This seems convincing as far as saying there are non-equivalent notions of probability. I wonder if there is a notion of quantum-random individual object (for a fixed quantum probability distribution) such as a quantum random real number? Maybe I should ask this as a separate question rather than a comment.

@Yemon Choi: "Lattice theory and universal algebra" is considered part of "rings and algebras" by the arXiv.

Greg Igusa's argument above shows by contradiction the existence of a polynomial $p$ whose solvability is independent of ZFC, but does not by itself tell us how to get our hands on such a $p$. The missing ingredient is the result that one can computably translate arbitrary existential ($\Sigma^0_1$) statements into statements about solvability of Diophantine equations. This yields both the uncomputability (Matiyasevich's theorem), and the fact that a specific $p$ can be found (as in Andreas Blass' comment and Alon Amit's answer) since $\neg$Con(ZFC) is a $\Sigma^0_1$ statement.