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Path: utzoo!dciem!ntt
From: ntt@dciem.UUCP (Mark Brader)
Newsgroups: net.math
Subject: Re: Interesting Numbers (New Proof)
Message-ID: <909@dciem.UUCP>
Date: Mon, 7-May-84 17:40:34 EDT
Article-I.D.: dciem.909
Posted: Mon May  7 17:40:34 1984
Date-Received: Mon, 7-May-84 18:49:18 EDT
References: <1281@uvacs.UUCP>
Organization: NTT Systems Inc., Toronto, Canada
Lines: 55

George Fisher (uvacs!gmf) presents: "another proof that all positive
integers are interesting which uses mathematical induction more
directly than my last one did."

He begins by explicitly stating:

		'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.

He then correctly proves that all numbers are interesting, and adds:

	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?

The answer is no.  You see, this assumption implies that all integers are
interesting (indeed, all real numbers), trivially and without requiring
mathematical induction.  Proof: for any number X, X is interesting because
it is the least number x with the property that x=X.

Well, the difficulty here obviously occurs because we have allowed too wide
a range of properties P.  The other option seems to be to restrict the list
so that not all integers are included.  We define a list of integers that are
"interesting without regard to being interesting or uninteresting", say,
"plain interesting".  An integer is plain interesting if it has property P1,
P2, P3, ..., Pn.  (Or the list could be infinite.)

In what follows I mean to refer to positive integers only.  I don't have
time to edit the message now.

Now,
	'Interesting Assumption 2'
says that an integer is interesting if it is plain interesting, or if it is
the least integer that is not plain interesting.  No contradiction here.

But the original problem really reduces to this:
	'Interesting Assumption 3':
An integer is interesting if it is plain interesting.  The least integer
that is not interesting is also interesting.

But the last sentence is an explicit contradiction if there is a least
non-interesting number.  Therefore, this alone is an indirect proof that
there is no such number, and mathematical induction is not involved.

If we try to weasel out of this by putting the property of being interesting
on the list P1,P2,..., then IA2 reduces to IA3.

Mark Brader