Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!wasatch!cs.utexas.edu!tut.cis.ohio-state.edu!bloom-beacon!mit-eddie!uw-beaver!cornell!rochester!pt.cs.cmu.edu!andrew.cmu.edu!ap1i+ From: ap1i+@andrew.cmu.edu (Andrew C. Plotkin) Newsgroups: comp.ai Subject: Re: Natural Paradox Message-ID: Date: 9 Feb 89 19:41:59 GMT References: <1706@tank.uchicago.edu> <9526@ihlpb.ATT.COM>, <44585@linus.UUCP> Organization: Class of '92, Carnegie Mellon, Pittsburgh, PA Lines: 37 In-Reply-To: <44585@linus.UUCP> / > It would be very interesting, though, to see a statement which / > was 'provable' yet not 'provable' that it's 'provable.' / /Consider the statement: / / Fermat's Last Theorem is provable. / / Many mathematicians believe the above statement to be true. / (Otherwise, they wouldn't continue searching for a proof of / Fermat's Last Theorem.) But as of this writing, there is no / proof that Fermat's Last Theorem is provable. / / Perhaps Fermat's Last Theorem is true but unprovable. In that case, the statement you made above is just false. We know lots of theorems that have been proved to be unprovable; F'sLT, so far, is not one of them. (A year ago or so, someone came up with a putative proof of the thing, which was being checked last I heard.) />How would you know it if you saw it? If you could point at it and say "That />statement is provable, but you can't prove it!" how would you know that the />first part of the claim is true, without invalidating the second part? / /Remember Rolle's theorem? /I don't know about how you learned Calculus, but when I went through it /the first time the book mentions Rolle's theorem (having to do with the /idea that a continuous function that is positive at one point and /negative at another must be zero at some point in-between) and says it's /gotta be true, but that they wouldn't dare prove it. Every other theorem /in a first-year calc book gets proven. Rolle's doesn't. I've never /seen a basic Calc book that proves it. Sitting here in the computer room, I asked a random person for a basic calc book; it had a proof of that theorem. (_Calculus_, James Stewart, Brooks/Cole Publishing). I vaguely recall having it proved in calculus. --Z