"integer" as a type is less offensive though, as it only has one intuitive deficiency compared the mathematical definition (finite range). Where as "real" as a type has many deficiencies... it simply does not contain irrational numbers, and it does not contain all rational numbers in 3 respects: range, precision and unrepresentable due to base2, and for integers it can represent a non-contiguous range!
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.
It is not a matter of modernity, but a matter of use-case. You don't need arbitrary-length integers for a coffee machine or a fridge. You don't even need FP to handle e.g. temperatures; fixed-point is often more than enough. So if you are making some sort of "portable assembler" for IOT devices, you can safely stick with simple integers.
