From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Numbers and sets
Organization: Boise State University
Date: Mon, 14 Dec 1992 16:24:38 GMT

In article <1992Dec13.162120.18457@husc3.harvard.edu> zeleny@husc10.harvard.edu (Michael Zeleny) writes:
>In article <1992Dec13.181447.354@guinness.idbsu.edu>
>holmes@opal.idbsu.edu (Randall Holmes) writes:
>
>>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...
>
>MZ:
>>>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.
>
>RH:
>>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.
>
>This is an enthymeme, with the hidden premiss that the axioms of ZFC
>constitute the sole basis of the corresponding notion of _set_.  Are
>you sure of not being a formalist, Randall?

The axioms of ZFC, other than choice and foundation, can be motivated
by a notion of set quite independent of the ordinals.  Choice asserts
roughly that the universe has (or can be given) a structure similar to
that of the ordinals; Foundation asserts that it has a structure
determined by the well-founded extensional relations -- these impose
the structure of the ordinals to a greater or lesser extent on the
universe.

>
>RH:
>>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".
>
>Your refinement is gratefully accepted.  Obviously, I was not thinking.
>
>RH:
>>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 choice, 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).
>
>Strike `conveniently', and replace `can' with `must', and I will
>gladly agree.  Before you drag in your favorite charity case, what do
>you see as an alternative to using the ordinals in describing the
>stages of ZFC?

Define the stages of the iterative hierarchy directly; they are quite
natural objects themselves.  There is, of course, a natural bijection
between them and the ordinals, but it is not necessary to mention the
ordinals in defining them.

>
>RH:
>>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.
>
>Randall, it is a brute sociological fact that, whenever anyone
>mentions a set in this forum, he means it as characterized by ZF-like
>theories, unless that someone is you.

I don't think so, necessarily.  This is a philosophy group; some know
little about technical set theory, probably, and so are speaking from
a naive notion of set.  Some are aware that there are other approaches
to set theory.

  But perhaps you can help me
>dispel some of my disdain for your pet theory.  My reading of Jensen
>and Forster indicates that NFU is equiconsistent with a fragment of
>PA.  If this is indeed so, can you give a good reason why I should be
>any more content with it as a foundational discipline, than with the
>wacky systems of Nelson, Yessenin-Volpin, and so on?

When I say NFU, I usually mean at least NFU + Infinity + Choice, which
is equiconsistent with the theory of types with infinity, and more
often a much stronger system (I am following a precedent here;
references to the theory of types usually implicitly include infinity;
TT without infinity is equiconsistent with the same fragment of PA).
There are extensions of NFU with equivalent consistency strength and
expressive power to any extension of ZFC in which we have confidence
(necessarily speaking roughly): this follows from Jensen's original
consistency results.  In fact, NFU is not a fundamentally different
approach to set theory (from a purely technical standpoint); it is
readily intertranslatable with Zermelo-style theories.  Doing set
theory in extensions of NFU is similar to doing set theory in ZFC; one
needs to learn a few technical maneuvers which are slightly different
from the ZFC technical maneuvers, and none of which would be
mysterious to someone familiar with type theory.

Positive advantages of the NFU approach may show up in extensions with
large cardinals; the relation between large cardinals and elementary
embeddings of the universe into itself might be brought out more
elegantly in a suitable extension of NFU.  The other advantage is
philosophical; "limitation of size", while adequate to avoid the
paradoxes, is seen to be stronger than is necessary; it is possible to
embed models of Zermelo-style set theory in models of extensions of
NFU without increasing consistency strength, for instance.  Of course,
there are other reasons to prefer one or the other than the avoidance
of the paradoxes.

NFU may provide an alternate foundation in areas where it is
technically convenient to have "big" objects, such as certain areas of
computer science or category theory.  It would work as a foundation
for general mathematical purposes just about exactly as well as the
usual set theory (so there is no reason to adopt it for such
purposes!).  There might be pedagogical advantages to having it
available as an alternative (one can comment that much of the early
reasoning which appeared dangerous was actually safe).  Where it is
used, no earth-shaking effects are to be expected, since it is
intertranslatable with the usual approaches.  It is actually
remarkable how little difference the superfically arresting
differences between the approaches make to real mathematical
applications.


>
>>-- 
>>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
Newsgroups: sci.math