Thanks, fun discussion. I think we understand each other. I’m not so worried about consistency. Even ZFC has the issue you describe. I found this paragraph fro

Thanks, fun discussion. I think we understand each other. I’m not so worried about consistency. Even ZFC has the issue you describe. I found this paragraph from Wikipedia about Gödels 2nd incompleteness theorem very interesting. > The corollary also indicates the epistemological relevance of the second incompleteness theorem. It would actually provide no interesting information if a system F proved its consistency. This is because inconsistent theories prove everything, including their consistency. Thus a consistency proof of F in F would give us no clue as to whether F really is consistent; no doubts about the consistency of F would be resolved by such a consistency proof. The interest in consistency proofs lies in the possibility of proving the consistency of a system F in some system F' that is in some sense less doubtful than F itself, for example weaker than F. For many naturally occurring theories F and F', such as F = Zermelo–Fraenkel set theory and F' = primitive recursive arithmetic, the consistency of F' is provable in F, and thus F' cannot prove the consistency of F by the above corollary of the second incompleteness theorem.