A NOTE ON FRAGMENTS OF UNIFORM REFLECTION IN SECOND ORDER ARITHMETIC

Emanuele Frittaion
{"title":"A NOTE ON FRAGMENTS OF UNIFORM REFLECTION IN SECOND ORDER ARITHMETIC","authors":"Emanuele Frittaion","doi":"10.1017/bsl.2022.23","DOIUrl":null,"url":null,"abstract":"Abstract We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory \n$T_0$\n extending \n$\\mathsf {RCA}_0$\n and axiomatizable by a \n$\\Pi ^1_{k+2}$\n sentence, and for any \n$n\\geq k+1$\n , \n$$\\begin{align*}T_0+ \\mathrm{RFN}_{\\varPi^1_{n+2}}(T) \\ = \\ T_0 + \\mathrm{TI}_{\\varPi^1_n}(\\varepsilon_0), \\end{align*}$$\n \n$$\\begin{align*}T_0+ \\mathrm{RFN}_{\\varSigma^1_{n+1}}(T) \\ = \\ T_0+ \\mathrm{TI}_{\\varPi^1_n}(\\varepsilon_0)^{-}, \\end{align*}$$\n where T is \n$T_0$\n augmented with full induction, and \n$\\mathrm {TI}_{\\varPi ^1_n}(\\varepsilon _0)^{-}$\n denotes the schema of transfinite induction up to \n$\\varepsilon _0$\n for \n$\\varPi ^1_n$\n formulas without set parameters.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Bulletin of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/bsl.2022.23","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 3

Abstract

Abstract We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending $\mathsf {RCA}_0$ and axiomatizable by a $\Pi ^1_{k+2}$ sentence, and for any $n\geq k+1$ , $$\begin{align*}T_0+ \mathrm{RFN}_{\varPi^1_{n+2}}(T) \ = \ T_0 + \mathrm{TI}_{\varPi^1_n}(\varepsilon_0), \end{align*}$$ $$\begin{align*}T_0+ \mathrm{RFN}_{\varSigma^1_{n+1}}(T) \ = \ T_0+ \mathrm{TI}_{\varPi^1_n}(\varepsilon_0)^{-}, \end{align*}$$ where T is $T_0$ augmented with full induction, and $\mathrm {TI}_{\varPi ^1_n}(\varepsilon _0)^{-}$ denotes the schema of transfinite induction up to $\varepsilon _0$ for $\varPi ^1_n$ formulas without set parameters.
二阶算法中均匀反射碎片的注释
在二阶算术理论的分析层次上,研究了公式的一致反射片段。主要结果是,对于任意二阶算术理论$T_0$扩展$\mathsf {RCA}_0$和公理化的$\Pi ^1_{k+2}$句子,以及对于任意$n\geq k+1$, $$\begin{align*}T_0+ \mathrm{RFN}_{\varPi^1_{n+2}}(T) \ = \ T_0 + \mathrm{TI}_{\varPi^1_n}(\varepsilon_0), \end{align*}$$$$\begin{align*}T_0+ \mathrm{RFN}_{\varSigma^1_{n+1}}(T) \ = \ T_0+ \mathrm{TI}_{\varPi^1_n}(\varepsilon_0)^{-}, \end{align*}$$,其中T为$T_0$增广的完全归纳,$\mathrm {TI}_{\varPi ^1_n}(\varepsilon _0)^{-}$表示对于$\varPi ^1_n$不设参数的公式$\varepsilon _0$的超越归纳模式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信