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I'm happy with theorems in $Z_2$ that are provably (or conceptually "obviously") harder to prove in $PA$. I would therefore agree with your restatement... and wish for some sort of "speedup theorem", or at least a plausible heuristic argument for such, for a "useful" theorem in number theory...
So are you saying that many techniques were originally developed for complete fields, after which completeness could be dropped and we were still left with useful techniques?
@ThomasRot This is very nice!! My peeve is that it doesn't seem to treat Cerf's work at all and merely outlines Hatcher's proof of the Smale Conjecture- but it looks wonderful nonetheless!