Journal of Symbolic Logic

On Diophantine Equations Solvable in Models of Open Induction

Margarita Otero

Abstract

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 $M$.

Article information

Source
J. Symbolic Logic, Volume 55, Issue 2 (1990), 779-786.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743331

Mathematical Reviews number (MathSciNet)
MR1056388

Zentralblatt MATH identifier
0703.03037

JSTOR