I would argue that even the rational numbers are unphysical in the same way that the integers are!

The idea that a quantity like 1/3 is meaningfully different than 333/1000 or 3333333/10000000 is not really that interesting on its own; only in the course of a physical process (a computation) would these quantities be interestingly different, and then only in the sense of the degree of approximation that is required for the computation.

The real numbers in the intuitionalist sense are the ground truth here in my opinion; the Cantorian real numbers are busted, and the rationals are too abstract.