@GeorgeMarangelis My construction is Exercise 6.15 in the book by Kechris. I'll see if I can come up with a better reference.

One minor thing. We are using $\mathrm{Col}(\omega,<\kappa)$, which is not countably closed, but preserves stationary subsets of $\kappa$ by the $\kappa$-c.c.

The question seems to assume GCH and AC have somewhat the same status, both as useful but controversial set theoretic principles, each of whose cases for axiomhood is strengthened by their applications. But AC is nearly universally accepted as a foundational truth with intrinsic appeal, and its usefulness stems from it already having been used, sometimes implicitly, in establishing mainstream mathematics. GCH is not viewed as intrinsically justified. There are other contradictory principles of arguably equal status. It hasn't been implicitly used in the mainstream.

I tried getting a situation like this from a huge cardinal but couldn’t quite work it out. You can find a stationary set of things that are all isomorphic to a fixed $V_\kappa$, but if there are extra constants in the language, the copies of $V_\kappa$ don’t all collapse to isomorphic models in the larger language.

Thanks for your answer. Could you say a bit about the consistency proof for the class of indestructibly sigma2 correct cardinals?

@AlexKruckman I wouldn’t call it a sociological question, because it’s not about their behavior in general. It’s just a question about the direction of current research.

What about any example of showing a statement independent of some axioms by constructing models? This can be quite explicit and non-diagonal. But maybe that’s not what you mean by decidability?

@PeterLeFanuLumsdaine This is not mathematical literature. This is an account of personal grievances.

First order logic is only a slight abstraction of things you can write on paper. It should be comprehended prior to the axioms of set theory.

@JoelDavidHamkins that’s right, thanks.

It seems one needs to find $\alpha$ such that the definition of $L_\xi$ is absolute to $L_\alpha$. Or is this not really an issue?