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On a diophantine representation of the predicate of provability

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2014

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Moroz, Boris Zelikovich

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https://doi.org/10.1007/s10958-014-1830-2
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Journal of Mathematical Sciences. 2014, 199(1), pp. 36-52. ISSN 1072-3374. eISSN 1573-8795. Available under: doi: 10.1007/s10958-014-1830-2

Zusammenfassung

Let P be the first-order predicate calculus with a single binary predicate letter. Making use of the techniques of Diophantine coding developed in the works on Hilbert’s tenth problem, we construct a polynomial F(t; x1, . . . , xn) with integral rational coefficients such that the Diophantine equation
F(t0;x1,…, xn)=0
is soluble in integers if and only if the formula of P numbered t0 in the chosen numbering of the formulas of P is provable in P. As an application of that construction, we describe a class of Diophantine equations that can be proved insoluble only under some additional axioms of the axiomatic set theory, for instance, assuming the existence of an inaccessible cardinal. Bibliography: 14 titles.

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510 Mathematik

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ISO 690CARL, Merlin, Boris Zelikovich MOROZ, 2014. On a diophantine representation of the predicate of provability. In: Journal of Mathematical Sciences. 2014, 199(1), pp. 36-52. ISSN 1072-3374. eISSN 1573-8795. Available under: doi: 10.1007/s10958-014-1830-2
BibTex
@article{Carl2014dioph-21343.2,
  year={2014},
  doi={10.1007/s10958-014-1830-2},
  title={On a diophantine representation of the predicate of provability},
  number={1},
  volume={199},
  issn={1072-3374},
  journal={Journal of Mathematical Sciences},
  pages={36--52},
  author={Carl, Merlin and Moroz, Boris Zelikovich}
}
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