Perhaps you would like to study constructive analysis, a la Bishop? You lose a lot of intuition in the hypotheses, which necessarily become more complicated—or perhaps I should say that you are forced to develop a different intuition about hypotheses—but you gain conclusions of exactly the sort that you want.
EDIT: Also, the fact that a structure is uncountable doesn't prevent you from operating algorithmically on it (not that you claimed it did!); it just means that there are some elements of it that cannot be singled out algorithmically.
https://en.wikipedia.org/wiki/Constructive_analysis
EDIT: Also, the fact that a structure is uncountable doesn't prevent you from operating algorithmically on it (not that you claimed it did!); it just means that there are some elements of it that cannot be singled out algorithmically.