{"title":"二阶算法中均匀反射碎片的注释","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":"1 2 1","pages":"451 - 465"},"PeriodicalIF":0.0000,"publicationDate":"2022-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"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\":\"1 2 1\",\"pages\":\"451 - 465\"},\"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}","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}
A NOTE ON FRAGMENTS OF UNIFORM REFLECTION IN SECOND ORDER ARITHMETIC
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.