From: zeleny@husc10.harvard.edu (Michael Zeleny)
Newsgroups: sci.math,sci.philosophy.tech
Subject: Re: Numbers and sets
Date: 19 Dec 92 19:09:25 GMT
Organization: The Phallogocentric Cabal

In article <1992Dec17.235906.13828@guinness.idbsu.edu> 
holmes@opal.idbsu.edu (Randall Holmes) writes:

>In article <1992Dec15.214124.7444@guinness.idbsu.edu>
>holmes@opal.idbsu.edu (Randall Holmes) writes:

Talking to yourself is a bad sign, Randall.

>>I repent!  I repent!  No >>>>>>>>>>>>>>>>>!
>>
>>My final answer to Zeleny is that I accept his point in a modified
>>form (which for him may not count as acceptance at all.)  I think that
>>the sets of Zermelo set theory and its extensions (with the Axiom of
>>Foundation) are best understood as being related to isomorphism types
>>of well-founded extensional relations.  Insofar as ordinals (best
>>understood as being related to isomorphism types of well-orderings)
>>are a prominent and structurally significant special case of this kind
>>of relation, I agree that the ordinals are essential to understanding
>>ZF.  This is sharpened when ZF is extended to ZFC by the addition of
>>the axiom of choice.  The particular approach of ZF makes it
>>impossible to choose canonical objects to serve as cardinal numbers
>>without using foundation or choice.  The concept of cardinal number
>>(although not the relations of having less or greater cardinality)
>>depends on foundation or choice for its expression.

This is the better part of my point, and I could not have said it
more succinctly myself.  Thank you for conceding this much of my claim.

>>But the axioms which mediate this relation with the ordinals (or more
>>generally with well-founded extensional relations) are strictly
>>foundation and choice.  ZF- is still a powerful set theory (it is no
>>weaker than ZFC in terms of consistency strength).  Its axioms of
>>separation and replacement embody a concept of _set_ (avoiding paradox
>>by application of the "limitation of size" doctrine) which is not
>>dependent on the ordinals.  ZF- is not studied much that I know of,
>>but the situation is the same in ZFA, in which extensionality is
>>weakened to allow atoms (I believe that Zermelo's original theory
>>allowed atoms, which argues against Zeleny's claim about the history
>>of the set idea!) without choice, and this theory is studied; it was
>>the first theory for which models without choice were constructed.
>>The iterative hierarchy is a powerful working hypothesis, but it is
>>not a prerequisite for the modern concept of set, but a refinement
>>thereof.  Notice that it _does_ appear in ZFA or ZF- as the structure
>>of _part_ of the universe.

Zermelo referred to urelements in the 1906 correspondence with
Poincar\'e; likewise, his 1908 and 1930 axiomatizations explicitly
admitted their existence.  Also, Fraenkel's 1922 independence proof of
AC (reprinted in van Heijenoort), as well as subsequent independence
results of Lindenbaum and Mostowski, used a technique which was to be
superseded by forcing, and required the existence of a denumerably and
non-denumerably many urelements, respectively.  Notably, the relative
consistency of NBG with urelements with respect to NBG was shown by
Mostowski in 1939.  However Fraenkel's own contemporary introduction of
his Axiom of Restriction (which was construed as "inversely analogous"
to Hilbert's model-maximality geometric Axiom of Completeness, and so
intended to yield categoricity by stipulating that the only sets to
exist were those implied by Zermelo's axioms and his own Axiom of
Replacement), implicitly excluded the existence of urelements (see
Moore's book on Zermelo, and especially Fraenkel _et alii_ on
foundations).  But of course this strong version of the limitation of
size doctrine is utterly antithetical to the profligate spirit of ZFC,
as embodied in Levy's reflexion principles, and thus would be only
satisfactory to those misguided souls of the Quinean ilk, who would
"round off smoothly" the set-theoretic ontology.

Having been provoked to undertake this tedious disquisition by your
parenthetical remark, I still cannot recall ever having claimed either
that ZFC was originaly construed as a pure set theory, or that such
historical considerations have any bearing whatsoever on the concept of
set.

>>By the way, Aczel's AFA is just as good as foundation; ZF- + AFA does
>>have well-defined canonical cardinal numbers.

True, but irrelevant: as Herb Enderton once observed in this venue, it
is unclear whether Aczel's entities are actually *sets*, as this term is
rightly and commonly understood.  I would add that the same is true of
the beasts populating the ontology of type-free theories of the
Quine-Church ilk.  The Axioms of Foundation and Choice are analytically
true of sets; this is the crux of our philosophical disagreement, and I
do not expect us to find a way to come to terms on its subject.

>>A definition of "stage of the iterative hierarchy":  let a "universe"
>>be a set which contains all elements of its elements and all subsets
>>of its elements.  Then a "stage of the iterative hierarchy" is a
>>universe which contains all its proper subsets which are universes.
>>The correctness of this definition is left as an exercise, but note
>>that it does not depend on the ordinals!
>
>Alas, this definition is incorrect.  All sets which have this property
>are stages, but not all stages have this property.  I was working by
>analogy with the definition of "ordinal" as "transitive set which
>contains all its transitive proper subsets"; I'm sure there is a
>refinement of this approach which does work, and I have another
>definition on tap which is correct.

Make up your mind, Randall.

>>-- 
>>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
>
>
>-- 
>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

What possesses the tab characters in your .sig file, that they do this
strange dance on the occasions of follow-ups?

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