Godel's proof of incompleteness and the real line?

Question

As far as I understand, Godel's incompleteness theorem uniquely numbers the possible statements that can be built and then applies the diagonalization argument of Cantor. But any system that can represent the real line will by default have a non countable list of true statements, right? It seems that a much more powerful form of incompleteness would be if you could omit a whole category of statements and their implications and still prove incompleteness. I am not yet familiar with the complete vocabulary of mathematical logic, but is my intuition right?