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From: ags@pucc-i (Seaman)
Newsgroups: net.math
Subject: Re: Pedantic Question - (nf)
Message-ID: <212@pucc-i>
Date: Sat, 18-Feb-84 13:30:17 EST
Article-I.D.: pucc-i.212
Posted: Sat Feb 18 13:30:17 1984
Date-Received: Mon, 20-Feb-84 07:16:19 EST
References: <5710@uiucdcs.UUCP>
Organization: Purdue University Computing Center
Lines: 33

It has been asked why the Cantor Diagonalization Process cannot be applied
to the natural numbers to show that there are uncountably many of them.
The basic question reduces to "why can't there be a number of the form
...1111111xyz, with an infinite number of leading ones?"

The natural numbers are defined by the Peano Axioms:

	(1) Zero is a number.

	(2) The successor of any number is a number.

	(3) No two distinct numbers have the same successor.

	(4) Zero is not the successor of any number.

	(5) Let S be a subset of the numbers which obeys the rules:
	    (a) Zero is in S.
	    (b) The successor of any number in S is also in S.
	    Then S contains all of the natural numbers.

Let S be the set of natural numbers whose binary representation has only
a finite number of digits.  By axiom (5), S is the complete set of natural 
numbers.  

Incidentally, the definition of an uncountable set is "one which cannot be 
placed into one-to-one correspondence with (any subset of) the natural numbers."
-- 

Dave Seaman
..!pur-ee!pucc-i:ags

"Against people who give vent to their loquacity 
by extraneous bombastic circumlocution."