Abstract
For any sufficiently strong theory of arithmetic, the set of Diophantine equations provably unsolvable in the theory is algorithmically undecidable, as a consequence of the MRDP theorem. In contrast, we show decidability of Diophantine equations provably unsolvable in Robinson’s arithmetic Q. The argument hinges on an analysis of a particular class of equations, hitherto unexplored in Diophantine literature. We also axiomatize the universal fragment of Q in the process.
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Jeřábek, E. Division by zero. Arch. Math. Logic 55, 997–1013 (2016). https://doi.org/10.1007/s00153-016-0508-5
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DOI: https://doi.org/10.1007/s00153-016-0508-5