The examples in this thread are interesting as curiosities, but while one never knows where fundamental explorations will lead, we should be skeptical of their physical applicability.
In physics, the same observable theory may have very different formulations. For example, we may have a formulation using a continuum, and a formulation using a limit of discrete approximations. Given the finite and approximate nature of observations, we should be skeptical if the discrete approximations do not converge to the continuum limit (or if singularities lead to nonrecursive computations). It would be remarkable if physical distance π is observably different from π-ε.
Now, consider two physical theories, where theory A depends on basic arithmetic and theory B depends on the Continuum Hypothesis (CH), with both A and B giving the same predictions (assuming CH). Even if the theory is experimentally confirmed, without more, we cannot say that the Continuum Hypothesis was resolved. What would have more value is a series of equivalences that point to a coherent vision of set theory, or even better, the ability to run experiments to test arbitrary set theoretical propositions of a certain type.
As for possible examples, mathematical consistency of a nontrivial relativistic 4-dimensional quantum field theory is an open problem. Thus, for all we know, it might be equiconsistent with a supercompact cardinal, though there is no present evidence to that effect. We also cannot yet rule out that the theory is consistent but only for nonrecursive configurations.