[chriswarbo]

Your theory will have countable models, but via Goedel (and assuming we're ignoring inconsistent theories) it will be incomplete, capturing only a subset of the reals' properties.

Any particular theory, capturing some aspect of the reals, can be given a countable model. Those "constructed reals" are countable, but they miss out almost all of the reals.

Think about it this way: a theory is like an interface, a theory of the reals provides an API to access the reals. The API is necessarily limited; not least because there are only countably many ways we can combine the operations.

Now consider a mock implementation of that interface: rather than passing around reals, we invent some (countable) dummy objects to use instead. Since the API is limited, we can always come up with a mock implementation which cannot be distinguished from the real implementation using that API/theory.

That doesn't mean that the reals are countable. It does mean that uncountability is an implementation convenience; regardless of what result we're after, we could get it using countable objects, but we might have to spend a lot of effort "out smarting" the algorithms we're using (i.e. coming up with our "mock" set).