I thought that independent (undecidable) statements are totally different than the Gödel sentences which demonstrate incompleteness. The latter is a statement which is true in the axiomatic system but which cannot be proven using the axiomatic system. The former is just a statement that essentially has no truth value in the axiomatic system.
The Godel sentences are of a different character than the continuum hypothesis (CH) because the Godel sentences are simple first-order arithmetic statements, while the CH is a higher-order, a.k.a. analytic statement. A Godel sentence can be assigned a truth meaning via Tarski's definition of truth independent of the axiom system in a way that is much harder to do with the CH.
Basically a Godel sentence says something about whether a given piece of software terminates when run on an ideal computer (specifically a piece of software that hunts for a proof of a contradiction within a specified axiom system). I'll argue that whether a specific piece of software would halt or not when run on an ideal machine has a definite truth value independent of any axiom system. Whereas CH doesn't really afford such a software interpretation.
I do respect the fact that there exist models of PA + ~Con(PA) but these models are non-standard and we don't use such models to reason about software, specifically because they are unsound in this sense.
You're quite right that CH is not an example of a Godel sentence which demonstrates incompleteness (examples of that are given in the linked article) -- I didn't mean to imply otherwise. I just wanted to share that this was a more practical example of incompleteness in our modern mathematical framework (ZFC) which folks might find interesting.
But as others have said, both statements are equally unprovable in ZFC. The Godel sentences demonstrating incompleteness are constructed in such a way that you could argue (outside of the axiomatic system) they are provably true or false, while CH is a case where reasonable mathematicians may disagree on whether it is true or false. But ultimately there is no proof in ZFC for either, so they are both examples of incompleteness in ZFC.
And note that the Godel sentence demonstrating incompleteness doesn't need to be true -- the inverse of the Godel sentence demonstrating incompleteness is also unprovable.
Something being true in a axiomatic system is the same thing as it being provable; that's what "true" means. While a Godel statement for X can be interpreted as "X does not prove this statement", that interpretation inherently relies on the semantic implied by X. The Godel construction is systematic way of generating independent statements without needing to know anything specific about the axiomatic system.
Ah yes, you're right, and my lazy wording in the former comment is inaccurate. A Gödel sentence is just a statement written in the syntax of whatever formal system we're dealing with: generally, it's a statement that there exists no natural number which satisfies a particular property. The formal system cannot prove or disprove that statement.
As you said, we tend to call the statement "true" because we know that the formal system itself was designed with the intention to describe natural numbers and arithmetic, and the statement was designed intentionally to refer indirectly to itself and claim its own unprovability. Since the statement is formally unprovable, we interpret it as being true. I had forgotten that Gödel actually showed that there are other interpretations of the formal system in which the Gödel statement is false.
