Fundamentals of Truth, Provability and Axioms by means of the Continuum Hypothesis Let S be the structure/language of ZFC (including PL 1). Let CH refer to the well-known continuum hypothesis. My claims are as follows and could u just say if it's true or wrong and why?

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*In S neither CH is true nor false because in S only tautologies and contradictions are already true/false and CH is not such.


*Let's say I assume only one axiom in S that says: there exists an empty set. Now in this scenario CH again is neither true nor false because we still can't talk about cardinalities of sets at all (and so we can't talk about CH).


*Let's say I assume ZFC in S. Now we can talk about cardinalities of sets. That means that here CH is a wff and so it is either true or false. But we cannot prove which one it is (Gödel, Cohen). But it means: CH is true xor false in ZFC in this very moment, we just don't know and we just will never know!


*If we just brutally assume CH to be true in ZFC (ZFC + CH), then there's no inconsistency (proof by Gödel), but if we take ZFC + ~CH we can prove there's no inconsistency either (Cohen), so ZFC is - loosely spoken - too general to catch CH's truth/falseness properly, just like a fisherman's net is sometimes too big to catch certain fishes.
 A: There are several issues here, which may not feel important at first but over time will cloud the (already quite nuanced) picture.

First of all, you're conflating structures, theories, and languages. In increasing order of complexity:

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*A language (also called a signature or vocabulary) is a set of non-logical symbols, such as $\{\in\}$ or $\{+,\times,0,1,<\}$.


*A theory is a set of first-order sentences, and for a language $\Sigma$ a $\Sigma$-theory is a theory consisting of sentences in the language $\Sigma$ - e.g. $\mathsf{ZFC}$ is an $\{\in\}$-theory and first-order $\mathsf{PA}$ is an $\{+,\times,0,1,<\}$-theory.


*A structure in a given language is a set together with an interpretation of the various symbols in that language in a precise sense.
Whether or not a particular string of symbols is a wff depends only on the language involved, not on what axioms we're considering nor on what structure (if any) we're specifically focusing on. $\mathsf{CH}$ is a wff in the language $\{\in\}$. What the empty $\{\in\}$-theory (your "$S$") can't do is prove basic things about $\mathsf{CH}$ and related sentences. So $S$ can talk about $\mathsf{CH}$, it just doesn't have much to say. This issue is implicit in $(1)$ and $(2)$, and explicit in $(3)$.

Now on to the more subtle point: truth and falsity. The satisfaction relation $\models$ connects structures and sentences/theories, with "$\mathcal{A}\models\varphi$" (resp. "$\mathcal{A}\models\Gamma$") being read as "$\varphi$ is true in $\mathcal{A}$" (resp. "Every sentence in $\Gamma$ is true in $\mathcal{A}$"). But we use the term "true" only in this context; when talking about theories, the relevant term is provable.
The main reason for reserving terms like "true" and "false" for structures as opposed to theories is that the standard properties of truth such as bivalence only hold of truth-in-a-structure, not provability-in-a-theory. By separating the terms we make it easier to be precise and avoid subtle errors. This is an issue in your point $(3)$, where truth and provability get mixed up. In particular, the statement

CH is true xor false in ZFC in this very moment, we just don't know and we just will never know

doesn't parse.
OK, unfortunately you will find people say that things are true/false in $\mathsf{ZFC}$. The connection is that a sentence is provable in a theory $T$ iff it is true in all models of $T$, so this isn't totally unjustified. But this is an abuse of terminology, and should be avoided until the fundamentals of the topic are mastered.

After shifting from truth to provability, point $(4)$ then is correct with one slight additional hypothesis: assuming $\mathsf{ZFC}$ is consistent in the first place, both $\mathsf{ZFC+CH}$ and $\mathsf{ZFC+\neg CH}$ are consistent.
