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From: ags@pucc-i (Seaman)
Newsgroups: net.math
Subject: Pedantic questions and circularity
Message-ID: <218@pucc-i>
Date: Thu, 23-Feb-84 09:35:48 EST
Article-I.D.: pucc-i.218
Posted: Thu Feb 23 09:35:48 1984
Date-Received: Fri, 24-Feb-84 02:42:06 EST
Organization: Purdue University Computing Center
Lines: 26

Now that we know how to prove that all natural numbers can be represented
with finitely many digits, would anyone care to prove that every natural
number is less than 2**n for some n?

I can think of an easy proof that begins by assuming that every number
has finitely many digits...

Seriously, neither of the above properties of natural numbers comes from
the definition.  If you want to avoid circularity, you had better start
with the Peano Axioms.  That is the only way you can know you are talking
about the natural numbers, and not about something else.

Here is a restatement of the Fifth Peano Axiom which I looked up last
night.  This version does not mention sets:

(5) Let P be a property which is true of zero, and which is true of the
    successor of any number which has the property.  Then P is true of
    all natural numbers.

-- 

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

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