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CONSERVATION THEOREMS ON SEMI-CLASSICAL ARITHMETIC
Published online by Cambridge University Press: 17 March 2022
Abstract
We systematically study conservation theorems on theories of semi-classical arithmetic, which lie in-between classical arithmetic $\mathsf {PA}$ and intuitionistic arithmetic
$\mathsf {HA}$. Using a generalized negative translation, we first provide a structured proof of the fact that
$\mathsf {PA}$ is
$\Pi _{k+2}$-conservative over
$\mathsf {HA} + {\Sigma _k}\text {-}\mathrm {LEM}$ where
${\Sigma _k}\text {-}\mathrm {LEM}$ is the axiom scheme of the law-of-excluded-middle restricted to formulas in
$\Sigma _k$. In addition, we show that this conservation theorem is optimal in the sense that for any semi-classical arithmetic T, if
$\mathsf {PA}$ is
$\Pi _{k+2}$-conservative over T, then
${T}$ proves
${\Sigma _k}\text {-}\mathrm {LEM}$. In the same manner, we also characterize conservation theorems for other well-studied classes of formulas by fragments of classical axioms or rules. This reveals the entire structure of conservation theorems with respect to the arithmetical hierarchy of classical principles.
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- © The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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