, Volume 52, Issue 1-2, pp 143-157

Real closures of models of weak arithmetic

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Abstract

D’Aquino et al. (J Symb Log 75(1):1–11, 2010) have recently shown that every real-closed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by IΔ0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss’ bounded arithmetic: PV or \({\Sigma^b_1-IND^{|x|_k}}\) . It also holds for IΔ0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the other hand, it fails for the extension of IOpen by an axiom expressing the Bézout property, even under the same assumption on cofinality.

Emil Jeřábek supported by RVO: 67985840, grant IAA100190902 of GA AV ČR, project 1M0545 of MŠMT ČR, and a grant from the John Templeton Foundation.
Leszek Aleksander Koł odziejczyk partially supported by grant N N201 382234 of the Polish Ministry of Science and Higher Education.