recursively impossible proof

Question

According to Godel’s theorems, we know that some propositions are expressible in a sufficiently expressive logic system, that cannot be proven or disproven

We also have proofs that some statements cannot be proven or disproven in certain theories (eg. as the continuum hypothesis in ZFC, or the axiom of choice in ZF).

My question is, is there a statement P such as

1. We cannot prove or disprove P
2. We cannot prove or disprove that we cannot prove and disprove P
3. we cannot prove or disprove (2.)
4. we cannot prove or disprove (3.)

$\ldots$

n. we cannot prove or disprove (n-1)

$\ldots$ and so on for all n

What would be necessary and/or sufficient conditions on a theory that has such statements?

Answer 1

We can construct an undecidable formula $\phi$ in the language of Peano Arithmetic (PA) such that

PA $\vdash\phi\leftrightarrow\neg\Box\phi$,

where $\Box$ is the provability operator.

Iterating on $\psi$ as $\phi\leftrightarrow\neg\Box\phi$

$\psi'$ as $\psi\leftrightarrow\neg\Box\psi$

$\psi''$ as $\psi'\leftrightarrow\neg\Box\psi'$ and so on,

we get a family of undecidable formulas on the assumption that PA is consistent.

Hence, any consistent and incomplete system that we can do these operations has to suffice. By an employment of Solovay's Arithmetical Completeness Theorem, the task might be simpler.