Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 55, Issue 2 (1990), 779-786.
On Diophantine Equations Solvable in Models of Open Induction
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 $|f(x,y)| < 1$ for every $(x,y) \in C$, 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 $M$.
J. Symbolic Logic, Volume 55, Issue 2 (1990), 779-786.
First available in Project Euclid: 6 July 2007
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Otero, Margarita. On Diophantine Equations Solvable in Models of Open Induction. J. Symbolic Logic 55 (1990), no. 2, 779--786. https://projecteuclid.org/euclid.jsl/1183743331
- See Correction: Margarita Otero. Corrigendum: On Diophantine Equations Solvable in Models of Open Induction. J. Symbolic Logic, Volume 56, Issue 3 (1991), 811--812.