Division by zero
(Submitted on 25 Apr 2016 (v1), last revised 24 Sep 2016 (this version, v3))
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.
| Comments: | 15 pages; fixed typos |
| Subjects: | Logic (math.LO); Logic in Computer Science (cs.LO) |
| MSC classes: | 03F30, 03F40 |
| Journal reference: | Archive for Mathematical Logic 55 (2016), no. 7, pp. 997--1013 |
| DOI : | 10.1007/s00153-016-0508-5 |
| Cite as: | arXiv:1604.07309 [math.LO] |
| (or arXiv:1604.07309v3 [math.LO] for this version) |
https://arxiv.org/abs/1604.07309
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