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. (shrink)

We show that IE1 proves that every element greater than 1 has a unique factorization into prime powers, although we have no way of recovering the exponents from the prime powers which appear. The situation is radically different in Bézout models of open induction. To facilitate the construction of counterexamples, we describe a method of changing irreducibles into powers of irreducibles, and we define the notion of a frugal homomorphism into Ẑ = ΠpZp, the product of the p-adic integers for (...) each prime p. (shrink)

We consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas. We prove that each model of IOpen can be embedded in a model where the equation x 2 1 + x 2 2 + x 2 3 + x 2 4 = a has a solution. The main lemma states that there is no polynomial f(x,y) with coefficients in a (nonstandard) DOR M such that $|f(x,y)| for every (x,y) ∈ C, where C is (...) the curve defined on the real closure of M by C: x 2 + y 2 = a and $a > 0$ is a nonstandard element of M. (shrink)

Models of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the language of ordered semirings. Here we study the problem of representability of an element a of a model M of NOI (in some extension of M) by a quadratic form of the type X2 + bY2 where b is a nonzero integer. Using either a trigonometric or a hyperbolic parametrization we prove (...) that except in some trivial cases, M[ x, y] with x2 + by2 = a can be embedded in a model of NOI. We also study quadratic extensions of a model M of NOI; we first prove some properties of the ring of Gaussian integers of M. Then we study the group of solutions of a Pell equation in NOI; we construct a model in which the quotient group by the squares has size continuum. (shrink)

We show that Intuitionistic Open Induction iop is not closed under the rule DNS(∃ - 1 ). This is established by constructing a Kripke model of iop + $\neg L_y(2y > x)$ , where $L_y(2y > x)$ is universally quantified on x. On the other hand, we prove that iop is equivalent with the intuitionistic theory axiomatized by PA - plus the scheme of weak ¬¬LNP for open formulas, where universal quantification on the parameters precedes double negation. We also show (...) that for any open formula φ(y) having only y free, $(PA^-)^i \vdash L_y\varphi(y)$ . We observe that the theories iop, i∀ 1 and iΠ 1 are closed under Friedman's translation by negated formulas and so under VR and IP. We include some remarks on the classical worlds in Kripke models of iop. (shrink)

We prove that the theory IΔ0, extended by a weak version of the Δ0-Pigeonhole Principle, proves that every integer is the sum of four squares (Lagrange's theorem). Since the required weak version is derivable from the theory IΔ0 + ∀x (xlog(x) exists), our results give a positive answer to a question of Macintyre (1986). In the rest of the paper we consider the number-theoretical consequences of a new combinatorial principle, the ‘Δ0-Equipartition Principle’ (Δ0EQ). In particular we give a new proof, (...) which can be formalized in IΔ0 + Δ0EQ, of the fact that every prime of the form 4n + 1 is the sum of two squares. (shrink)

Models of normal open induction are those normal discretely ordered rings whose nonnegative part satisfy Peano's axioms for open formulas in the language of ordered semirings. (Where normal means integrally closed in its fraction field.) In 1964 Shepherdson gave a recursive nonstandard model of open induction. His model is not normal and does not have any infinite prime elements. In this paper we present a recursive nonstandard model of normal open induction with an unbounded set of infinite prime elements.

The theory , axiomatized by the induction scheme for sharply bounded formulae in Buss’ original language of bounded arithmetic , has recently been unconditionally separated from full bounded arithmetic S2. The method used to prove the separation is reminiscent of those known from the study of open induction.We make the connection to open induction explicit, showing that models of can be built using a “nonstandard variant” of Wilkie’s well-known technique for building models of IOpen. This makes it possible to transfer (...) many results and methods from open to sharply bounded induction with relative ease.We provide two applications: the Shepherdson model of IOpen can be embedded into a model of , which immediately implies some independence results for ; extended by an axiom which roughly states that every number has a least 1 bit in its binary notation, while significantly stronger than plain , does not prove the infinity of primes. (shrink)