Archive for Mathematical Logic

, Volume 52, Issue 1–2, pp 143–157 | Cite as

Real closures of models of weak arithmetic

  • Emil Jeřábek
  • Leszek Aleksander Kołodziejczyk
Article

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 IE2) 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.

Keywords

Bounded arithmetic Real-closed field Recursive saturation Tennenbaum’s theorem Models of arithmetic 

Mathematics Subject Classification

Primary 03F30 Secondary 03C62 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Emil Jeřábek
    • 1
  • Leszek Aleksander Kołodziejczyk
    • 2
  1. 1.Institute of Mathematics of the Academy of SciencesPraha 1Czech Republic
  2. 2.Institute of MathematicsUniversity of WarsawWarszawaPoland

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