And even if the theorem prover has no errors, the math behind the theorem prover could be inconsistent according to Gödel's incompleteness theorem. So even in principle, we can never be completely certain about the soundness of Rust, though the more theoretical concerns tend to be dwarfed by basic human error.

I think it's a good thing to keep in mind, that the actual implementation of Rust almost certainly isn't sound. Even an "ideal Rust" still has unsafe, which can affect the soundness of even faraway safe code. But I don't think it's nessesarily wrong to talk about Rust as if it were safe. Much of our communication has implicit assumptions, and the way we talk about Rust is no different. It would be exhausting to qualify every discussion of saftey with "assuming the compiler has no bugs", "assuming all unsafe code follows the (not fully specified!) rules", etc.

Ultimately it's like any other implementation of theory into practice. The theoretical model of Rust lets us reason about and design an implementation to achieve certain practical goals even though the model is a simplification of reality. In Rust's case, practical use has demonstrated it achieves design goals like reducing bugs and reducing development time (compared to peers like C++) to a substantial degree.

This moving of goal posts whenever correctness through type systems is being discussed is a bit tiresome, because it quickly leads to such ad absurdum corner cases.

Yes, you are all correct. The compiler may have a bug. And if it does not, then the type system may have a problem. And if it is formally verified, then the theorem prover might have issues. And if it does not, then the very math it relies on may be wrong.

But what do we contrast this to, what is the alternative to correctness through the type system? Almost every time, when it isn't "nothing", it's "tests". We rely on tests to test whether the program behaves correctly.

And as we all know, quite in contrast to type systems and their checkers, tests are always error-free, complete, and sensible?

Or more likely, an error in the formalization. "What you proved is true, but it's not sufficient for what you care about." See for example WPA2 KRACK attacks after WPA2 was proven secure.