I’ve had this paper downloaded for about a decade and haven’t gotten around to reading it, but thinking about it, especially if space and time are quantized (an undetermined question last I checked and almost certainly still so), there would exist numbers in ℝ that cannot be expressed as physical quantities, even with an infinite universe. It’s possible that even the algebraic numbers include numbers that are non-physical (although it might be a larger subset of numbers than the constructible numbers depending on the structure of space-time’s quantization).

If reality is quantized and there is a smallest number that is physically relevant, you don't need the reals to break it. Take that smallest number and divide it by two, and now you have a physically meaningless number using only the rationals.

This isn't fair for what quantization means in reality, but I'm just pointing out that you don't have to introduce the real numbers to get physically meaningless quantities.