{"title":"VARIANTS OF KREISEL’S CONJECTURE ON A NEW NOTION OF PROVABILITY","authors":"P. G. Santos, R. Kahle","doi":"10.1017/bsl.2021.68","DOIUrl":null,"url":null,"abstract":"Abstract Kreisel’s conjecture is the statement: if, for all \n$n\\in \\mathbb {N}$\n , \n$\\mathop {\\text {PA}} \\nolimits \\vdash _{k \\text { steps}} \\varphi (\\overline {n})$\n , then \n$\\mathop {\\text {PA}} \\nolimits \\vdash \\forall x.\\varphi (x)$\n . For a theory of arithmetic T, given a recursive function h, \n$T \\vdash _{\\leq h} \\varphi $\n holds if there is a proof of \n$\\varphi $\n in T whose code is at most \n$h(\\#\\varphi )$\n . This notion depends on the underlying coding. \n${P}^h_T(x)$\n is a predicate for \n$\\vdash _{\\leq h}$\n in T. It is shown that there exist a sentence \n$\\varphi $\n and a total recursive function h such that \n$T\\vdash _{\\leq h}\\mathop {\\text {Pr}} \\nolimits _T(\\ulcorner \\mathop {\\text {Pr}} \\nolimits _T(\\ulcorner \\varphi \\urcorner )\\rightarrow \\varphi \\urcorner )$\n , but , where \n$\\mathop {\\text {Pr}} \\nolimits _T$\n stands for the standard provability predicate in T. This statement is related to a conjecture by Montagna. Also variants and weakenings of Kreisel’s conjecture are studied. By the use of reflexion principles, one can obtain a theory \n$T^h_\\Gamma $\n that extends T such that a version of Kreisel’s conjecture holds: given a recursive function h and \n$\\varphi (x)$\n a \n$\\Gamma $\n -formula (where \n$\\Gamma $\n is an arbitrarily fixed class of formulas) such that, for all \n$n\\in \\mathbb {N}$\n , \n$T\\vdash _{\\leq h} \\varphi (\\overline {n})$\n , then \n$T^h_\\Gamma \\vdash \\forall x.\\varphi (x)$\n . Derivability conditions are studied for a theory to satisfy the following implication: if , then \n$T\\vdash \\forall x.\\varphi (x)$\n . This corresponds to an arithmetization of Kreisel’s conjecture. It is shown that, for certain theories, there exists a function h such that \n$\\vdash _{k \\text { steps}}\\ \\subseteq\\ \\vdash _{\\leq h}$\n .","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Bulletin of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/bsl.2021.68","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract Kreisel’s conjecture is the statement: if, for all
$n\in \mathbb {N}$
,
$\mathop {\text {PA}} \nolimits \vdash _{k \text { steps}} \varphi (\overline {n})$
, then
$\mathop {\text {PA}} \nolimits \vdash \forall x.\varphi (x)$
. For a theory of arithmetic T, given a recursive function h,
$T \vdash _{\leq h} \varphi $
holds if there is a proof of
$\varphi $
in T whose code is at most
$h(\#\varphi )$
. This notion depends on the underlying coding.
${P}^h_T(x)$
is a predicate for
$\vdash _{\leq h}$
in T. It is shown that there exist a sentence
$\varphi $
and a total recursive function h such that
$T\vdash _{\leq h}\mathop {\text {Pr}} \nolimits _T(\ulcorner \mathop {\text {Pr}} \nolimits _T(\ulcorner \varphi \urcorner )\rightarrow \varphi \urcorner )$
, but , where
$\mathop {\text {Pr}} \nolimits _T$
stands for the standard provability predicate in T. This statement is related to a conjecture by Montagna. Also variants and weakenings of Kreisel’s conjecture are studied. By the use of reflexion principles, one can obtain a theory
$T^h_\Gamma $
that extends T such that a version of Kreisel’s conjecture holds: given a recursive function h and
$\varphi (x)$
a
$\Gamma $
-formula (where
$\Gamma $
is an arbitrarily fixed class of formulas) such that, for all
$n\in \mathbb {N}$
,
$T\vdash _{\leq h} \varphi (\overline {n})$
, then
$T^h_\Gamma \vdash \forall x.\varphi (x)$
. Derivability conditions are studied for a theory to satisfy the following implication: if , then
$T\vdash \forall x.\varphi (x)$
. This corresponds to an arithmetization of Kreisel’s conjecture. It is shown that, for certain theories, there exists a function h such that
$\vdash _{k \text { steps}}\ \subseteq\ \vdash _{\leq h}$
.