Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10 5/3/83; site ssc-vax.UUCP
Path: utzoo!linus!decvax!tektronix!uw-beaver!ssc-vax!tjj
From: tjj@ssc-vax.UUCP (T J Jardine)
Newsgroups: net.math
Subject: Re: A new paradox?
Message-ID: <519@ssc-vax.UUCP>
Date: Thu, 15-Sep-83 16:55:34 EDT
Article-I.D.: ssc-vax.519
Posted: Thu Sep 15 16:55:34 1983
Date-Received: Fri, 16-Sep-83 15:30:43 EDT
References: <406@5941ux.UUCP>
Organization: Boeing Aerospace, Seattle
Lines: 32

In response to Dave Ellis' "paradox", I first restate the problem:

	Theorem:  In any set of N elements (N >= 1), all elements in the set
		  are equal.

	Proof:  By induction on N.  

	For N=1, the theorem is clearly true.  If N>1, then we can inductively
	assume that the theorem is true for all sets of N-1 elements.  Select
	a set of N elements x(1), ..., x(N).  The elements x(1), ..., x(N-1)
	constitute a set of N-1 elements, which by the inductive hypothesis
	are all equal.  Similarly, the elements x(2), ..., x(N) constitute a
	set of N-1 elements, which by the inductive hypothesis are all equal.
	Consequently, by the transitive property of equality, all the elements
	x(1), x(2), ..., x(N-1), x(N) are equal.  This completes the proof.

	Where's the fallacy?

The fallacy is in the difference between the method of proof used and the
method of mathematical induction.  Mathematical induction states that if a
sequence of elements in a set are such that a statement can be proven for
one element AND if true for a preceding element in sequence it can be shown
to be true for the element in question, then the statement is true for all
elements of the set.  The step of showing that the property is true for
N=1 corresponds to the first part; however, in attempting the second part
of the proof the possibility that the two sets {x(1) ... x(N-1)} and
{x(2) ... x(N)} are disjoint is totally ignored.  In the case that they
are (as for N=2) there is no middle element to which transitivity can be
applied.

TJ (with Amazing Grace) The Piper
...!uw-beaver!ssc-vax!tjj