[h0mEDw]

Yup, it is possible. That was just a hand wavy example to show where the issue is. Symbolic real numbers need not be represented as small programs/objects generating a sequence of digits. But all of those implementations/representations will fail to compare some numbers.

Edit: for instance, you could imagine a representation (think object of a virtual class in c++) that gives you a fraction and a 1/2^n bound for error, where you can query the number for whatever n you see fit. Still, in some cases it's undecidable to determine which number is greater.