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Sniffnoy comments on No coinductive datatype of integers - Less Wrong
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I'm really at a loss as to why such a thing should be intuitive. The additional condition seems to me to be highly unnatural; Ramsey's theorem is a purely graph-theoretic result, and this strengthened version involves comparing the number of vertices used to numbers that the vertices happen to correspond to, a comparison we would ordinarily consider meaningless.
If I'm following cousin it, the idea doesn't have anything really to do with the statement about Ramsey numbers. As I understand it, if in some system that is only slightly stronger than PA we can show some statement S of the form A x in N, P(x), then we should believe that the correct models of PA are those which have S being true. Or to put it a different way, we should think PA + S will do a better job telling us about reality than PA + ~S would. I'm not sure this can be formalized beyond that. Presumably if it he had a way to formalize this, cousin it wouldn't have an issue with it.