> Since Peano arithmetic cannot prove its own consistency by Gödel's second incompleteness theorem, this shows that Peano arithmetic cannot prove the strengthened finite Ramsey theorem.

This seems quite circular, where the (claims mine) contrived statements enabling godel 2 incompletude theorem, make it unable to prove a non-contrived statement?

Thanks for the reference though, I'll admit that is a topic I'm not familiar enough with to fully challenge it.

> I must say, be careful with "reasoning". For example, once a philosophy PhD minimized mathematics eloquently by saying all mathematical results are "tautological", hence uninteresting. Hence, why bother studying it, and in your case, why bother thinking Gödel did anything special.

Yeah that's a classic, it's a good thought experiment but that only prove that one must not be careful with the pruning that can enable high level reasoning but more with being careful of fallacious or misleading reasonings. Yes in theory, mathematical chain proofs are only tautologies derived from ZFC/higher order logic, so yes mathematics doesn't say anything new. However in practice, the task of unfolding reasoning chains and being able to refer to past lemnas as abtractions/objects, enable the reader to optimize for cognition, and hence the more proofs advances, the more we can discover, retain, understand and refer, useful tautological yet transitive knowledge.