• RFA, rudimentary function arithmetic.
• 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.
• RCATemplate:Su, a second-order form of EFA sometimes used in reverse mathematics.
• WKLTemplate:Su, 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 ${ℰ}^n$ 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

• ${\mathrm{\Pi }}_1^1\text{-}{𝖢𝖠}_0$, Π₁¹ comprehension has a rather large proof-theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams",Template:Sfnp and which is bounded by ψ₀(Ω_ω) in Buchholz's notation. It is also the ordinal of ID_ω, the theory of finitely iterated inductive definitions. And also the ordinal of MLW, Martin-Löf type theory with indexed W-Types.Template:Sfnp
• 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 ${\mathrm{\Sigma }}_2^1\text{-}𝖠𝖢+𝖡𝖨$.
• KPi, an extension of Kripke–Platek set theory based on a recursively inaccessible ordinal, has a very large proof-theoretic ordinal $\psi ({\epsilon }_{I+1})$ described in a 1983 paper of Jäger and Pohlers, where I is the smallest inaccessible. This ordinal is also the proof-theoretic ordinal of ${\mathrm{\Delta }}_2^1\text{-}𝖢𝖠+𝖡𝖨$.
• KPM, an extension of Kripke–Platek set theory based on a recursively Mahlo ordinal, has a very large proof-theoretic ordinal θ.Template:Sfnp
• TTM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof-theoretic ordinal ${\psi }_{{\mathrm{\Omega }}_1}({\mathrm{\Omega }}_{M+\omega })$.
• $𝖪𝖯+{\mathrm{\Pi }}_3-Ref$ has a proof-theoretic ordinal equal to $\mathrm{\Psi }({\epsilon }_{K+1})$, where $K$ refers to the first weakly compact, due to (Rathjen 1993)
• $𝖪𝖯+{\mathrm{\Pi }}_{\omega }-Ref$ has a proof-theoretic ordinal equal to ${\mathrm{\Psi }}_X^{{\epsilon }_{\mathrm{\Xi }+1}}$, where $\mathrm{\Xi }$ refers to the first ${\mathrm{\Pi }}_0^2$-indescribable and $𝕏=({\omega }^{+};P_0;ϵ,ϵ,0)$.Template:Sfnp
• $𝖲𝗍𝖺𝖻𝗂𝗅𝗂𝗍𝗒$ has a proof-theoretic ordinal equal to ${\mathrm{\Psi }}_{𝕏}^{{\epsilon }_{\mathrm{{\rm Y}}+1}}$ where $\mathrm{{\rm Y}}$ is a cardinal analogue of the least ordinal $\alpha$ which is $(\alpha +\beta )$-stable for all $\beta <\alpha$ and $𝕏=({\omega }^{+};P_0;ϵ,ϵ,0)$.Template:Sfnp

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 ${\mathrm{\Pi }}_2^1-C{A}_0$, full second-order arithmetic (${\mathrm{\Pi }}_{\mathrm{\infty }}^1-C{A}_0$) 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
$\omega$	$𝖰$, ${𝖯𝖠}^{-}$
${\omega }^2$	$𝖱𝖥𝖠$, ${𝖨\Delta }_0$
${\omega }^3$	$𝖤𝖥𝖠$, ${𝖨\Delta }_0^{\mathsf{+}}$, $𝖡{\mathrm{\Sigma }}_1$Theorem 4.1	${𝖱𝖢𝖠}_0^{\ast }$, ${𝖶𝖪𝖫}_0^{\ast }$
${\omega }^n$Template:Ref	${𝖤𝖥𝖠}^{𝗇}$, ${𝖨\Delta }_0^{𝗇\mathsf{+}}$
${\omega }^{\omega }$	$𝖯𝖱𝖠$, ${𝖨\Sigma }_1$p. 13	${𝖱𝖢𝖠}_0$p. 13, ${𝖶𝖪𝖫}_0$p. 13		$𝖢𝖯𝖱𝖢$
${\omega }^{{\omega }^{\omega }}$	$𝖨{\mathrm{\Sigma }}_2$p. 13
${\omega }^{{\omega }^{{\omega }^{\omega }}}$	$𝖨{\mathrm{\Sigma }}_3$p. 13	${𝖱𝖢𝖠}_0+({\mathrm{\Pi }}_2^0{)}^{-}\mathsf{-}𝖨𝖭𝖣$Template:Rp
$\omega \uparrow \uparrow (n+1)$Template:Ref	$𝖨{\mathrm{\Sigma }}_n$p. 13
${\epsilon }_0$	$𝖯𝖠$p. 13	${𝖠𝖢𝖠}_0$p. 13, ${\Sigma }_1^1{\mathsf{-}𝖠𝖢}_0$p. 13, $\text{R-}\widehat{𝐄𝜴}$p. 8, $𝖱𝖢𝖠$p. 148, $𝖶𝖪𝖫$p. 148, ${\Delta }_1^1{\mathsf{-}𝖢𝖠}_0$Template:Sfnp	${\mathrm{K}\mathrm{P}\mathrm{u}}^r$Template:Sfnp			${𝖤𝖬}_0$
${\epsilon }_{\omega }$		${𝖠𝖢𝖠}_0+𝗂𝖱𝖳$, ${𝖱𝖢𝖠}_0+\mathrm{\forall }Y\mathrm{\forall }n\mathrm{\exists }X(\text{TJ}(n,X,Y))$Template:Rp
${\epsilon }_{{\epsilon }_0}$		$𝖠𝖢𝖠$p. 959
${\zeta }_0$		${𝖠𝖢𝖠}_0+\mathrm{\forall }X\mathrm{\exists }Y(\text{TJ}(\omega ,X,Y))$, ${𝗉}_1({𝖠𝖢𝖠}_0)$,Template:Rp ${𝖱𝖥𝖭}_0$p. 17, ${𝖠𝖢𝖠}_0+(𝖡𝖱)$p. 5
$\phi (2,{\epsilon }_0)$		$𝖱𝖥𝖭$, $𝖠𝖢𝖠+\mathrm{\forall }X\mathrm{\exists }Y(\text{TJ}(\omega ,X,Y))$p. 52
$\phi (\omega ,0)$	${𝖨𝖣}_1\mathrm{\#}$, ${𝖳𝖨𝖣}_0$p. 137	${\Delta }_1^1\mathsf{-}𝖢𝖱$, ${\mathrm{\Sigma }}_1^1{\mathsf{-}𝖣𝖢}_0$				${𝖤𝖬}_0\mathsf{+}𝖩𝖱$
$\phi ({\epsilon }_0,0)$	${\widehat{𝖨𝖣}}_1$, $𝖪𝖥𝖫$p. 17, $𝖪𝖥$p. 17	${\Delta }_1^1\mathsf{-}𝖢𝖠$p. 140, ${\Sigma }_1^1\mathsf{-}𝖠𝖢$p. 140, ${\Sigma }_1^1\mathsf{-}𝖣𝖢$p. 140, $\text{W-}\widehat{𝐄𝜴}$p. 8	${\mathrm{K}\mathrm{P}\mathrm{u}}^r+({\mathrm{I}\mathrm{N}\mathrm{D}}_N)$Template:Sfnp	${𝖬𝖫}_1$		${𝖤𝖬}_0\mathsf{+}𝖩$
$\phi ({\epsilon }_{{\epsilon }_0},0)$		$\widehat{𝐄𝜴}$p. 27, ${\widehat{𝐄𝐈𝐃}}_1$p. 27
$\phi (\phi (\omega ,0),0)$	$\mathrm{P}\mathrm{R}\mathrm{S}\omega$p. 9
$\phi (<\mathrm{\Omega },0)$Template:Ref	$𝖠𝗎𝗍\mathsf{(}𝖨𝖣\mathrm{\#}\mathsf{)}$
${\mathrm{\Gamma }}_0$	${\widehat{𝖨𝖣}}_{<\omega }$, $𝖴\mathsf{(}𝖯𝖠\mathsf{)}$, ${𝐊𝐅𝐋}^{\ast }$p. 22, ${𝐊𝐅}^{\ast }$p. 22, $𝒰(\mathrm{N}\mathrm{F}\mathrm{A})$, ${𝖳𝖨𝖣}_0^{+}$p. 137	${𝖠𝖳𝖱}_0$, ${\Delta }_1^1\mathsf{-}𝖢𝖠\mathsf{+}𝖡𝖱$, ${\mathrm{\Delta }}_1^1{\mathrm{-}\mathrm{C}\mathrm{A}}_0+\mathrm{(}\mathrm{S}\mathrm{U}\mathrm{B}\mathrm{)}$, ${\mathrm{F}\mathrm{P}}_0$p. 26	${𝖪𝖯𝗂}^0$,Template:Sfnp ${𝖪𝖯𝗎}^0+(\mathrm{B}\mathrm{R})$Template:Sfnp	${𝖬𝖫}_{<\omega }$, $𝖬𝖫𝖴$
${\mathrm{\Gamma }}_{{\omega }^{\omega }}$			${𝖪𝖯𝖨}^0+({{\Sigma }_{\mathrm{𝟣}}\mathsf{-}𝖨}_{\omega })$p. 13
${\mathrm{\Gamma }}_{{\epsilon }_0}$	${\widehat{𝖨𝖣}}_{\omega }$	$𝖠𝖳𝖱$	${𝖪𝖯𝖨}^0{\mathsf{+}𝖥\mathsf{-}𝖨}_{\omega }$
$\phi (1,\omega ,0)$	${\widehat{𝖨𝖣}}_{<{\omega }^{\omega }}$	${𝖠𝖳𝖱}_0+({\Sigma }_1^1\mathsf{-}𝖣𝖢)$Template:Rp	${𝖪𝖯𝗂}^0{\mathsf{+}{\Sigma }_{\mathrm{𝟣}}\mathsf{-}𝖨}_{\omega }$
$\phi (1,{\epsilon }_0,0)$	${\widehat{𝖨𝖣}}_{<{\epsilon }_0}$	$𝖠𝖳𝖱+({\Sigma }_1^1\mathsf{-}𝖣𝖢)$Template:Rp	${𝖪𝖯𝗂}^0{\mathsf{+}𝖥\mathsf{-}𝖨}_{\omega }$
$\phi (1,{\mathrm{\Gamma }}_0,0)$	${\widehat{𝖨𝖣}}_{<{\mathrm{\Gamma }}_0}$			$𝖬𝖫𝖲$
$\phi (2,0,0)$	$𝖠𝗎𝗍\mathsf{(}\widehat{𝖨𝖣}\mathsf{)}$, ${𝖥𝖳𝖱}_0$	$A{x}_{{\mathrm{\Sigma }}_1^1\mathsf{-}𝖠𝖢}{𝖳𝖱}_0$p. 1167, $A{x}_{𝖠𝖳𝖱+{\mathrm{\Sigma }}_1^1\mathsf{-}𝖣𝖢}{𝖱𝖥𝖭}_0$p. 1167	${𝖪𝖯𝗁}^0$	$𝖠𝗎𝗍\mathsf{(}𝖬𝖫\mathsf{)}$
$\phi (2,0,{\epsilon }_0)$	$𝖥𝖳𝖱$	$A{x}_{{\mathrm{\Sigma }}_1^1\mathsf{-}𝖠𝖢}𝖳𝖱$p. 1167, $A{x}_{𝖠𝖳𝖱+{\mathrm{\Sigma }}_1^1\mathsf{-}𝖣𝖢}𝖱𝖥𝖭$p. 1167
$\phi (2,{\epsilon }_0,0)$			${𝖪𝖯𝗁}_0+({𝖥\mathsf{-}𝖨}_{\omega })$Template:Rp
$\phi (\omega ,0,0)$		$({\mathrm{\Pi }}_2^1\mathsf{-}𝖱𝖥𝖭{)}_0^{{\mathrm{\Sigma }}_1^1\mathsf{-}𝖣𝖢}$p. 233, ${\mathrm{\Sigma }}_1^1{\mathsf{-}𝖳𝖣𝖢}_0$p. 233	${𝖪𝖯𝗆}^0$p. 276	$𝖤𝖬𝖠$p. 276
$\phi ({\epsilon }_0,0,0)$		$({\mathrm{\Pi }}_2^1\mathsf{-}𝖱𝖥𝖭{)}^{{\mathrm{\Sigma }}_1^1\mathsf{-}𝖣𝖢}$p. 233, ${\mathrm{\Sigma }}_1^1\mathsf{-}𝖳𝖣𝖢$	${𝖪𝖯𝗆}^0+({ℒ}^{\ast }{\mathsf{-}𝖨}_{𝖭})$p. 277	$𝖤𝖬𝖠+(𝕃{\mathsf{-}𝖨}_{𝖭})$p. 277
$\phi (1,0,0,0)$		${𝗉}_1({\mathrm{\Sigma }}_1^1{\mathsf{-}𝖳𝖣𝖢}_0)$Template:Rp
${\psi }_{{\mathrm{\Omega }}_1}({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})$		${𝖱𝖢𝖠}_0^{\ast }+{\mathrm{\Pi }}_1^1{\mathsf{-}𝖢𝖠}^{-}$, ${𝗉}_3({𝖠𝖢𝖠}_0)$Template:Rp
$\vartheta ({\mathrm{\Omega }}^{\mathrm{\Omega }})$	$𝖳𝖨𝖣$, ${𝖳𝖨𝖣}_1$p. 171	${𝗉}_1({𝗉}_3({𝖠𝖢𝖠}_0))$Template:Rp		$𝖥𝖨𝖳$p. 171
${\psi }_0({\epsilon }_{\mathrm{\Omega }+1})$Template:Ref	${𝖨𝖣}_1$	$\text{W-}\widetilde{𝐄𝜴}$p. 8	$𝖪𝖯$, $𝖪𝖯\omega$, $\mathrm{K}\mathrm{P}\mathrm{u}$Template:Sfnp	${𝖬𝖫}_1𝖵$	$𝖢𝖹𝖥$	$𝖤𝖮𝖭$
$\psi ({\epsilon }_{\mathrm{\Omega }+{\epsilon }_0})$		$\widetilde{𝐄𝜴}$p. 31, ${\widetilde{𝐄𝐈𝐃}}_1$p. 31, $𝐀𝐂𝐀+({\mathrm{\Pi }}_1^1\text{-CA}{)}^{-}$p. 31
$\psi ({\epsilon }_{\mathrm{\Omega }+\mathrm{\Omega }})$		$({𝖨𝖣}_1^2{)}_0+𝖡𝖱$
$\psi ({\epsilon }_{{\epsilon }_{\mathrm{\Omega }+1}})$		$𝐄𝜴$p. 33, ${𝐄𝐈𝐃}_1$p. 33, $𝐀𝐂𝐀+({\mathrm{\Pi }}_1^1\text{-CA}{)}^{-}+({\mathrm{B}\mathrm{I}}_{\mathrm{P}\mathrm{R}}{)}^{-}$p. 33
${\psi }_0({\mathrm{\Gamma }}_{\mathrm{\Omega }+1})$Template:Ref	${𝖴\mathsf{(}𝖨𝖣}_1\mathsf{)}$, ${\widehat{𝖨𝖣}}_{<\omega }^{\bullet }$p. 26, ${\mathrm{\Sigma }}_1^1{\mathsf{-}𝖣𝖢}_0^{\bullet }+({𝖲𝖴𝖡}^{\bullet })$p. 26, ${𝖠𝖳𝖱}_0^{\bullet }$p. 26, ${\mathrm{\Sigma }}_1^1{\mathsf{-}𝖠𝖢}_0^{\bullet }+({𝖲𝖴𝖡}^{\bullet })$p. 26, $𝒰({𝖨𝖣}_1)$p. 26	${𝖥𝖯}_0^{\bullet }$p. 26, ${𝖠𝖳𝖱}_0^{\bullet }$p. 26
${\psi }_0(\phi (<\mathrm{\Omega },0,\mathrm{\Omega }+1))$	$𝖠𝗎𝗍\mathsf{(}𝖴\mathsf{(}𝖨𝖣\mathsf{)}\mathsf{)}$
${\psi }_0({\mathrm{\Omega }}_{\omega })$	${𝖨𝖣}_{<\omega }$p. 28	${\Pi }_1^1{\mathsf{-}𝖢𝖠}_0$p. 28, ${\Delta }_2^1{\mathsf{-}𝖢𝖠}_0$		$𝖬𝖫𝖶$		$𝖲𝖴𝖲+(𝖲-{𝖨}_{𝖭})$p. 27
${\psi }_0({\mathrm{\Omega }}_{\omega }{\omega }^{\omega })$		${\mathrm{\Pi }}_1^1{\mathsf{-}𝖢𝖠}_0+{\mathrm{\Pi }}_2^1\mathsf{-}𝖨𝖭𝖣$
${\psi }_0({\mathrm{\Omega }}_{\omega }{\epsilon }_0)$	${𝖶\mathsf{-}𝖨𝖣}_{\omega }$	${\Pi }_1^1\mathsf{-}𝖢𝖠$Template:Sfnp
${\psi }_0({\mathrm{\Omega }}_{\omega }\mathrm{\Omega })$		${\mathrm{\Pi }}_1^1\mathsf{-}𝖢𝖠\mathsf{+}𝖡𝖱$
${\psi }_0({\mathrm{\Omega }}_{\omega }^{\omega })$		${\mathrm{\Pi }}_1^1{\mathsf{-}𝖢𝖠}_0+{\mathrm{\Pi }}_2^1\mathsf{-}𝖡𝖨$
${\psi }_0({\mathrm{\Omega }}_{\omega }^{{\omega }^{\omega }})$		${\mathrm{\Pi }}_1^1{\mathsf{-}𝖢𝖠}_0+{\mathrm{\Pi }}_2^1\mathsf{-}𝖡𝖨+{\mathrm{\Pi }}_3^1\mathsf{-}𝖨𝖭𝖣$
${\psi }_0({\epsilon }_{{\mathrm{\Omega }}_{\omega }+1})$Template:Ref	${𝖨𝖣}_{\omega }$	${\Pi }_1^1\mathsf{-}𝖢𝖠\mathsf{+}𝖡𝖨$	$𝖪𝖯𝖨$
${\psi }_0({\mathrm{\Omega }}_{{\omega }^{\omega }})$	${𝖨𝖣}_{<{\omega }^{\omega }}$	${\Delta }_2^1\mathsf{-}𝖢𝖱$p. 28				$𝖲𝖴𝖲+(𝖭-{𝖨}_{𝖭})$p. 27
${\psi }_0({\mathrm{\Omega }}_{{\epsilon }_0})$	${𝖨𝖣}_{<{\epsilon }_0}$	${\Delta }_2^1\mathsf{-}𝖢𝖠$p. 28, ${\Sigma }_2^1\mathsf{-}𝖠𝖢$	$𝖶\mathsf{-}𝖪𝖯𝗂$			$𝖲𝖴𝖲+(\mathrm{L}-{𝖨}_{𝖭})$p. 27
${\psi }_0({\mathrm{\Omega }}_{\mathrm{\Omega }})$	$𝖠𝗎𝗍\mathsf{(}𝖨𝖣\mathsf{)}$Template:Ref
${\psi }_{{\mathrm{\Omega }}_1}({\epsilon }_{{\mathrm{\Omega }}_{\mathrm{\Omega }}+1})$	${𝖨𝖣}_{{\prec }^{\ast }}$, ${𝖡𝖨𝖣}^{2\ast }$, ${𝖨𝖣}^{2\ast }+𝖡𝖨$		${𝖪𝖯𝗅}^{\ast }$, ${𝖪𝖯𝗅}_{\mathrm{\Omega }}^r$
${\psi }_0({\mathrm{\Phi }}_1(0))$		${\mathrm{\Pi }}_1^1{\mathsf{-}𝖳𝖱}_0$, ${\mathrm{\Pi }}_1^1{\mathsf{-}𝖳𝖱}_0+{\mathrm{\Delta }}_2^1{\mathsf{-}𝖢𝖠}_0$, ${\mathrm{\Delta }}_2^1\mathsf{-}𝖢𝖠\mathsf{+}𝖡𝖨\mathsf{(}𝗂𝗆𝗉𝗅\mathsf{-}{\mathrm{\Sigma }}_2^1)$,${\mathrm{\Delta }}_2^1\mathsf{-}𝖢𝖠\mathsf{+}𝖡𝖱\mathsf{(}𝗂𝗆𝗉𝗅\mathsf{-}{\mathrm{\Sigma }}_2^1)$,${𝐀𝐔𝐓\mathbf{-}𝐈𝐃}_0^{pos}$, ${𝐀𝐔𝐓\mathbf{-}𝐈𝐃}_0^{mon}$Template:Rp	${𝖪𝖯𝗂}^w+𝖥𝖮𝖴𝖭𝖣𝖱(𝗂𝗆𝗉𝗅\mathsf{-})\mathrm{\Sigma })$,Template:Rp ${𝖪𝖯𝗂}^w+𝖥𝖮𝖴𝖭𝖣(𝗂𝗆𝗉𝗅\mathsf{-})\mathrm{\Sigma })$,Template:Rp ${𝐀𝐔𝐓\mathbf{-}𝐊𝐏𝐥}^r$, ${𝐀𝐔𝐓\mathbf{-}𝐊𝐏𝐥}^r+{𝐊𝐏𝐢}^r$Template:Rp
${\psi }_0({\mathrm{\Phi }}_1(0){\epsilon }_0)$		${\mathrm{\Pi }}_1^1\mathsf{-}𝖳𝖱$, ${𝐀𝐔𝐓\mathbf{-}𝐈𝐃}^{pos}$, ${𝐀𝐔𝐓\mathbf{-}𝐈𝐃}^{mon}$Template:Rp	${𝐀𝐔𝐓\mathbf{-}𝐊𝐏𝐥}^w$Template:Rp
${\psi }_0({\epsilon }_{{\mathrm{\Phi }}_1(0)+1})$		${\mathrm{\Pi }}_1^1\mathsf{-}𝖳𝖱+(𝖡𝖨)$, ${𝐀𝐔𝐓\mathbf{-}𝐈𝐃}_2^{pos}$, ${𝐀𝐔𝐓\mathbf{-}𝐈𝐃}_2^{mon}$Template:Rp	$𝐀𝐔𝐓\mathbf{-}𝐊𝐏𝐥$Template:Rp
${\psi }_0({\mathrm{\Phi }}_1({\epsilon }_0))$		${\mathrm{\Pi }}_1^1\mathsf{-}𝖳𝖱+{\mathrm{\Delta }}_2^1\mathsf{-}𝖢𝖠$, ${\mathrm{\Pi }}_1^1\mathsf{-}𝖳𝖱+{\mathrm{\Sigma }}_2^1\mathsf{-}𝖠𝖢$Template:Rp	${𝐀𝐔𝐓\mathbf{-}𝐊𝐏𝐥}^w+{𝐊𝐏𝐢}^w$Template:Rp
${\psi }_0({\mathrm{\Phi }}_{\omega }(0))$		${\mathrm{\Delta }}_2^1{\mathsf{-}𝖳𝖱}_0$, ${\mathrm{\Sigma }}_2^1{\mathsf{-}𝖳𝖱𝖣𝖢}_0$, ${\mathrm{\Delta }}_2^1{\mathsf{-}𝖢𝖠}_0+({\mathrm{\Sigma }}_2^1\mathsf{-}𝖡𝖨)$Template:Rp	${𝐊𝐏𝐢}^r+(\mathrm{\Sigma }\mathsf{-}𝖥𝖮𝖴𝖭𝖣)$, ${𝐊𝐏𝐢}^r+(\mathrm{\Sigma }\mathsf{-}𝖱𝖤𝖢)$Template:Rp
${\psi }_0({\mathrm{\Phi }}_{{\epsilon }_0}(0))$		${\mathrm{\Delta }}_2^1\mathsf{-}𝖳𝖱$, ${\mathrm{\Sigma }}_2^1\mathsf{-}𝖳𝖱𝖣𝖢$, ${\mathrm{\Delta }}_2^1\mathsf{-}𝖢𝖠+({\mathrm{\Sigma }}_2^1\mathsf{-}𝖡𝖨)$Template:Rp	${𝐊𝐏𝐢}^w+(\mathrm{\Sigma }\mathsf{-}𝖥𝖮𝖴𝖭𝖣)$, ${𝐊𝐏𝐢}^w+(\mathrm{\Sigma }\mathsf{-}𝖱𝖤𝖢)$Template:Rp
$\psi ({\epsilon }_{I+1})$Template:Ref		${\Delta }_2^1\mathsf{-}𝖢𝖠\mathsf{+}𝖡𝖨$p. 28, ${\Sigma }_2^1\mathsf{-}𝖠𝖢\mathsf{+}𝖡𝖨$	$𝖪𝖯𝗂$		$𝖢𝖹𝖥\mathsf{+}𝖱𝖤𝖠$	${𝖳}_0$
$\psi ({\mathrm{\Omega }}_{I+\omega })$				${𝖬𝖫}_1𝖶$Template:Rp
$\psi ({\mathrm{\Omega }}_L)$Template:Ref			$𝖪𝖯𝗁$	${𝖬𝖫}_{<\omega }𝖶$
$\psi ({\mathrm{\Omega }}_{L^{\ast }})$Template:Ref				$𝖠𝗎𝗍\mathsf{(}𝖬𝖫𝖶\mathsf{)}$
${\psi }_{\mathrm{\Omega }}({\chi }_{{\epsilon }_{M+1}}(0))$Template:Ref		${\Delta }_2^1\mathsf{-}𝖢𝖠\mathsf{+}𝖡𝖨\mathsf{+}\mathsf{(}𝖬\mathsf{)}$	$𝖪𝖯𝖬$		$𝖢𝖹𝖥𝖬$
$\psi ({\mathrm{\Omega }}_{M+\omega })$Template:Ref			${𝖪𝖯𝖬}^{+}$	$𝖳𝖳𝖬$
${\mathrm{\Psi }}_{\mathrm{\Omega }}^0({\epsilon }_{K+1})$Template:Ref			${𝖪𝖯\mathsf{+}\Pi }_3-𝖱𝖾𝖿$Template:Sfnp
${\mathrm{\Psi }}_{({\omega }^{+};P_0,ϵ,ϵ,0)}^{{\epsilon }_{\mathrm{\Xi }+1}}$Template:Ref			${𝖪𝖯\mathsf{+}\Pi }_{\omega }-𝖱𝖾𝖿$Template:Sfnp
${\mathrm{\Psi }}_{({\omega }^{+};P_0,ϵ,ϵ,0)}^{{\epsilon }_{\mathrm{{\rm Y}}+1}}$Template:Ref			$𝖲𝗍𝖺𝖻𝗂𝗅𝗂𝗍𝗒$Template:Sfnp
${\psi }_{{\omega }_1^{CK}}({\epsilon }_{{𝕊}^{+}+1})$			$𝖪𝖯\omega +{\mathrm{\Pi }}_1^1-𝖱𝖾𝖿$, $𝖪𝖯\omega +(M{\prec }_{{\mathrm{\Sigma }}_1}V)$
${\psi }_{{\omega }_1^{CK}}({\epsilon }_{𝕀+1})$		${\mathrm{\Sigma }}_3^1\mathsf{-}𝖣𝖢\mathsf{+}𝖡𝖨$, ${\mathrm{\Sigma }}_3^1\mathsf{-}𝖠𝖢\mathsf{+}𝖡𝖨$	$𝖪𝖯\omega +{\mathrm{\Pi }}_1-𝖢𝗈𝗅𝗅𝖾𝖼𝗍𝗂𝗈𝗇+(V=L)$
${\psi }_{{\omega }_1^{CK}}({\epsilon }_{{𝕀}_N+1})$		${\mathrm{\Sigma }}_{N+2}^1\mathsf{-}𝖣𝖢\mathsf{+}𝖡𝖨$, ${\mathrm{\Sigma }}_{N+2}^1\mathsf{-}𝖠𝖢\mathsf{+}𝖡𝖨$	$𝖪𝖯\omega +{\mathrm{\Pi }}_N-𝖢𝗈𝗅𝗅𝖾𝖼𝗍𝗂𝗈𝗇+(V=L)$
${\psi }_{{\omega }_1^{CK}}({𝕀}_{\omega })$	$𝖯𝖠+\bigcup _{N<\omega }𝖳𝖨[{\mathrm{\Pi }}_0^{1-},{\psi }_{{\omega }_1^{CK}}({\epsilon }_{{𝕀}_N+1})]$	${𝐙}_2$, ${\mathrm{\Pi }}_{\mathrm{\infty }}^1-𝖢𝖠$, Bar	$𝖪𝖯+{\mathrm{\Pi }}_{\omega }^{\text{set}}-𝖲𝖾𝗉𝖺𝗋𝖺𝗍𝗂𝗈𝗇$	$\lambda 2$	$𝖢𝖹𝖥\mathsf{+}𝖲𝖾𝗉$Template:Sfnp

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). All proof-theoretic ordinals are countable.
• $𝕊$ 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 $L_{{𝕀}_N}\models 𝖪𝖯\omega +{\mathrm{\Pi }}_N-𝖢𝗈𝗅𝗅𝖾𝖼𝗍𝗂𝗈𝗇+(V=L)$; N is a variable that defines a series of ordinal analyses of the results of ${\mathrm{\Pi }}_N-𝖢𝗈𝗅𝗅𝖾𝖼𝗍𝗂𝗈𝗇$ forall $1\le N<\omega$. when N=1, ${\psi }_{{\omega }_1^{CK}}({\epsilon }_{{𝕀}_1+1})={\psi }_{{\omega }_1^{CK}}({\epsilon }_{𝕀+1})$
• Additional symbols can be found in the notes.

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

• First-order arithmetic
  • $𝖰$ is Robinson arithmetic
  • ${𝖯𝖠}^{-}$ 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.
  • ${𝖨\Delta }_0^{\mathsf{+}}$ 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 ${ℰ}^n$ of the Grzegorczyk hierarchy is total.
  • ${𝖨\Delta }_0^{𝗇\mathsf{+}}$ is ${𝖨\Delta }_0^{\mathsf{+}}$ augmented by an axiom ensuring that each element of the n-th level ${ℰ}^n$ 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.
  • ${𝖨𝖣}_{\nu }\mathrm{\#}$ is ${\widehat{𝖨𝖣}}_{\nu }$ but with induction only for positive formulas.
  • ${\widehat{𝖨𝖣}}_{\nu }$ extends PA by ν iterated fixed points of monotone operators.
  • $𝖴\mathsf{(}𝖯𝖠\mathsf{)}$ is not exactly a first-order arithmetic system, but captures what one can get by predicative reasoning based on the natural numbers.
  • $𝖠𝗎𝗍\mathsf{(}\widehat{𝖨𝖣}\mathsf{)}$ is autonomously iterated ${\widehat{𝖨𝖣}}_{\nu }$ (in other words, once an ordinal is defined, it can be used to index a new series of definitions.)
  • ${𝖨𝖣}_{\nu }$ extends PA by ν iterated least fixed points of monotone operators.
  • ${𝖴\mathsf{(}𝖨𝖣}_{\nu }\mathsf{)}$ 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.
  • $𝖠𝗎𝗍\mathsf{(}𝖴\mathsf{(}𝖨𝖣\mathsf{)}\mathsf{)}$ is autonomously iterated ${𝖴\mathsf{(}𝖨𝖣}_{\nu }\mathsf{)}$.
  • ${𝖶\mathsf{-}𝖨𝖣}_{\nu }$ is a weakened version of ${𝖨𝖣}_{\nu }$ based on W-types.
  • $𝖳𝖨[{\mathrm{\Pi }}_0^{1-},\alpha ]$ is a transfinite induction of length α no more than ${\mathrm{\Pi }}_0^1$-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.

• • ${𝖱𝖢𝖠}_0^{\ast }$ is a second order form of $𝖤𝖥𝖠$ sometimes used in reverse mathematics.
  • ${𝖶𝖪𝖫}_0^{\ast }$ 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.
  • $𝖳𝖩(n,X,Y)$ is the predicate "the nth Turing jump of X is Y".
  • ${𝖠𝖳𝖱}_0$ is arithmetical transfinite recursion.
  • $𝖠𝖳𝖱$ is ${𝖠𝖳𝖱}_0$ plus the full second-order induction scheme.
  • $𝖡𝖨$ is the bar induction axiom.
  • ${\Delta }_2^1\mathsf{-}𝖢𝖠\mathsf{+}𝖡𝖨\mathsf{+}\mathsf{(}𝖬\mathsf{)}$ is ${\Delta }_2^1\mathsf{-}𝖢𝖠\mathsf{+}𝖡𝖨$ plus the assertion "every true ${\Pi }_3^1$-sentence with parameters holds in a (countable coded) $\beta$-model of ${\Delta }_2^1\mathsf{-}𝖢𝖠$".
• 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 $\omega$.
  • $𝖶\mathsf{-}𝖪𝖯𝖨$ is a weakened version of $𝖪𝖯𝖨$ based on W-types.
  • $𝖪𝖯𝖨$ asserts that the universe is a limit of admissible sets.
  • $𝖶\mathsf{-}𝖪𝖯𝗂$ 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.
  • ${𝖪𝖯\mathsf{+}\Pi }_{𝗇}-𝖱𝖾𝖿$ is $𝖪𝖯$ augmented by a certain first-order reflection scheme.
  • $𝖲𝗍𝖺𝖻𝗂𝗅𝗂𝗍𝗒$ is KPi augmented by the axiom $\mathrm{\forall }\alpha \mathrm{\exists }\kappa \ge \alpha (L_{\kappa }{⪯}_1{L}_{\kappa +\alpha })$.
  • ${𝖪𝖯𝖬}^{+}$ is KPI augmented by the assertion "at least one recursively Mahlo ordinal exists".
  • $𝖪𝖯\omega +(M{\prec }_{{\mathrm{\Sigma }}_1}V)$ is $𝖪𝖯\omega$ with an axiom stating that 'there exists a non-empty and transitive set M such that $M{\prec }_{{\mathrm{\Sigma }}_1}V$'.

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.
  • $𝖠𝗎𝗍\mathsf{(}𝖬𝖫\mathsf{)}$ is type theory without W-types and with autonomously iterated universes.
  • ${𝖬𝖫}_1𝖵$ 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.
  • $𝖠𝗎𝗍\mathsf{(}𝖬𝖫𝖶\mathsf{)}$ is type theory with W-types and with autonomously iterated universes.
  • $𝖳𝖳𝖬$ is type theory with a Mahlo universe.
  • $\lambda 2$ is System F, also polymorphic lambda calculus or second-order lambda calculus.
• Constructive set theory
  • $𝖢𝖹𝖥$ is Aczel's constructive set theory.
  • $𝖢𝖹𝖥\mathsf{+}𝖱𝖤𝖠$ is $𝖢𝖹𝖥$ plus the regular extension axiom.
  • ${𝖢𝖹𝖥\mathsf{+}𝖱𝖤𝖠\mathsf{+}𝖥𝖹}_2$ is $𝖢𝖹𝖥\mathsf{+}𝖱𝖤𝖠$ plus the full-second order induction scheme.
  • $𝖢𝖹𝖥𝖬$ is $𝖢𝖹𝖥$ with a Mahlo universe.
• Explicit mathematics
  • ${𝖤𝖬}_0$ is basic explicit mathematics plus elementary comprehension
  • ${𝖤𝖬}_0\mathsf{+}𝖩𝖱$ is ${𝖤𝖬}_0$ plus join rule
  • ${𝖤𝖬}_0\mathsf{+}𝖩$ is ${𝖤𝖬}_0$ plus join axioms
  • $𝖤𝖮𝖭$ is a weak variant of the Feferman's ${𝖳}_0$.
  • ${𝖳}_0$ is ${𝖤𝖬}_0\mathsf{+}𝖩\mathsf{+}𝖨𝖦$, where $𝖨𝖦$ is inductive generation.
  • $𝖳$ is ${𝖤𝖬}_0{\mathsf{+}𝖩\mathsf{+}𝖨𝖦\mathsf{+}𝖥𝖹}_2$, 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.Template:NoteFor $1<n\le \omega$

2.Template:NoteThe Veblen function $\phi$ with countably infinitely iterated least fixed points.Template:Clarification needed

3.Template:NoteCan also be commonly written as $\psi ({\epsilon }_{\mathrm{\Omega }+1})$ in Madore's ψ.

4.Template:NoteUses Madore's ψ rather than Buchholz's ψ.

5.Template:NoteCan also be commonly written as $\psi ({\epsilon }_{{\mathrm{\Omega }}_{\omega }+1})$ in Madore's ψ.

6.Template:Note$K$ represents the first recursively weakly compact ordinal. Uses Arai's ψ rather than Buchholz's ψ.

7.Template:NoteAlso the proof-theoretic ordinal of $𝖠𝗎𝗍\mathsf{(}𝖶\mathsf{-}𝖨𝖣\mathsf{)}$, as the amount of weakening given by the W-types is not enough.

8.Template:Note$I$ represents the first inaccessible cardinal. Uses Jäger's ψ rather than Buchholz's ψ.

9.Template:Note$L$ represents the limit of the $\omega$-inaccessible cardinals. Uses (presumably) Jäger's ψ.

10.Template:Note$L^{\ast }$represents the limit of the $\mathrm{\Omega }$-inaccessible cardinals. Uses (presumably) Jäger's ψ.

11.Template:Note$M$ represents the first Mahlo cardinal. Uses Rathjen's ψ rather than Buchholz's ψ.

12.Template:Note$K$ represents the first weakly compact cardinal. Uses Rathjen's Ψ rather than Buchholz's ψ.

13.Template:Note$\mathrm{\Xi }$ represents the first ${\mathrm{\Pi }}_0^2$-indescribable cardinal. Uses Stegert's Ψ rather than Buchholz's ψ.

14.Template:Note$Y$ is the smallest $\alpha$ such that $\mathrm{\forall }\theta <Y\mathrm{\exists }\kappa <Y($'$\kappa$ is $\theta$-indescribable') and $\mathrm{\forall }\theta <Y\mathrm{\forall }\kappa <Y($'$\kappa$ is $\theta$-indescribable $\to \theta <\kappa$'). Uses Stegert's Ψ rather than Buchholz's ψ.

15.Template:Note$M$ represents the first Mahlo cardinal. Uses (presumably) Rathjen's ψ.

16.Template:NoteFor $n\in \mathbb{N},n\ge 1$. $\omega \uparrow \uparrow n(n\in \mathbb{N})$ represents tetration (denoted as ${\omega }_n$ in the source)

References

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