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From: dje@5941ux.UUCP
Newsgroups: net.math
Subject: A new paradox?
Message-ID: <406@5941ux.UUCP>
Date: Wed, 14-Sep-83 12:00:16 EDT
Article-I.D.: 5941ux.406
Posted: Wed Sep 14 12:00:16 1983
Date-Received: Thu, 15-Sep-83 06:16:14 EDT
Lines: 20

Here's a little paradox based on mathematical induction.

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?

Dave Ellis / Bell Labs, Piscataway NJ
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