Here's a quote that I think you may find interesting, as it seems to be along the lines of what you are saying: "A meta-mathematical proof of the consistency o

Here's a quote that I think you may find interesting, as it seems to be along the lines of what you are saying: "A meta-mathematical proof of the consistency of arithmetic is not excluded by...Goedel's analysis. In point of fact, meta-mathematical proofs of the consistency of arithmetic have been constructed, notably by Gerhard Gentzen, a member of the Hilbert school, in 1936. But such proofs are in a sense pointless if, as can be demonstrated, they employ rules of inference whose own consistency is as much open to doubt as is the formal consistency of arithmetic itself. Thus, Gentzen used the so-called "principle of transfinite mathematical induction" in his proof. But the principle in effect stipulates that a formula is derivable from an infinite class of premises. Its use therefore requires the employment of nonfinitistic meta - mathematical notions, and so raises once more the question which Hilbert's original program was intended to resolve." -Ernest Nagel and James Newman Taken from https://www.mathpages.com/home/kmath347/kmath347.htm https://www.mathpages.com/home/kmath347/kmath347.htm