The real (no pun intended!) distinction is between computable and noncomputable reals, as first identified by Alan Turing in "On Computable Numbers". The integ

The real (no pun intended!) distinction is between computable and noncomputable reals, as first identified by Alan Turing in "On Computable Numbers". The integers are a subset of the rationals, which are a subset of the algebraic numbers, which are a subset of the computable numbers, which (most mathematicians would accept) are a subset of the reals. There are lots of subtleties about this. In fact Scott Aaronson just started a series that is in some sense about some of those subtleties, which I hope to be able to understand some day!