[gowld]

Getting back to the OP: in a very real sense, the set of Real numbers does not exist. Therefore, it is impossible to know its size via experience. The only possible way is to derive it from chosen axioms.

Note that even your conception of 3 is reliant on axioms. How do you "know" that 3 ducks is the same 3 as 3 fingers? Only via axioms.

[danharaj]

That's sounds a little absurd. A five year old child knows that 3 fingers is the same 3 as 3 ducks. Certainly 5000 years ago people understood 3 without ever knowing what an axiom is. The peano axioms didn't create arithmetic, arithmetic created the peano axioms. S(S(S(0))) follows | | |, not the other way around.