A NOTE ON FRAGMENTS OF UNIFORM REFLECTION IN SECOND ORDER ARITHMETIC

Emanuele Frittaion
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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。
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