From: zeleny@husc10.harvard.edu (Michael Zeleny)
Newsgroups: sci.math,sci.philosophy.tech
Subject: Re: Numbers and sets
Date: 13 Dec 92 03:34:07 GMT
 <1992Dec10.124223.18352@husc3.harvard.edu> <1992Dec11.160146.23727@guinness.idbsu.edu>
Organization: The Phallogocentric Cabal

In article <1992Dec11.160146.23727@guinness.idbsu.edu>
holmes@opal.idbsu.edu (Randall Holmes) writes:

>In article <1992Dec10.124223.18352@husc3.harvard.edu>
>zeleny@husc10.harvard.edu (Michael Zeleny) writes:

>>In article <1992Dec5.155535.6854@sun0.urz.uni-heidelberg.de>
>>gsmith@lauren.iwr.uni-heidelberg.de (Gene W. Smith) writes:

>>>In article <Byqo93.FCv@mentor.cc.purdue.edu>
>>>hrubin@pop.stat.purdue.edu (Herman Rubin) writes:

HR:
>>>>Is the cardinal interpretation or the ordinal interpretation more 
>>>>"natural"?  Which can be more easily understood?  Which is more 
>>>>suitable to the appropriate extensions?  These questions are non-
>>>>trivial.

GWS:
>>>An ordinal number has structure--it is a well-ordering.  Up to
>>>isomorphism, a cardinal number is any set, and any set can
>>>serve as a cardinal number.  So I think cardinality is a lot
>>>more basic and much simpler conceptually.

MZ:
>>I am surprised that no one has observed the well-known fundamental
>>problem involved in this approach, that the concept of a set, and, _a
>>fortiori_, the concept of a cardinal number, both logically depend on
>>the concept of the ordinals.  (Consider the structure of V.)

RH:
>This is ridiculous.  I won't even trot out NFU.  Read the axioms of
>ZFC, Mikhail.  See what order the definitions come in.  Ordinals are
>defined as being particular sets and their properties are deduced
>using the axioms of set theory.  The structure of V is described using
>ordinals, but ordinals are not a primitive notion of ZFC; they are
>defined as sets in set-theoretic terms from axioms which refer only to
>sets, and their properties, as well as the structure of V you refer
>to, follow from these same axioms, which do not mention ordinals.  And
>if you appeal to the history of the ideas involved, I can point out
>the genetic fallacy just as well as you can...

Randall, you are way off the mark here; I do, however, appreciate your
not dragging in NFU, which may be the only reasonable part of your
response.  As you undoubtedly know, the canonical definition of cardinal
is an ordinal, which is not injectible into any smaller ordinal.  (See
the books by Hatcher, Bell & Machover, Drake, or Fraenkel, Bar-Hillel,
and Levy.)  More importantly, the mere fact that the axioms of ZFC make
no mention of the ordinals, should not impress any card-carrying
mathematical realist; a moment's contemplation of the intended model of
ZFC (choice is needed for the above definition, though a less elegant
version, due to Scott, may be given independently of it and the
ordinals, -- see Drake) should convince you that the iterative hierarchy
is not only *described* using the ordinals, but *depends* on their
ontological priority for its meta-theory.  Surely any restriction of the
question of priority to the object language is arbitrary for anyone who
allows the existence of content of the language in question.  History
has nothing to do with the question, which was just my point.

>-- 
>The opinions expressed		|     --Sincerely,
>above are not the "official"	|     M. Randall Holmes
>opinions of any person		|     Math. Dept., Boise State Univ.
>or institution.			|     holmes@opal.idbsu.edu

cordially,
mikhail zeleny@husc.harvard.edu
"Le cul des femmes est monotone comme l'esprit des hommes."