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Division by zero

Archive for Mathematical Logic 55 (7-8):997-1013 (2016)

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On diophantine equations solvable in models of open induction.Margarita Otero - 1990 - Journal of Symbolic Logic 55 (2):779-786.details 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 (...) Download Export citation Bookmark 3 citations
Hilbert's tenth problem for weak theories of arithmetic.Richard Kaye - 1993 - Annals of Pure and Applied Logic 61 (1-2):63-73.details Hilbert's tenth problem for a theory T asks if there is an algorithm which decides for a given polynomial p() from [] whether p() has a root in some model of T. We examine some of the model-theoretic consequences that an affirmative answer would have in cases such as T = Open Induction and others, and apply these methods by providing a negative answer in the cases when T is some particular finite fragment of the weak theories IE1 or IU-1. Download Export citation Bookmark 1 citation

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