I haven't seen a formalisation of "higher order" relative independencies in ZFC, but it probably exists somewhere, I just haven't found it yet. For the sake of this question, let an n-independent statement be one such that its (relative) independence's independence's... independence is independence (n times), beginning at index 1 for the singularly (relatively) independent statements (e.g. AoC, Continuum hypothesis, etc).
My two (main) questions are as follows:
Do >1-independent statements exist?
If so, does there exist a statement which is "infinitely"-independent?
I haven't been able to construct an example of a 2-undecidable statement yet, I'm still trying to wrap my head around the concept of it existing (if it even does?), and what it even means to be doubly undecidable.
I'd like to hear other peoples' ideas when it comes to this concept!
