The idea is we can't actually prove a non-computable real number exists without purposefully having axioms that allow for deriving non-computable things. (We can't prove they don't exist either, without making some strong assumptions).
Sorry, what do you mean?
The real numbers are uncountable. (If you're talking about constructivism, I guess it's more complicated. There's some discussion at https://mathoverflow.net/questions/30643/are-real-numbers-co... . But that is very niche.)
The set of things we can compute is, for any reasonable definition of computability, countable.
In this case, to actually prove the statement internally that "not every real number is computable", you'd need some non-constructive principle (usually added to the logical system rather than the theory itself). But, the absence of that proof doesn't make its negation provable either ("every real number is computable"). While some schools of constructivism want the negation, others prefer to live in the ambiguity.
Formal reasoning is so powerful you can pretend these things actually exist, but they don’t!
I see you are already familiar with subcountability so you know the rest.
Doesn't that formal string of symbols exist?
Seems like allowing formal string of symbols that don't necessarily "exist" (or well useful for physics) can still lead you to something computable at the end of the day?
Like a meta version of what happens in programming - people often start with "infinite" objects eg `cycle [0,1] = [0,1,0,1...]` but then extract something finite out of it.
List functions like that need to be handled carefully to ensure termination. Summations of infinite series deal are a better example, consider adding up a geometric series. You need to add “all” the terms to get the correct result.
Of course you don’t actually add all the terms, you use algebra to determine a value.