-----Original Message----- From: David Gale Sent: Monday, August 09, 2004 6:05 PM To: math-fun Subject: RE: [math-fun] Re: More innumeracy in high places

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Theorem: For any N, there is a prime > N. Proof: Consider any prime factor of N!+1

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This is very nice, but recall, we started out talkng about the man/woman on the street and if I know them they may not find it completely obvious that,

Corollary: There are infinitely many primes

In my experience, even mathematically unsophisticated people do find this part to be obvious. Even if they don't, the proof by contradiction that starts

Proof: Hmm.?? We don't want to back down at his point and say "If there were only a finite number then. . .".

seems to me to be a bit of a cheat. After all, why should it follow from the fact that the set S of primes is finite (that is, cannot be put into 1-1 correspondence with a proper subset) that there exists an N that is greater than any element of S?

I think the solution would be for Socrates to show up just in the nick of time and say "Look, John/Mary, What does it mean to say that there are an INFINITE number of whatevers?" Perhaps using his celebrated method he could get them to agree that infinite MEANS more than N for any N.

Right. And I don't think this is a cheat. if we want to express the statement that "there are infinitely many n such that P(n)" in the language of the Peano postulates, rather than in the language of set theory, the standard way to do so is "For all N there exists an M such that M > N and P(M)" Andy Latto andy.latto@pobox.com