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
- We cannot prove or disprove P
- We cannot prove or disprove that we cannot prove and disprove P
- we cannot prove or disprove (2.)
- we cannot prove or disprove (3.)
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?