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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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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CARL, 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/s10958-014-1830-2BibTex
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title={On a diophantine representation of the predicate of provability},
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<dcterms:abstract xml:lang="eng">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<br />F(t<sub>0</sub>;x<sub>1</sub>,…, x<sub>n</sub>)=0<br />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.</dcterms:abstract>
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