This is the topic from a recent article in the New Scientist
devoted to philosophical implications of existence of uncomputable real numbers. The article introduces Chaitin's Omega, the probability that a (Turing machine) program, chosen at random from among all the possible programs, will halt. This is a real number between 0 and 1, but it is uncomputable. Given a well-known connection between Turing machines and Diophantine equations, Chaitin discovered (a very long) Diophantine equation with a parameter N. It turns out that fixing N and asking a question if the equation has an infinite number of solutions gives the N-th bit of Omega. Given that Omega is
uncomputable, there does not exist an algorithm that can tell if the
equation has an infinite number of solutions as a function of N. Thus
in general the question about the number of solutions of a Diophantine
equation is undecidable. Even in the number theory, there are "simple" questions that simply do not have any answer.
As the article says, "Chaitin has shown that there are an infinite number of mathematical facts but, for the most part, they are unrelated to each other and impossible to tie together with unifying theorems. If mathematicians find any connections between these facts, they do so by luck. "Most of mathematics is true for no particular reason," Chaitin says. "Maths is true by accident."