Ordinal analysis

Ordinal analysis is a special way to measure how strong a mathematical theory is. In proof theory, each theory gets a number called an ordinal. These numbers help us understand the theories better.

If two theories have the same ordinal number, they are often just as strong and can prove the same things. If one theory has a bigger ordinal number than another, it can usually show that the smaller theory is consistent, meaning it doesn’t contain any contradictions.

Ordinal analysis also gives us extra information about the theory we are studying. For example, it can help us understand what kinds of functions the theory can prove are well-behaved. This work helps mathematicians compare different theories and see how they relate to each other.

History

The field of ordinal analysis started in 1934. Gerhard Gentzen used a method called cut elimination. He showed that Peano arithmetic, a basic math system, has a special strength level. This level is called its proof-theoretic ordinal, and it is ε₀. This work helped prove that Peano arithmetic is consistent. This means it does not contain contradictions.

Definition

Ordinal analysis studies special math ideas that help us understand numbers in an organized way. These ideas let us see how strong a math system is by giving it a special number called a proof-theoretic ordinal. This number shows how far the theory can go in checking and organizing statements.

Some theories need help to talk about very big numbers. We use something called ordinal notations to help them. These notations let the theory discuss big numbers clearly, so we can see how strong each theory is.

Upper bound

In proof theory, we can measure how strong a mathematical theory is by using special numbers called ordinals. The proof-theoretic ordinal of any theory cannot be larger than a special number called the Church–Kleene ordinal.

For some types of theories, there is always a recursive ordering that the theory cannot prove is well-organized. This helps show that the proof-theoretic ordinal of such a theory will always be a smaller, countable ordinal.

Examples

Theories with proof-theoretic ordinal ω

• Q, Robinson arithmetic
• 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.
• RCA^*
  ₀, a second-order form of EFA sometimes used in reverse mathematics.
• WKL^*
  ₀, a second-order form of EFA sometimes used in reverse mathematics.

Friedman's grand conjecture suggests that much "ordinary" mathematics can be proved in weak systems having this as their proof-theoretic ordinal.

Theories with proof-theoretic ordinal ωⁿ (for n = 2, 3, ... ω)

• IΔ₀ or EFA augmented by an axiom ensuring that each element of the n-th level E n {\displaystyle {\mathcal {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

Unsolved problem in mathematics

What is the proof-theoretic ordinal of full second-order arithmetic?

More unsolved problems in mathematics

• Π 1 1 - C A 0 {\displaystyle \Pi _{1}^{1}{\mbox{-}}{\mathsf {CA}}_{0}} , Π₁¹ comprehension has a rather large proof-theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams", and which is bounded by ψ₀(Ω_ω) in Buchholz's notation. It is also the ordinal of I D 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 Σ 2 1 - A C + B I {\displaystyle \Sigma _{2}^{1}{\mbox{-}}{\mathsf {AC}}+{\mathsf {BI}}} .
• KPi, an extension of Kripke–Platek set theory based on a recursively inaccessible ordinal, has a very large proof-theoretic ordinal ψ ( ε 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 Δ 2 1 - C A + B I .
• KPM, an extension of Kripke–Platek set theory based on a recursively Mahlo ordinal, has a very large proof-theoretic ordinal θ, which was described by Rathjen (1990).
• TTM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof-theoretic ordinal ψ Ω 1 ( Ω M + ω ) .
• K P + Π 3 − R e f has a proof-theoretic ordinal equal to Ψ ( ε K + 1 ) , where K refers to the first weakly compact, due to (Rathjen 1993)
• K P + Π ω − R e f has a proof-theoretic ordinal equal to Ψ X ε Ξ + 1 , where Ξ refers to the first Π 0 2 -indescribable and X = ( ω + ; P 0 ; ϵ , ϵ , 0 ) , due to (Stegert 2010).
• S t a b i l i t y has a proof-theoretic ordinal equal to Ψ X ε Υ + 1 where Υ is a cardinal analogue of the least ordinal α which is ( α + β ) -stable for all β Table of proof-theoretic ordinalsOrdinalFirst-order arithmeticSecond-order arithmeticKripke–Platek set theoryType theoryExplicit mathematicsωQ, P A −ω 2R F A, I Δ 0ω 3E F A, I Δ 0 + , B Σ 1 ^{Theorem 4.1}R C A 0 ∗, W K L 0 ∗ω nE F A n, I Δ 0 n +ω ωP R A, I Σ 1 ^p. 13R C A 0 ^p. 13, W K L 0 ^p. 13C P R Cω ω ωI Σ 2 ^p. 13ω ω ω ωI Σ 3 ^p. 13R C A 0 + ( Π 2 0 ) − − I N D ^: 40ω ↑↑ ( n + 1 )I Σ n ^p. 13ε 0P A ^p. 13A C A 0 ^p. 13, Σ 1 1 − A C 0 ^p. 13, R- E Ω ^ , R C A ^p. 148, W K L ^p. 148, Δ 1 1 − C A 0K P u r ^p. 869E M 0ε ωA C A 0 + i R T , R C A 0 + ∀ Y ∀ n ∃ X ( T J ( n , X , Y ) ) ^: 8ε ε 0A C A ^p. 959ζ 0A C A 0 + ∀ X ∃ Y ( T J ( ω , X , Y ) ) , p 1 ( A C A 0 ) ^: 7 R F N 0 ^p. 17, A C A 0 + ( B R ) ^p. 5φ ( 2 , ε 0 )R F N, A C A + ∀ X ∃ Y ( T J ( ω , X , Y ) ) ^p. 52φ ( ω , 0 )I D 1 # , T I D 0 ^p. 137Δ 1 1 − C R, Σ 1 1 − D C 0 E M 0 + J Rφ ( ε 0 , 0 )I D ^ 1, K F L ^p. 17, K F ^p. 17Δ 1 1 − C A ^p. 140, Σ 1 1 − A C ^p. 140, Σ 1 1 − D C ^p. 140, W- E Ω ^ ^p. 8K P u r + ( I N D N ) ^p. 870M L 1E M 0 + Jφ ( ε ε 0 , 0 )E Ω ^ ^p. 27, E I D ^ 1 ^p. 27φ ( φ ( ω , 0 ) , 0 )P R S ω ^p. 9φ ( A u t ( I D # )Γ 0I D ^ , U ( P A ) , K F L ∗ ^p. 22, K F ∗ ^p. 22, U ( N F A ) , T I D 0 + ^p. 137A T R 0, Δ 1 1 − C A + B R, Δ 1 1 − C A 0 + ( S U B ) , F P 0 ^p. 26K P i 0 ^p. 878, K P u 0 + ( B R ) ^p. 878M L UΓ ω ωK P I 0 + ( Σ 1 − I ω ) ^p. 13Γ ε 0I D ^ ωA T R K P I 0 + F − I ωφ ( 1 , ω , 0 )I D ^A T R 0 + ( Σ 1 1 − D C ) ^: 7K P i 0 + Σ 1 − I ωφ ( 1 , ε 0 , 0 )I D ^A T R + ( Σ 1 1 − D C ) ^: 7K P i 0 + F − I ωφ ( 1 , Γ 0 , 0 )I D ^M L Sφ ( 2 , 0 , 0 )A u t ( I D ^ ), F T R 0 A x Σ 1 1 − A C T R 0 ^p. 1167, A x A T R + Σ 1 1 − D C R F N 0 ^p. 1167K P h 0A u t ( M L )φ ( 2 , 0 , ε 0 )F T R A x Σ 1 1 − A C T R ^p. 1167, A x A T R + Σ 1 1 − D C R F N 0 ^p. 1167φ ( 2 , ε 0 , 0 )K P h 0 + ( F − I ω ) ^: 11φ ( ω , 0 , 0 )( Π 2 1 − R F N ) 0 Σ 1 1 − D C ^p. 233, Σ 1 1 − T D C 0 ^p. 233K P m 0 ^p. 276E M A ^p. 276φ ( ε 0 , 0 , 0 )( Π 2 1 − R F N ) Σ 1 1 − D C ^p. 233, Σ 1 1 − T D CK P m 0 + ( L ∗ − I N ) ^p. 277E M A + ( L − I N ) ^p. 277φ ( 1 , 0 , 0 , 0 )p 1 ( Σ 1 1 − T D C 0 ) ^: 7ψ Ω 1 ( Ω Ω ω )R C A 0 ∗ + Π 1 1 − C A − , p 3 ( A C A 0 ) ^: 7ϑ ( Ω Ω )T I D, T I D 1 ^p. 171p 1 ( p 3 ( A C A 0 ) ) ^: 7F I T ^p. 171ψ 0 ( ε Ω + 1 )I D 1W- E Ω ~ ^p. 8K P, K P ω, K P u ^p. 869M L 1 VE O Nψ ( ε Ω + ε 0 )E Ω ~ ^p. 31, E I D ~ 1 ^p. 31, A C A + ( Π 1 1 -CA ) − ^p. 31ψ ( ε Ω + Ω )( I D 1 2 ) 0 + B R ψ ( ε ε Ω + 1 )E Ω, E I D 1 ^p. 33, A C A + ( Π 1 1 -CA ) − + ( B I P R ) − ^p. 33ψ 0 ( Γ Ω + 1 )U ( I D 1 ), I D ^ ^p. 26, Σ 1 1 − D C 0 ∙ + ( S U B ∙ ) ^p. 26, A T R 0 ∙ ^p. 26, Σ 1 1 − A C 0 ∙ + ( S U B ∙ ) ^p. 26, U ( I D 1 ) ^p. 26F P 0 ∙ ^p. 26, A T R 0 ∙ ^p. 26ψ 0 ( Φ 1 ( 0 ) )A u t ( U ( I D ) )ψ 0 ( Ω ω )I D ^p. 28Π 1 1 − C A 0, Δ 2 1 − C A 0M L WS U S + ( S − I N ) ^p. 27ψ 0 ( Ω ω ω ω )Π 1 1 − C A 0 + Π 2 1 − I N D ψ 0 ( Ω ω ε 0 )W − I D ωΠ 1 1 − C A ^p. 14W − K P Iψ 0 ( Ω ω Ω )Π 1 1 − C A + B R ψ 0 ( Ω ω ω )Π 1 1 − C A 0 + Π 2 1 − B I ψ 0 ( Ω ω ω ω )Π 1 1 − C A 0 + Π 2 1 − B I + Π 3 1 − I N D ψ 0 ( ε Ω ω + 1 )I D ωΠ 1 1 − C A + B IK P Iψ 0 ( Ω ω ω )I DΔ 2 1 − C R ^p. 28S U S + ( N − I N ) ^p. 27ψ 0 ( Ω ε 0 )I DΔ 2 1 − C A ^p. 28, Σ 2 1 − A CW − K P iS U S + ( L − I N ) ^p. 27ψ 0 ( Ω Ω )A u t ( I D ) ψ Ω 1 ( ε Ω Ω + 1 )I D ≺ ∗, B I D 2 ∗, I D 2 ∗ + B I K P l ∗, K P l Ω rψ 0 ( Φ 1 ( 0 ) )Π 1 1 − T R 0, Π 1 1 − T R 0 + Δ 2 1 − C A 0, Δ 2 1 − C A + B I ( i m p l − Σ 2 1 ), Δ 2 1 − C A + B R ( i m p l − Σ 2 1 ), A U T − I D 0 p o s, A U T − I D 0 m o n ^: 72K P i w + F O U N D R ( i m p l − ) Σ ),^: 72 K P i w + F O U N D ( i m p l − ) Σ )^: 72
  A U T − K P l r, A U T − K P l r + K P i r ^: 72
  ψ 0 ( Φ 1 ( 0 ) ε 0 )Π 1 1 − T R, A U T − I D p o s, A U T − I D m o n ^: 72A U T − K P l w ^: 72ψ 0 ( ε Φ 1 ( 0 ) + 1 )Π 1 1 − T R + ( B I ), A U T − I D 2 p o s, A U T − I D 2 m o n ^: 72A U T − K P l ^: 72ψ 0 ( Φ 1 ( ε 0 ) )Π 1 1 − T R + Δ 2 1 − C A, Π 1 1 − T R + Σ 2 1 − A C ^: 72A U T − K P l w + K P i w ^: 72ψ 0 ( Φ ω ( 0 ) )Δ 2 1 − T R 0, Σ 2 1 − T R D C 0, Δ 2 1 − C A 0 + ( Σ 2 1 − B I ) ^: 72K P i r + ( Σ − F O U N D ), K P i r + ( Σ − R E C ) ^: 72ψ 0 ( Φ ε 0 ( 0 ) )Δ 2 1 − T R, Σ 2 1 − T R D C, Δ 2 1 − C A + ( Σ 2 1 − B I ) ^: 72K P i w + ( Σ − F O U N D ), K P i w + ( Σ − R E C ) ^: 72ψ ( ε I + 1 )Δ 2 1 − C A + B I ^p. 28, Σ 2 1 − A C + B IK P iT 0ψ ( Ω I + ω )M L 1 W ^: 38ψ ( Ω L )K P hM Lψ ( Ω L ∗ )A u t ( M L W )ψ Ω ( χ ε M + 1 ( 0 ) )Δ 2 1 − C A + B I + ( M ) K P Mψ ( Ω M + ω )K P M +T T M Ψ Ω 0 ( ε K + 1 )K P + Π 3 − R e f Ψ ( ω + ; P 0 , ϵ , ϵ , 0 ) ε Ξ + 1K P + Π ω − R e f Ψ ( ω + ; P 0 , ϵ , ϵ , 0 ) ε Υ + 1S t a b i l i t y ψ ω 1 C K ( ε S + + 1 )K P ω + Π 1 1 − R e f, K P ω + ( M ≺ Σ 1 V ) ψ ω 1 C K ( ε I + 1 )Σ 3 1 − D C + B I, Σ 3 1 − A C + B IK P ω + Π 1 − C o l l e c t i o n + ( V = L )ψ ω 1 C K ( ε I N + 1 )Σ N + 2 1 − D C + B I, Σ N + 2 1 − A C + B IK P ω + Π N − C o l l e c t i o n + ( V = L )ψ ω 1 C K ( I ω )P A + ⋃ N Z 2, Π ∞ 1 − C A, BarK P + Π ω s e t − S e p a r a t i o nλ 2

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 {\displaystyle \mathbb {S} } !{\displaystyle \mathbb {S} } is an ordinal term denoting a stable ordinal, and S + {\displaystyle \mathbb {S} ^{+}} !{\displaystyle \mathbb {S} ^{+}} the least admissible ordinal above S {\displaystyle \mathbb {S} } !{\displaystyle \mathbb {S} } .
• I N {\displaystyle \mathbb {I} _{N}} !{\displaystyle \mathbb {I} _{N}} is an ordinal term denoting an ordinal such that L I N ⊨ K P ω + Π N − C o l l e c t i o n + ( V = L ) {\displaystyle L_{\mathbb {I} _{N}}\models {\mathsf {KP}}\omega +\Pi _{N}-{\mathsf {Collection}}+(V=L)} !{\displaystyle L_{\mathbb {I} _{N}}\models {\mathsf {KP}}\omega +\Pi _{N}-{\mathsf {Collection}}+(V=L)} ; N is a variable that defines a series of ordinal analyses of the results of Π N − C o l l e c t i o n {\displaystyle \Pi _{N}-{\mathsf {Collection}}} !{\displaystyle \Pi _{N}-{\mathsf {Collection}}} forall 1 ≤ N 0-predicates without any axiom asserting that exponentiation is total.
  • E F A {\displaystyle {\mathsf {EFA}}} !{\displaystyle {\mathsf {EFA}}} is elementary function arithmetic.
  • I Δ 0 + {\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}} !{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}} is arithmetic with induction restricted to Δ₀-predicates augmented by an axiom asserting that exponentiation is total.
  • E F A n {\displaystyle {\mathsf {EFA}}^{\mathsf {n}}} !{\displaystyle {\mathsf {EFA}}^{\mathsf {n}}} is elementary function arithmetic augmented by an axiom ensuring that each element of the n-th level E n {\displaystyle {\mathcal {E}}^{n}} !{\displaystyle {\mathcal {E}}^{n}} of the Grzegorczyk hierarchy is total.
  • I Δ 0 n + {\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {n+}}} !{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {n+}}} is I Δ 0 + {\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}} !{\displaystyle {\mathsf {I\Delta }}_{0}^{\mathsf {+}}} augmented by an axiom ensuring that each element of the n-th level E n {\displaystyle {\mathcal {E}}^{n}} !{\displaystyle {\mathcal {E}}^{n}} of the Grzegorczyk hierarchy is total.
  • P R A {\displaystyle {\mathsf {PRA}}} !{\displaystyle {\mathsf {PRA}}} is primitive recursive arithmetic.
  • I Σ 1 {\displaystyle {\mathsf {I\Sigma }}_{1}} !{\displaystyle {\mathsf {I\Sigma }}_{1}} is arithmetic with induction restricted to Σ₁-predicates.
  • P A {\displaystyle {\mathsf {PA}}} !{\displaystyle {\mathsf {PA}}} is Peano arithmetic.
  • I D ν # {\displaystyle {\mathsf {ID}}_{\nu }\#} !{\displaystyle {\mathsf {ID}}_{\nu }\#} is I D ^ ν {\displaystyle {\widehat {\mathsf {ID}}}_{\nu }} !{\displaystyle {\widehat {\mathsf {ID}}}_{\nu }} but with induction only for positive formulas.
  • I D ^ ν {\displaystyle {\widehat {\mathsf {ID}}}_{\nu }} !{\displaystyle {\widehat {\mathsf {ID}}}_{\nu }} extends PA by ν iterated fixed points of monotone operators.
  • U ( P A ) {\displaystyle {\mathsf {U(PA)}}} !{\displaystyle {\mathsf {U(PA)}}} is not exactly a first-order arithmetic system, but captures what one can get by predicative reasoning based on the natural numbers.
  • A u t ( I D ^ ) {\displaystyle {\mathsf {Aut({\widehat {ID}})}}} !{\displaystyle {\mathsf {Aut({\widehat {ID}})}}} is autonomously iterated I D ^ ν {\displaystyle {\widehat {\mathsf {ID}}}_{\nu }} !{\displaystyle {\widehat {\mathsf {ID}}}_{\nu }} (in other words, once an ordinal is defined, it can be used to index a new series of definitions.)
  • I D ν {\displaystyle {\mathsf {ID}}_{\nu }} !{\displaystyle {\mathsf {ID}}_{\nu }} extends PA by ν iterated least fixed points of monotone operators.
  • U ( I D ν ) {\displaystyle {\mathsf {U(ID}}_{\nu }{\mathsf {)}}} !{\displaystyle {\mathsf {U(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.
  • A u t ( U ( I D ) ) {\displaystyle {\mathsf {Aut(U(ID))}}} !{\displaystyle {\mathsf {Aut(U(ID))}}} is autonomously iterated U ( I D ν ) {\displaystyle {\mathsf {U(ID}}_{\nu }{\mathsf {)}}} !{\displaystyle {\mathsf {U(ID}}_{\nu }{\mathsf {)}}} .
  • W − I D ν {\displaystyle {\mathsf {W-ID}}_{\nu }} !{\displaystyle {\mathsf {W-ID}}_{\nu }} is a weakened version of I D ν {\displaystyle {\mathsf {ID}}_{\nu }} !{\displaystyle {\mathsf {ID}}_{\nu }} based on W-types.
  • T I [ Π 0 1 − , α ] {\displaystyle {\mathsf {TI}}[\Pi _{0}^{1-},\alpha ]} is a transfinite induction of length α no more than Π 0 1 {\displaystyle \Pi _{0}^{1}} !{\displaystyle \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.

• • R C A 0 ∗ {\displaystyle {\mathsf {RCA}}_{0}^{*}} !{\displaystyle {\mathsf {RCA}}_{0}^{*}} is a second order form of E F A {\displaystyle {\mathsf {EFA}}} !{\displaystyle {\mathsf {EFA}}} sometimes used in reverse mathematics.
  • W K L 0 ∗ {\displaystyle {\mathsf {WKL}}_{0}^{*}} !{\displaystyle {\mathsf {WKL}}_{0}^{*}} is a second order form of E F A {\displaystyle {\mathsf {EFA}}} !{\displaystyle {\mathsf {EFA}}} sometimes used in reverse mathematics.
  • R C A 0 {\displaystyle {\mathsf {RCA}}_{0}} !{\displaystyle {\mathsf {RCA}}_{0}} is recursive comprehension.
  • W K L 0 {\displaystyle {\mathsf {WKL}}_{0}} !{\displaystyle {\mathsf {WKL}}_{0}} is weak Kőnig's lemma.
  • A C A 0 {\displaystyle {\mathsf {ACA}}_{0}} ![{\displaystyle {\mathsf {ACA}}_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c859a47f5d58a4af830b23

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