The cardinality of the natural number set is the same as the cardinality of the rational number set. So, in some sense, you are saying that we know that the natural counting numbers are "real," which is a self-evident truth. In another sense, you are saying that fractions of an inch/cm/etc are "real" which is another self-evident truth.
The question is whether uncountable infinity somehow exits in nature (in the case of real numbers, it is a question about the infinitesimally small scale). To say that it's a "polite fiction to make some proofs works" is too strong a statement, we do not know the answer to that question. At the same time, our measurement instrument will always be discrete and bounded, so the question is seemingly beyond science itself.