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From: ags@pucc-i (Seaman)
Newsgroups: net.math
Subject: Numbers, Diagonalization and Exponentiation
Message-ID: <221@pucc-i>
Date: Sat, 25-Feb-84 09:37:09 EST
Article-I.D.: pucc-i.221
Posted: Sat Feb 25 09:37:09 1984
Date-Received: Mon, 27-Feb-84 04:20:54 EST
Organization: Purdue University Computing Center
Lines: 47

There is a way to define exponentiation which sidesteps some of the problems
I outlined in my previous note.  I think it is worth mentioning, because it
leads to a simple proof that n < 2**n for all n, using a variation of
Cantor Diagonalization, which is what started the whole discussion.

For openers, let's consider the Zermelo-Fraenkel model of the natural numbers.
According to this model,

  (0) Zero is the empty set, 0 = {}.

  (1) One is the set {0} = {{}}.

  (2) Two is the set {0,1} = {{}, {{}}}.

  (3) Three is the set {0,1,2} = {{} ,{{}} ,{{{},{{}}}}.

  ...

  (n) Is the set {0,...,n-1}.

In this model, numbers are sets of a particular form.  The successor of a
number n is the set n+{n}, "n union singleton n", where + represents set union.
It is rather pleasant to see how everything can be built out of nothing.

Exercise:  Show that the Peano Axioms are satisfied by this model.

Exercise:  Show that any finite set can be placed in one-to-one correspondence
	   with exactly one of the natural numbers, viewed as a set.  This
	   number is called the CARDINALITY of the set.  The concept of
	   cardinality can be carried to transfinite sets, but we don't
	   need that here.

Exercise:  Show that multiplication of natural numbers can be defined in
	   terms of the cardinality of the CARTESIAN PRODUCT of the two
	   numbers, viewed as sets.  (If you don't know what a Cartesian
	   product is, skip this exercise.  It's just a curiosity which
	   won't be needed.)

Exercise:  Think of a similarly clever way to define exponentiation in terms
	   of sets.  Answer to follow in next article.
-- 

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

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