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From: gmf@uvacs.UUCP
Newsgroups: net.math
Subject: Interesting Numbers (New Proof)
Message-ID: <1281@uvacs.UUCP>
Date: Sat, 5-May-84 17:17:07 EDT
Article-I.D.: uvacs.1281
Posted: Sat May  5 17:17:07 1984
Date-Received: Sun, 6-May-84 07:38:30 EDT
Lines: 36


Here is another proof that all positive integers are interesting which
uses mathematical induction more directly than my last one did.

First, though:
'Interesting Assumption':
Assume that for any property P (or any property in a class which
contains the properties used below), if x is the least positive
integer with property P, then x is interesting.  Also assume that
no positive integer x is both interesting and uninteresting.

Proof.  1 has the property of being a positive integer, and is the
least positive integer with this property, so 1 is interesting.
Assume for purposes of induction that n is a positive integer, and
n is interesting.  Suppose, by way of contradiction, that n+1 were
uninteresting.  Then n+1 has the property of being the least positive
integer > n which is uninteresting.  Hence by the 'Interesting
Assumption', n+1 is interesting.  But n+1 can't be both uninteresting
and interesting.  This contradiction shows that n+1 is interesting.
Thus n+1 is interesting whenever n is interesting.  Hence by the
axiom of mathematical induction, any positive integer is interesting.

In case the first sentence of the proof is too paradoxical sounding
for someone, we could assume that if x is the least * number * (say
in the real numbers) with property P, then x is interesting.  Then
1 is the least number with the property of being a positive integer.

I think this proof is not fallacious once the 'Interesting Assumption'
is accepted.  But one can reasonably reject the assumption that a
number which is the least in a class (defined by a property) is
interesting just by virtue of the fact that it is the least number
in that class.  Or one can reject the axiom of mathematical induction.

Question:  Is the 'Interesting Assumption' an interesting assumption?

     Gordon Fisher