# Justifying the use of real numbers for measuring length

I am not sure if this is the most appropriate place to post this but here goes nothing:

Assume we were trying to come up with system of numbers $S$ to model our intuition of length. We want $S$ to have these properties intuitively at least:

1. S is an abelian group under some operation (say +). Abelian because we want it count stuff.
2. It should be a $\mathbb{Q}$ - vector space. This corresponds to our notion of divisibility of length, given a ruler, we can imagine $1/k$ th of that ruler.
3. It should be archimedean. No observation contradicts it.
4. We should have a notion of limits in it through something like the intuition behind Zeno's paradoxes. Essentially, we are asking for completeness.

I think these are all our intuitions in mathematical language. Of course, these are also enough to force our system to be the familiar reals. I am not sure if anyone has done something similar to this before and I have a few questions.

Does the other stuff we model by the reals(temperature, probabilities, entropy etc) also follow the same/similar intuitions? Is there any reason that all our measurements have these properties?

If not, do we measure other properties in physics by other systems? The only one I can think of is complex numbers in Quantum Mechanics but I don't know anything about that.

Finally, is it coincidence that we have a uniqueness theorem for exactly those properties that model our intuitions about the world?

I am sorry that my questions are vague/philosophical/physical. This seemed like an interesting enough phenomenon to post anyway.