But I can build a computer that would accept any pair of integers you sent it, add them and return the result. Granted, they'd have to be streamed over a network, in little-endian form, but unbounded integer addition is truly doable. And the restriction is implicit: you can only send it finite integers, and given finite time it will return another finite answer.

You can't say the same about most real numbers, all the ones that we postulate must exist but that have infinite complexity. You can't ever construct a single one of them.