Can we go further and ditch the reals, relying instead on rational numbers or even IEEE floats? After all, the computers that we use for predicting empirical results all run on integers.
Can you? Most likely. Should you? You’ll need to reprove more than a handful of theorems, and for what? What advantage does using rationals instead of reals get you? You might enjoy taking courses in real & complex analysis, the general purpose of which is to impart upon the receiver an understanding of why we’ve constructed those particular number systems and how despite the names they both describe things which are…
Having implemented exact real computation to better understand reals, I think of a real number as a kind of machine that generates infinite streams. Operations on them instantiate new machines which query their real operands, computating until there's sufficient information to emit a next term of the stream. When the next term needs an infinite amount of information to decide what to spit out next, it results in an "unproductive" infinite loop.
Rational numbers are interesting, more realistic, because they always terminate. In the real world, measurement tolerances and physical limits means at some point having to extract a rational. When we work with reals we are really only working with rational approximations or symbols with associated properties and relations.
Reals are a powerful and elegant tool to rigorously reason about mathematical spaces and operations on algebraic objects but trying to work with them in reality in their exact form is a fun and visceral lesson on the nature of undecidability. It's hard to go two steps without tripping over a non-terminating loop (such as any operation that starts with an irrational and results in a rational or equality testing in general).
This is an observation on our interface with reality and not on its true nature, which may or may not admit reals (although my non-serious guess is that black holes form whenever you try to do something that requires a proper real number).