Is the set of all mathematical truths countable or uncountable?

A theorem is, by definition, a finite string of symbols that can be derived by some specified proof system. Because there are countably many finite strings of symbols, there are at most countably many theorems in any given theory.

Speaking of "a set of mathematical truths", where the elements of that set is supposed to be something different from symbolic representations, is not well defined. What kind of object is "a mathematical truth" to you such that you can put them into a set and count them? There is no definition of such a thing in general use, except for the formulas of symbolic logic.

In model theory one can speak of theories where the set of possible symbols are uncountable, such as a symbol for each real number. Such a theory, of course, has uncountably many theorems of the form $c=c$. However, these theories are generally considered artificial objects of study. Studying them can be useful as a stepping stone in proving things about ordinary countable theories, but their formulas are not considered to directly represent "mathematical truth".

If you have countable many theorems $T_1,T_2,T_3...$, you can construct an uncountable set of new theorems by stating that every subset of {$T_1,T_2,T_3...$} is a theorem.

Additionally there are true finite statements which in a given formal system is only provable by a countable infinite number of theorems. (eg. Goodstein's theorem in PA).