You at exactly right. The "paradox" is that people naively assume that the Real ("Real" = math theory, "real" = CS turing-computable and physics "real univers

You at exactly right. The "paradox" is that people naively assume that the Real ("Real" = math theory, "real" = CS turing-computable and physics "real universe") numbers are a smooth continuum, but if you follow the actual definition, you discover that it is too powerful -- the Real numbers could construct physically impossible objects, which proves that real numbera aren't real. In reality, we must confine ourselves to countable sets, which has ugly asymmetries: we must distinguish a set of "nameable" numbers as more "real" than the others, but we can choose any small-enough subset of Real numbers we want, we can choose every single Real number we practically encounter, but we can't choose all of them at the same time.