A proof-theoretical analysis of ptykes

Catlow, J. R. G.

doi:10.1007/bf01275470

Cited by 3 publications

(9 citation statements)

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“…Nonetheless, even without knowing what |𝑇| Π 1 2 is for a given 𝑇, we can deduce a fair amount of information about and from it. In particular, we obtain the following extensional description which ties the theory developed here with the work of [4] and [20].…”

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confidence: 95%

The Π21$\Pi ^1_2$ consequences of a theory

Aguilera

Pakhomov

2022

Journal of London Math Soc

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We develop the abstract framework for a proof‐theoretic analysis of theories with scope beyond the ordinal numbers, resulting in an analog of ordinal analysis aimed at the study of theorems of complexity normalΠ21$\Pi ^1_2$. This is done by replacing the use of ordinal numbers by particularly uniform, wellfoundedness preserving functors on the category of linear orders. Generalizing the notion of a proof‐theoretic ordinal, we define the functorial normalΠ21$\Pi ^1_2$ norm of a theory and prove its existence and uniqueness for normalΠ21$\Pi ^1_2$‐sound theories. From this, we further abstract a definition of the normalΣ21$\Sigma ^1_2$‐ and normalΠ21$\Pi ^1_2$‐soundness ordinals of a theory; these quantify, respectively, the maximum strength of true normalΣ21$\Sigma ^1_2$ theorems and minimum strength of false normalΠ21$\Pi ^1_2$ theorems of a given theory. We study these ordinals, developing a proof‐theoretic classification theory for recursively enumerable extensions of sans-serifACAsans-serif0${\mathsf {ACA_0}}$. Using techniques from infinitary and categorical proof theory, generalized recursion theory, constructibility, and forcing, we prove that an admissible ordinal is the normalΠ21$\Pi ^1_2$‐soundness ordinal of some recursively enumerable extension of sans-serifACAsans-serif0${\mathsf {ACA_0}}$ if and only if it is not parameter‐free normalΣ11$\Sigma ^1_1$‐reflecting. We show that the normalΣ21$\Sigma ^1_2$‐soundness ordinal of sans-serifACAsans-serif0${\mathsf {ACA_0}}$ is ω1CK$\omega _1^{CK}$ and characterize the normalΣ21$\Sigma ^1_2$‐soundness ordinals of recursively enumerable, normalΣ21$\Sigma ^1_2$‐sound extensions of normalΠ11$\Pi ^1_1$‐sans-serifCAsans-serif0${\mathsf {CA_0}}$.

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confidence: 95%

The Π21$\Pi ^1_2$ consequences of a theory

Aguilera

Pakhomov

2022

Journal of London Math Soc

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“…Nonetheless, even without knowing what |T | Π 1 2 is for a given T , we can deduce a fair amount of information about and from it. In particular, we prove the following extensional description which ties the theory developed here with the work of [4] and [19].…”

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confidence: 58%

“…The fact that the set of all dilators is Π 1 2 -complete can be proved by a tree-construction similar to the proof of Shoenfield absoluteness, using Kleene-Brouwer orderings. Hence, this fact is provable in ACA 0 (this result is due to Catlow [4]). A coded dilator D is recursive if its code is recursive; equivalently, if the functions…”

Section: Fix Some Construction Of Orders D(a) For All Orders Amentioning

confidence: 81%

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The $Π^1_2$ Consequences of a Theory

Aguilera,

Pakhomov

2021

Preprint

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We develop the abstract framework for a proof-theoretic analysis of theories with scope beyond ordinal numbers, resulting in an analog of Ordinal Analysis aimed at the study of theorems of complexity Π 1 2 . This is done by replacing the use of ordinal numbers by particularly uniform, wellfoundedness preserving functors in the category of linear orders.Generalizing the notion of a proof-theoretic ordinal, we define the functorial Π 1 2 norm of a theory and prove its existence and uniqueness for Π 1 2 -sound theories. From this, we further abstract a definition of the Σ 1 2 -and Π 1 2 -soundness ordinals of a theory; these quantify, respectively, the maximum strength of true Σ 1 2 theorems and minimum strength of false Π 1 2 theorems of a given theory. We study these ordinals, developing a proof-theoretic classification theory for recursively enumerable extensions of ACA 0Using techniques from infinitary and categorical proof theory, generalized recursion theory, constructibility, and forcing, we prove that an admissible ordinal is the Π 1 2 -soundness ordinal of some recursively enumerable extension of ACA 0 if and only if it is not parameter-free Σ 1 1 -reflecting. We show that the Σ 1 2 -soundness ordinal of ACA 0 is ω ck 1 and characterize the Σ 1 2 -soundness ordinals of recursively enumerable, Σ 1 2 -sound extensions of Π 1 1 −CA 0 .

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“…In [2], bounds were found for those ptykes (see [3]) of level 1 and 2 which can be proved to be ptykes in the second-order theory of arithmetic ACAo. A question which was frequently asked in relation to that paper was, 'are there corresponding results in first-order arithmetic?'…”

Section: Introductionmentioning

confidence: 97%