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A NOTE ON FRAGMENTS OF UNIFORM REFLECTION IN SECOND ORDER ARITHMETIC

Part of: General logic Proof theory and constructive mathematics

Published online by Cambridge University Press:  09 June 2022

EMANUELE FRITTAION Open the ORCID record for EMANUELE FRITTAION [Opens in a new window]
EMANUELE FRITTAION*
Affiliation:
DEPARTMENT OF MATHEMATICS TECHNISCHE UNIVERSITÄTDARMSTADT, GERMANYE-mail: emanuelefrittaion@gmail.com
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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.

Keywords

second order arithmeticuniform reflectionω-arithmeticpredicative cut elimination

MSC classification

Primary: 03F03: Proof theory, general
Secondary: 03F05: Cut-elimination and normal-form theorems 03F30: First-order arithmetic and fragments 03F35: Second- and higher-order arithmetic and fragments 03B30: Foundations of classical theories (including reverse mathematics)
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 28 , Issue 3 , September 2022 , pp. 451 - 465
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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