From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Numbers and sets
Organization: Boise State University
Date: Sun, 13 Dec 1992 18:14:47 GMT

In article <1992Dec12.223409.18446@husc3.harvard.edu> zeleny@husc10.harvard.edu (Michael Zeleny) writes:
>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.

You claimed above that the notion of _set_ depends on the ordinals.
Considerations about the axioms already cited show that this is not
the case.

On cardinals, you could have made your case stronger with a little
thought, as the alternate definition of the cardinality of a set A is
"the collection of all sets equinumerous with A and of minimal rank".
It is interesting to observe that, while this does work in the absence
of choice, it does not work in the absence of both foundation, and it
has been shown that it is impossible to define cardinal number (in the
sense of choosing a canonical object to represent each cardinal) in
ZF- (where neither choice nor foundation is present).  Thus, the
definition of the notion of cardinal number in the usual set theory
_does_ depend on the presence of either choice (with an obvious
relationship to the concept of ordinals) or the hierarchical structure
of the universe, as provided by foundation.  So you are right, in a
sense, but the priority does not belong to the ordinals _per se_
(although they are convenient) but to the stages of the iterative
hierarchy (which can be conveniently indexed by the ordinals, of
course).

But this is an accident of the particular approach used in ZF.  In
NFU, Frege's definition allows us to define cardinals without
difficulty; there is no dependence of the structure of the universe or
the notion of cardinality on the ordinals.  Also, from the standpoint
of NFU (or of type theory!!!), Zermelo-style set theory is seen to be
the theory of isomorphism types of well-founded extensional relations,
and the special role played by the stages of the iterative hierarchy
is clearly seen to have nothing to do with the nature of sets or
cardinals per se.  Please note that I still don't need to talk about
NFU; type theory inteprets Zermelo-style theories in the same way.

>
>>-- 
>>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."


-- 
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
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