TL;DR
This paper explores the structure of Pi-1-2 sentences conservative over certain reverse mathematics theories, revealing their Pi_2 completeness and analyzing the impact of induction on formulas within these frameworks.
Contribution
It identifies new elements in the conservative sets and proves their Pi_2 completeness, advancing understanding of the logical strength of reverse mathematics theories.
Findings
Pi_2 completeness of conservative sets established
New elements of Pi-1-2 conservative sentences identified
Induction for Sigma-n formulas has minimal impact on Delta-(n+1) formulas
Abstract
We investigate the set of Pi-1-2 sentences which are Pi-1-1 conservative over the theories of reverse mathematics RCA0+ISigma_n and ACA0. We exhibit new elements of these sets and conclude that the sets are Pi_2 complete. Along the way, we show that, over the theory RCA, induction for Sigma-n formulas has essentially no consequences for Delta-(n+1) formulas.
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