Gödel's Incompleteness Theorems and Independent Statements I am a physics student currently taking a detour into mathematical logic, and some concepts about Gödel's Incompleteness Theorems confuse me. I have self-studied a bit of axiomatic set theory, but I am still quite new to deep results like the incompleteness theorems, and at this point I only aim to understand them conceptually.
I understand the first incompleteness theorem shows that, for a system rich enough to develop arithmetic, we can always construct a statement that reads 'I am unprovable', which must be true if the system is consistent. This means there exists an unprovable truth within mathematics. Gödel managed to construct this statement using his ingenious numbering system.
What confuses me is the connection between 'unprovable truths' and independent statements. I understand independent statements are sentences that cannot be derived from a list of axioms, like how ZFC can't prove or disprove the Continuum Hypothesis, and how ZF can't prove or disprove the Axiom of Choice. We know these statements are independent because we can construct models of ZFC where CH can be true or false, and we can construct models of ZF where AC can be true or false.
But does Gödel's theorem say anything about the existence of these independent statements? From the articles I've read, it seems like people use CH as an example to illustrate Gödel's theorem, but it seems to me like Gödel's 'I am unprovable' is quite different from CH. For 'I am unprovable', this statement is true, but we don't have a proof for it from the axioms. Meanwhile, does it make even sense to speak about the truth value of CH?
 A: Informally, "I'm unprovable" is an independent statement under some consistency hypothesis. As you said, this statement must be true, therefore there cannot be a proof thanks to consistency. But there cannot be a disproof either, otherwise the theory is not $\omega$-consistent (or some weaker version of consistency at least) -- roughly, even if we have a proposition $P(n)$ such that $P(0), P(1), P(2), \cdots$ are all false, this is not in direct conflict with the statement $\exists n\in \mathbb N, P(n)$, because to exhibit a contradiction, we need to put infinitely many statements together; and a theory is $\omega$-consistent if this doesn't happen.
Godel's second incompleteness theorem gives a more explicit independent statement, i.e. the consistency of the theory itself (more precisely the statement that codes its consistency) is independent. Historically, it was briefly wondered whether independent statements must have some meta-flavor that somehow codes a reference to the system itself. The independence of axiom of choice, CH, etc has disproved this thesis.
In logic, it doesn't make sense at all to say something is true or not, but only provable or not. One can only discuss "Truth" for specific models. But we cannot really construct a canonical model of ZFC (though Godel's universe is a good candidate, in which GCH holds), and then discuss the truth value of CH in that model -- after all, we can only do "forcing" (in this sense, perhaps we should accept GCH, as it's much easier to show ZFC+GCH is consistent than ZFC+ $\neg$ GCH).
That said, I had overheard some expert in the field expressing the hope that eventually the mathematical community will agree with each other about the truth value of CH, just like we have (almost) all accepted axiom of choice.  After all, set theory is just a tool to express our intuitions about sets, and there might be some kind of platonic existence of sets that we are trying to study/characterize.
