In ordinary mathematics there are plenty of ways to define the real numbers, but they're all equivalent; you can translate from one to another and if anything's

In ordinary mathematics there are plenty of ways to define the real numbers, but they're all equivalent; you can translate from one to another and if anything's true of one then its translation will be true of another. With the restrictions being used here, though, the different definitions aren't all equivalent, and apparently it turns out that one version of "the real numbers" is necessarily countable and another isn't (because it's countable "in a particular topos"). "Countable" and "enumerable" are the same thing. (But there's a separate term, "recursively enumerable", that means something else.) One way to see that the rational numbers are countable: write them down in a sort of square where m/n goes in position (m,n). You'll have e.g. 2/3 and 4/6 and 6/9 and ..., all in different places, but that's harmless. Then start at (1,1) and walk along the finite diagonals: 1/1, then 1/2, 2/1, then 1/3, 2/2, 3/1, etc. Boom, you've counted the rational numbers :-).