From: apollo@math.berkeley.edu (Apollo Hogan)
Subject: Re: Ordinals of Infinite Complexity?
Date: 4 May 2000 05:13:06 GMT
Newsgroups: sci.math
Summary: Cantor Normal Form for ordinals

In article <5244-390E1044-41@storefull-108.iap.bryant.webtv.net>,
DWIII <dwiiiandspam@webtv.net> wrote:
>Does there exist any (transfinite) ordinals or cardinals which require
>an infinite amount of information to explicitly describe?
>
>Let S be the set of all ordinals of the form:
>
>... + w^5*b5 + w^4*b4 + w^3*b3 + w^2*b2 + w*b1 + b0
>
>where "w" is the ordinal omega, and b0, b1, b2, ... are
>arbitrarily-chosen elements of N (naturals), for each and every w^c such
>that c is an element of N.

A fact which you might find of interest is that _every_ ordinal can
be represented in the following form ("Cantor Normal Form")
   w^{a_1}*n1 + w^{a_2}*n2 + w^{a_3}*n3 + ... + w^{a_k}*nk

Where  k  is a finite number, each  ni  is a finite ordinal and
the  a_i's are arbitrary ordinals such that
	a_1 > a_2 > a_3 > ... > a_k

Thus every ordinal can be represented as finite 'polynomial'.

(This doesn't really answer your question, since we may have for
example, ordinals  a  such that  a = w^a, so the cantor normal form
is just the ordinal itself...)

--Apollo Hogan
UC Berkeley