Naive logic on which mathematics should be relevant to physics?

Question

Background

So I am under the impression that there is no physical theory which makes use of any kind of mathematics which has a set of solution space greater than the cardinality of the real numbers.

My reasoning is as follows:

The experimentalist has to measure this theory. The set of solutions when continuous is (or part of a) real number line.

Any theory must have a solution space which can be mapped to experiment. Hence, the cardinality must be the same.

Question

Is there an exception I have not heard of? Is the above reasoning correct? I'm sure there are mathematics which work in larger cardinalities and while mathematicians may not mind if their work meets physics (I'm sure it's a delight when it does), does such mathematics find application?

P.S: I use the word "naive" as I don't think that our imagination exceeds that of the universe.

Answer 1

Conceivably it might happen. The possibility of the existence of some wave-function might be shown to depend on the existence of some subset of N with certain properties. E.g. the existence of a measurable cardinal implies the existence of a thing called $O$^# (Oh-Sharp). The existence of $O$^# (i) is equivalent to the existence of a certain kind of subset of N, and (ii) implies the consistency of a proper class of inaccessible cardinals.