VARIANTS OF KREISEL’S CONJECTURE ON A NEW NOTION OF PROVABILITY

P. G. Santos, R. Kahle
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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 $\vdash _{k \text { steps}}\ \subseteq\ \vdash _{\leq h}$ .
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关于可证明性新概念的克瑞塞尔猜想的变体
Kreisel猜想是这样的陈述:如果,对于所有$n\in \mathbb {N}$, $\mathop {\text {PA}} \nolimits \vdash _{k \text { steps}} \varphi (\overline {n})$,那么$\mathop {\text {PA}} \nolimits \vdash \forall x.\varphi (x)$。对于算术T的理论,给定递归函数h,如果T中存在$\varphi $的证明,其代码最多为$h(\#\varphi )$,则$T \vdash _{\leq h} \varphi $成立。这个概念依赖于底层编码。${P}^h_T(x)$是t中$\vdash _{\leq h}$的一个谓词,证明存在一个句子$\varphi $和一个总递归函数h,使得$T\vdash _{\leq h}\mathop {\text {Pr}} \nolimits _T(\ulcorner \mathop {\text {Pr}} \nolimits _T(\ulcorner \varphi \urcorner )\rightarrow \varphi \urcorner )$,但是,其中$\mathop {\text {Pr}} \nolimits _T$代表t中的标准可证明性谓词。这个命题与Montagna的一个猜想有关。本文还研究了Kreisel猜想的变体和弱化。通过使用反射原理,我们可以得到一个理论$T^h_\Gamma $,它扩展了T,使得Kreisel猜想的一个版本成立:给定一个递归函数h和$\varphi (x)$一个$\Gamma $ -公式(其中$\Gamma $是一个任意固定的公式类),对于所有$n\in \mathbb {N}$, $T\vdash _{\leq h} \varphi (\overline {n})$,然后$T^h_\Gamma \vdash \forall x.\varphi (x)$。研究了一个理论的可导性条件,以满足以下蕴涵:如果,则$T\vdash \forall x.\varphi (x)$。这相当于克瑞塞尔猜想的一个算术化。结果表明,对于某些理论,存在一个函数h使得$\vdash _{k \text { steps}}\ \subseteq\ \vdash _{\leq h}$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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POUR-EL’S LANDSCAPE CATEGORICAL QUANTIFICATION POINCARÉ-WEYL’S PREDICATIVITY: GOING BEYOND A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF John MacFarlane, Philosophical Logic: A Contemporary Introduction, Routledge Contemporary Introductions to Philosophy, Routledge, New York, and London, 2021, xx + 238 pp.
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