Archive for Mathematical Logic

, Volume 55, Issue 7–8, pp 997–1013 | Cite as

Division by zero

  • Emil Jeřábek


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.


Robinson arithmetic Diophantine equation Decidability Universal theory 

Mathematics Subject Classification



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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute of Mathematics of the Czech Academy of SciencesPraha 1Czech Republic

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