From: zeleny@husc10.harvard.edu (Michael Zeleny)
Newsgroups: sci.math,sci.philosophy.tech
Subject: Re: Numbers and sets
Date: 15 Dec 92 18:50:27 GMT
Article-I.D.: husc3.1992Dec15.135030.18526
Organization: The Phallogocentric Cabal

In article <1992Dec14.162438.15591@guinness.idbsu.edu>
holmes@opal.idbsu.edu (Randall Holmes) writes:

>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.

MZ:
>>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?

RH:
>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.

Why do you suppose that your "can be" is of any relevance to this
argument?  The intended interpretation of ZFC is the iterative
hierarchy, which, as you observe, directly validates Foundation, and
indirectly supports AC.  Any other interpretation of its axioms is of
no more intrinsic interest to the study of sets than the nonstandard
models of PA are to the study of the integers.

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".

MZ:
>>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).

MZ:
>>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?

RH:
>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.

Indeed, the stages of V may be defined directly in the object language
(see Potter's introductory text, based on Scott's axiomatization of
the iterative hierarchy); however a good rationalist remains under the
obligation to question the natural bijection between the stages of V
and its ordinals, which becomes evident in the meta-theory.  There are
no "mere" coincidences in such matters.  And I suspect that you will
have to recapitulate the definition of the ordinals in the
metalanguage to account for the facts of the matter.

May I take it that you have conceded my original point?

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.

MZ:
>>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.

RH:
>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.

Based on the preponderance of ZF-style sets among the afficionados of
abstract objects, I say that the naive use of the terms refers to the
same, under the hypothesis of the division of linguistic labor.

MZ:
>>					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?

RH:
>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).

OK.

RH:
>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.

The question is, why bother.

RH:
>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.

That's precisely my gripe: the genetic variety of the limitation of
size hypothesis is well-understood, whilst the semantic foundations of
NF-style theories remain only marginally less obscure than they were
in 1968, when they were fairly characterized as nonexistent.  Yes, I
am saying this with a copy of Forster in my hands.  It may well turn
out that the "untyped universe" is yet to meet its Zermelo; but
clearly this has not yet happened.

RH:
>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.

		...which is a nice way of saying that most working
mathematicians don't give a fuck about foundations.  It was not always
so: witness Hilbert, Brouwer, Poincar\'e, Von Neumann, Tarski, Ulam,
Bishop, and so on.  Pigeons on the grass, alas, alas...

>-- 
>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."
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