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The known 1970 solution to the 10th Hilbert problem says that no algorithm is possible that would decide whether a given Diophantine equation has a solution. In set terms, this means that not all Diophantine sets are decidable. In a posting to the Foundations of Mathematica mailing list, Timothy Y. Chow asked for possible formal justification for his impression that most Diophantine equations are not decidable. One such possible justification is presented in this paper. 1 Formulation of the Problem Decidability of Diophantine equations: a brief reminder. A Diophantine equation is an equation of the type P (x1, . . . , xn, c1, . . . , cm) = 0, where P is a polynomial with integer coefficients, x1, . . . , xn are natural-valued unknowns, and c1, . . . , cm are natural-valued parameters. For many equations: • for some values of the parameters c = (c1, . . . , cm), there is a solution, while • for other values c = (c1, . . . , cm), the equation has no solution. For some Diophantine equations, an algorithm is known that, given the tuples of parameters c = (c1, . . . , cm), tell us whether the corresponding equation has a solution. One of the problems that David Hilbert formulated in 1900 – as challenges for the 20 century mathematicians – was to find a general algorithm for telling whether a given instance of a Diophantine equation has a solution; this was Problem No. 10; see, e.g., [1, 2]. The answer to this problem turned out to be negative: in 1970, Yuri Matiyasevich proved that no such general algorithm is possible; see, e.g., [3]. Moreover, he proved the following stronger result.

Almost all Diophantine sets are undecidable