From: zeleny@husc10.harvard.edu (Michael Zeleny)
Newsgroups: sci.math,sci.philosophy.tech
Subject: Re: Numbers and sets
Date: 27 Dec 92 08:54:10 GMT
Article-I.D.: husc3.1992Dec27.035413.18857
Expires: December 31, 1999
Organization: The Phallogocentric Cabal

In article <1992Dec23.175145.18528@guinness.idbsu.edu> 
holmes@opal.idbsu.edu (Randall Holmes) writes:

>In article <Bzosz1.FMx@cantua.canterbury.ac.nz>
>wft@math.canterbury.ac.nz (Bill Taylor) writes:

>>In article <1992Dec19.140927.18700@husc3.harvard.edu>,
>>zeleny@husc10.harvard.edu (Michael Zeleny) writes:

MZ:
>>>The Axioms of Foundation and Choice are analytically
>>>true of sets;

BT:
>>What does this sentence mean ? 
>>Could someone please explain further ?
>>Why can it be so blithely asserted ?

RH:
>What the sentence purports to mean is that the axioms of foundation
>and choice are true of sets for the same kind of reason that bachelors
>are necessarily unmarried; that they are included in the definition of
>the notion of "set".  The sentence is false.

Wow!  Check out this self-assured bluster!  Randall, what are you
trying to insinuate by your implied distinction between meaning and
*purporting* to mean?  I assure you that I am fully capable of meaning
what I say!  And what makes you so sure that my sentense is false?
Are you suggesting that it it false on *any* conception of sets? is it
even false on the overwhelmingly *prevalent* conception of sets?

RH:
>The power of blithe assertion is independent of the truth or
>falsehood, plausibility or implausibility, of the sentences asserted.

Indeed this is so, and equally true of yourself.

RH:
>The axiom of foundation asserts that each set is disjoint from at
>least one of its elements; it ensures that sets are constructed in an
>orderly fashion starting with the empty set or perhaps with atoms as
>well, each set being a set of previously constructed sets.  This
>prevents such oddities as sets which are elements of themselves
>(which, I hasten to point out to the uninformed, are _not_ paradoxical
>and do turn up in some set theories).

Better yet: any non-empty set contains a member which is disjoint from
it; as stated, your "axiom" is falsified by the empty set.

Self-containing sets do turn up in *some* "set" theories; the salient
question is whether these theories are explicative of the correct
notion of sets.  But we have been over this ground before, without
making much progress.

RH:
>The axiom of choice asserts that, given any collection of disjoint
>sets, there is some set which consists of exactly one element of each
>element of the collection of disjoint sets; as Lord Russell put it, if
>we have infinitely many pairs of socks, we may choose one sock from
>each pair and form a set (in many different ways).

Or that the cartesian product of any family of non-empty sets is
itself non-empty, or that every set can be well-ordered, or that every
set is equipollent to an ordinal number, and so on, in varying degrees
of strength.  See the Rubin and Rubin book, _Equivalents of the Axiom
of Choice, II_, or Moore's _Zermelo's Axiom of Choice_, on the various
technical and historical aspects of AC.

A tiny correction, -- unless you feel absolutely compelled to make a
superfluous gesture of obeisance to nobility, you should note that, at
the time he made his comment, Bertie was ever so far from inheriting
his title.  In any case, his point had to do with the fact that
choosing a shoe set could be accomplished by means of pure logic,
merely by stipulating a definite description "the left/right shoe in
the Nth pair"; whilst, they being presumably indistinguishable by
podaic laterality, doing so for socks would require an essential
application of AC.

RH:
>Both axioms have powerful common-sense arguments behind them, but it
>is also the case that there are good arguments against both of them.
>I don't think that either of them is part of the _definition_ of the
>concept "set".  

Again, this issue deserves more than my glib sloganeering, or your
equally superficial gainsaying.  As a consequence of the Axiom of
Foundation, we obtain that if there is some element which fulfils a
given condition, then there is an \epsilon-minimal element, fulfilling
the same condition; it can be easily shown that this is equivalent to
the proposition that all sets are well-founded, i.e. belong to some
initial segment of the iterative hierarchy.  Exhaustive discussion of
these and related issues will be found in Fraenkel, Bar-Hillel, and
Levy's _Foundations of Set Theory_.

To defer to Drake's superb exposition once again, Foundation may be
justified by taking a non-empty set, and proceeding upwards in the
cumulative hierarchy, until we reach the first level at which it has
at least one member.  Any such member must be disjoint from the set
itself, since any members of the former will have occurred on the
lower levels.

Like the Axiom of Comprehension, Foundation is a principle that
stipulates that all sets have certain structural properties; all the
other axioms of ZFC can be interpreted as weakened instances of the
unrestricted comprehension scheme.  So the assumptions exemplified in
this axiom are (i) that members of any set occurring on a given level,
must have occurred at the lower levels; and (ii) that if we proceed
upwards through the cumulative types, we shall find a *first* one with
any given property, assuming that there are any in the first place.

It is obvious that, as a structural principle, the axiom *is* analytic
on any conception of sets, which is based on the Zermelo-G\"odel idea
of the cumulative rank hierarchy, as it is commonly understood.  On an
alternative idea of a set as the extension of a property, it is hard
to deny the argument of Kreisel, that the only natural formulation
thereof would call for unrestricted comprehension scheme, and the
concomitant abandonment of classical logic, demanded by any realist
philosophy of mathematics.  Once again, your Quinian preferences will
undoubtedly allow you to disagree with this claim; but we both know
that such disagreement will not get us anywhere.  So I propose that
you stick to your views, whilst I continue holding onto mine.

RH:
>		For the record, my official set theory includes choice
>and denies foundation; I'm a professional set theorist, so you might
>want to take this into account in evaluating Zeleny's claim.

Randall, it is my fervent wish that, should I ever acquire exalted
institutional status similar to your own, I would manage to retain a
bare modicum of good sense, sufficient to compel me to abstain from
adducing it in support of my philosophical arguments.  Especially if,
as in your present case, I would be tempted to adduce it as my *only*
argument.

>>--------------------------------------------
>>Bill Taylor.       wft@math.canterbury.ac.nz 
>>Bill Trylor. que  rwft@maih.casterkury.aa.n! 
>>Tiel Tryloco quer rwst@maihuc sterkesy.ga.n!
>>Thelworyd co quer rwsi@mvihus strikesy.gain!
>>The world conqueror sig-virus strikes again!
>>--------------------------------------------

>-- 
>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."