Newsgroups: sci.math
From: hoey@aic.nrl.navy.mil (Dan Hoey)
Date: 1995/07/10
Subject: Re: What is 0 to the power 0???

t...@cco.caltech.edu (Toby Bartels) writes:

> The natural number 0 is the empty set {}.
> The integer 0 is the diagonal set {(0, 0), (1, 1), ...} of the
> natural numbers. ...

No, I'm sorry, but this just won't do.  We definitely want the natural
numbers to be a subset of the integers, and so this sort of naive
construction is just not good enough.

The usual technique is after you form this set of proto-integers Z*, a
partition of NxN, you have to show that there is a subset Z*[N] that
acts just like the integers.  Then define Z = ( Z* \ Z*[N] ) U N, with
the natural isomorphism to Z*.  This is called "embedding N in Z* to
form Z".  Similarly, you have to embed Z into Q* to form Q, and Q into
R* to form R, and R into C* to form C.

Of course, this stupid, wasteful exercise is only given to quibble
artists who try to assert some stupid, wasteful distinction between
e.g. the natural number 0 and the real number 0.  Those of us who knew
they were the same all along don't need that sort of noise.

Dan Hoey                        ZERO is REAL unless declared INTEGER.
Hoey@AIC.NRL.Navy.Mil           NOUGHT is INTEGER unless declared REAL.