[danharaj]

This is an extremely unusual conception of epistemology, even for mathematics. This knowledge that is contingent on axioms isn't at all convincing. At the beginning of the 20th century, this was a raging debate. It wasn't quite resolved, it just didn't quite matter to practicing mathematicians so it kind of faded into the background.

There is a massive gap between 3 and the cardinality of the continuum. 3 is directly examinable. If I count some collection of objects, my fingers say, I will immediately grasp 3.

On the other hand, the set of real numbers is a highly pathological, abstract concept. Everything we know about the reals suffers from two severe deficiencies: One, it depends on infinitary axioms which presuppose facts about the phenomenon we would like to investigate. Two, even what we can infer from such axioms is always indirect evidence. That's not surprising: All reasoning about infinity is indirect.

In certain mathematical settings it is even true that the Cauchy sequence definition of real numbers and the Dedekind cut definition do not agree. At the very least, the reals isn't even a thing: there are many reals. That you say that "we know the cardinality of the reals" because we know CH is independent of ZFC is... preposterous to say the least. "If you assume you know the cardinality of the reals, then you know the cardinality of the reals; therefore we know the cardinality of the reals" is basically what such axiomatic acrobatics boils down to. Axioms are not knowledge.

[no_identd]

Here, another interesting construction of the real numbers:

https://arxiv.org/abs/math/0405454 - The Eudoxus Real Numbers

https://ncatlab.org/nlab/show/Eudoxus+real+number

https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...

(Surprisingly enough, ncat & Wikipedia complement each other here in their explanations of it.)

[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.