Pat Hayes wrote: (01)
> It is kind of fun to see if there is a coherent alternative, though. How
> could one make sense of the thesis that there are only finitely many
> integers? The counterproof seems very simple and is probably one of the
> oldest mathematical proofs ever devised.
> 1 Suppose there were a largest integer.
> 2 Call it N.
> 3 Consider N+1.
> 4 N+1 is larger than N: contradiction.
> 5 Ergo, there is no largest integer. (02)
This reminded me of an interchange not long back. In providing my
contact information, I mentioned that, unlike my NIST colleagues who had
telephone numbers like 3535, I had an "uninteresting number". A French
mathematician in the group immediately observed: (03)
"But there are no uninteresting numbers. Proof:
1. Any set of positive integers is bounded below.
2. All telephone numbers are positive integers.
3. If the set of uninteresting telephone numbers is non-empty,
it is bounded below.
4. Therefore, there would be a 'least uninteresting number'
5. But, by virtue of having that property, the 'least uninteresting
number' would be interesting! Contradiction!
6. Ergo, the set of uninteresting telephone numbers must be empty." (04)
I offer the above by way of adding further levity to a conversation
which Chris Menzel just accurately described as "hopping down the
Cantorian bunny trail". :-) (05)
Edward J. Barkmeyer Email: edbark@xxxxxxxx
National Institute of Standards & Technology
Manufacturing Systems Integration Division
100 Bureau Drive, Stop 8263 Tel: +1 301-975-3528
Gaithersburg, MD 20899-8263 FAX: +1 301-975-4694 (07)
"The opinions expressed above do not reflect consensus of NIST,
and have not been reviewed by any Government authority." (08)
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