It's been proven that the Continuum Hypothesis is independent of ZFC, yet some people still talk about it being "true" or "false", or that we need to search for a non-mathematical justification for it. But I'm not really sure what all of this means.
The way I see it, the fact that CH is independent just means that there's two versions of the concept of sets, but both are equally valid, similar to the way the independence of the parallels postulate means that there are two versions of geometry (Euclidean and Hyperbolic).
But if I think about the natural numbers, and take some statement $P$ independent of Peano Arithmetic, I wouldn't say "there's two versions of the natural numbers, both of which are equally valid", because I have some "physical model" of the natural numbers (by drawing dots on a sheet of paper, for example). I would say that it's possible to justify either $P$ or $\lnot P$, and that if we assumed the other one, it wouldn't be right to call the objects we're talking about "natural numbers".
So my question is, when people talk about something like the CH being true/false, do they have some "intuitive model" of what a set is, and if yes, what does it look like?