> If I can accept that there are more reals than reals with finite descriptions, why can't there be integers without finite descriptions/positions?

Because the integers are defined as the finite successors to the number 0 and their additive opposites. See, for instance, the construction section of: https://en.m.wikipedia.org/wiki/Integer

You can start talking about numbers that can't be constructed in this way, but then you've ceased to be talking about the integers.