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From: bwk@mbunix.mitre.org (Barry W. Kort)
Newsgroups: comp.ai
Subject: Re: Natural Paradox
Summary: Can you believe this?
Keywords: Counterexample, Evidence, Belief, Proof, Provability
Message-ID: <44803@linus.UUCP>
Date: 11 Feb 89 16:04:03 GMT
References: <1706@tank.uchicago.edu> <9526@ihlpb.ATT.COM> <44585@linus.UUCP> <5898@sdcsvax.UUCP>
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Reply-To: bwk@mbunix.mitre.org (Barry Kort)
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In article <5898@sdcsvax.UUCP> demers@beowulf.UCSD.EDU
(David E Demers) writes:

 > We already know that {under appropriate circumstances} there
 > are TRUE but UNPROVABLE statements; however, are there PROVABLE
 > statements about which one cannot PROVE they are provable?  

It depends on what you mean by "one" in the above sentence.

Consider if you will the assertion,

	"The integer formed by 317 repetitions of
	 the digit 1 is prime."

I claim that the above assertion is true (but I don't
know how to prove it).

I also claim that the assertion is not only true, but provable. 
(But I don't know how to prove that claim, either.)

 > If the statement is provable, then one simply needs 
 > to exhibit the proof in order to prove its provability.

Evidently I have failed to supply the needed proof.  So
I clearly haven't proven that the assertion is provable.

Nevertheless, I still claim it is the case that the assertion
is both true and provavble.

--Barry Kort