On a Diophantine Representation of the Predicate of Provability

Let $\mathcal{P}$ 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

$$ F\left( {{t_0};{x_1},\ldots,\ {x_n}} \right)=0 $$

is soluble in integers if and only if the formula of $\mathcal{P}$ numbered t0 in the chosen numbering of the formulas of $\mathcal{P}$ is provable in $\mathcal{P}$. 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.