Archive for Mathematical Logic 52 (1-2):143-157 (2013)

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
Keywords Bounded arithmetic  Real-closed field  Recursive saturation  Tennenbaum’s theorem  Models of arithmetic
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DOI 10.1007/s00153-012-0311-x
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Bootstrapping, Part I.Sedki Boughattas & J. -P. Ressayre - 2010 - Annals of Pure and Applied Logic 161 (4):511-533.
Bounded Existential Induction.George Wilmers - 1985 - Journal of Symbolic Logic 50 (1):72-90.

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