In what follows consistency is assumed through-out (to avoid adding a qualification to every sentence). At the very first level the following can be thought of as an answer to the question:
https://www.physicsforums.com/threa...xiom-or-a-theorem.1053712/page-2#post-6913946
I will quote the relevant part here:
SSequence said:
If we have an incomplete theory under consideration [as often happens to be the case] then (assuming consistency) every statement exactly falls into one of the following three categories:
(i) provable in the theory (or a theorem of the theory)
(ii) disproveable in the theory
(iii) independent
And once again the statements in category-(ii) that would be disproveable (in the theory) would have their negation as a theorem [theorem of the theory that is].
So in that sense CH just falls into category-(iii).
==============================Here is a somewhat more detailed way of looking at it (and also sort of a heuristic way of thinking about it). But it is still quite basic of course and lacks depth and detailed justification for (too many) finer points (which I don't know either). It is more of a heuristic description as a way of looking at it. However, lacking detailed understanding, I still found it fairly useful.
Essentially we think of "world of sets" as given to us
[or perhaps in other words "collection of all sets"
]. This is denoted by ##V## (also called cumulative hierarchy). It can be thought of as the power-set operation running through the ordinals. We set ##V_{0}## as the empty set. Next we set ##V_{\alpha+1}=\mathcal{P}(V_{\alpha})##. For limit ordinals ##\alpha##, ##V_{\alpha}## is defined as the union of all lower levels. The following two points are quite basic but are worth mentioning because of their importance:
(a) For every set ##A##, there would exist some (smallest) ordinal ##\alpha## such that ##A \in V_{\alpha}##.
(b) For all ordinals ##\alpha##, ##V_{\alpha}## will be a set. However, ##V## itself is not a set.Now when we talk about models, there are two kinds of "models":
(i) set models
(ii) class models
A more complete description is well-beyond my own understanding/knowledge. However, knowing a few points about models is fairly useful (to get a very rough picture):
(a) A set model is, as the name implies, just a set.
(b) A class model can be thought of as a "collection" in some sense. However, they aren't sets. That's because class models pick elements from ##V_{\alpha}## with ##\alpha## taking arbitrarily large values in ##\mathrm{Ord}##. In an informal sense, they are too big to be sets.
(c) Normally texts often just "write" model with the context of whether a "set model" or "class model" is being talked about as
understood from the context.
(d) There is a certain (precise) sense which makes qualifies a "set/class model" as a "model". Informally it is said that a model satisfies all the axioms of ZF(C)
[and I think the sense is probably slightly different for set models and class models
]. However, to be honest, I don't really know what that means in a more precise sense. I had quite vague sense of it few years back, but I have forgotten it. If you look up for it you might be able to find some descriptions at least.
Nevertheless, if you picked up any specific set from ##V##, it will either be a (set) model or it won't.
(e) Quite importantly, the "world of sets" ##V## itself is a class model.
Sorry this got a bit long. But now coming to the question, what does this have to do with CH? You already kind of described it in your question. Basically every statement/question that can posed in ZF(C)
[and basically CH is one of them
] has either a true or false value in a specific model
[be it a set model or a class model
]. Now this is how it relates to your initial question. When we think about any specific ##V##, the value of CH will be either true or false in the given ##V##. There will be no two ways about it.
For example, suppose that our ##V## is what is called "constructible universe"
[note that word "universe" is a just a mathematical usage
]. Then CH will be true in such a ##V##. In fact, GCH is also true in such a ##V##.
However, it is also possible to have a ##V## where ##CH## is false. But it is perfectly possible that we could have class models
[that, very informally, select a sub-collection of the elements in ##V## using something like a additional axiom I think
] for such a ##V## where ##CH## could be true.
And that is just about the limit of the depth of my understanding for the specific question at hand :P.
