Everyone likes to debate the philosophy of whether the reals are “real”, but for me there is a much more practical question at hand: does the existence of something within a mathematical theory (i.e., derivability of a “∃ [...]” sentence) reflect back on our ability to predict the result of symbolic manipulations of arbitrary finite strings according to an arbitrary finite rule set over an arbitrary finite period of time?

For AC and CH, the answer is provably “no” as these axioms have been shown to say nothing about the behavior of halting problems, which any question about the manipulation of symbols can be phrased in terms of (well, any specific question—more general cases move up the arithmetical hierarchy).

If it’s not reflective in this precise sense, then the derivation of, e.g., a set-theoretic ∃ in some instances has no effect on any prediction of known physics (i.e., we are aware of no method of falsification).