From: cmenzel@tamsun.tamu.edu (Chris Menzel)
Newsgroups: sci.philosophy.meta,sci.logic
Subject: Re: Russell's Paradox (was re: Of apples and their skins)
Date: 11 Oct 92 17:22:25 GMT
Organization: Texas A&M University, College Station

In article <1992Oct10.011243.16219@husc3.harvard.edu> zeleny@husc10.harvard.edu (Michael Zeleny) writes:
>
>[Putatively shameful remarks by Randall Holmes omitted...]
>
>Shame on you, Randall!  Just because your favorite set theory has been
>found inconsistent in its earlier incarnation does not entitle you to cast
>aspersions on Cantor in blatant disregard of the historical record.  As you
>know perfectly well, Russell discovered his paradox by applying Cantor's
>proof to Frege's theory of properties.  In fact, contrary to the popular
>misconception you are perpetuating, NO CONTRADICTION HAS EVER BEEN FOUND
>WITHIN CANTORIAN SET THEORY.  Indeed, Cantor discovered Burali-Forti's
>argument two years earlier than Burali-Forti, and remained nonplussed by
>it.  

It should be put even more strongly still, since this way of wording
things might suggest that Cantor "discovered" the BF paradox, but
missed it ramifications and just didn't think much of it.  To the
contrary, Cantor never even *saw* a paradox, since he had already
essentially made the distinction between sets and proper classes
("absolutely infinite" collections).  The "paradox" for him was
nothing but a reductio proof (laid out in his 1899 letter to Dedekind,
translated in van Heijenoort) that the class of ordinals is absolutely
infinite.  The focus of the letter, after all, is the thesis that all
cardinal numbers are in the sequence of alephs, which, upon proving,
he intended to use in service of a proof of his ultimate--and
ultimately elusive--prey, the continuum hypothesis.  (It would be
good, after all, to eliminate the possibility that 2^aleph_0 isn't
even an aleph, let alone aleph_1, at the outset.)  Cantor used the
absolute infinity of the class of ordinals as a lemma in an
(erroneous, as it turns out) proof of the thesis that all cards are
alephs.  His reference to absolutely infinite collections as
"inconsistent multiplicities" reflects only the fact that any attempt
to consider them ordinary sets leads to contradiction.  Nowhere does
he even hint at anything like the existence of paradox or
inconsistency in the foundations of set theory.

>Since his philosophical principles excluded the existence of the
>universal set, he simply interpreted the result as a proof of non-existence
>of the set of all ordinals.

Better, its absolute infinity.  Cantor didn't make strict
terminological distinctions in his use of various collective terms
(Menge, Vielheit, Mannigfaltigkeit, etc.), and in particular on
occasion uses the preferred term `Menge' to refer to absolutely
infinite collections as late as 1895.

>To claim, as you did, that it seems prefectly acceptable that for any
>property P there is a set of all things with property P, is simply
>ludicrous on any intelligent analysis of the notion of set, regardless of
>considerations specific to the paradox.  Recall that a set is "defined" by
>Cantor as a multiplicity presented to the mind as a whole.  Now, there are
>many reasons why, on the classical view first advanced by Aristotle,
>revived by the scholastics, and further elaborated by Cantor, the
>absolutely infinite, as opposed to the Cantorian transfinite, is
>intrinsically self-contradictory.  One of them, articulated in Cantor's
>1887-8 papers on transfinite numbers, is that, insofar as this putative
>object would not admit of any augmentation or diminution, it _eo ipso_
>would not lend itself to quantitative characterization as an absolute
>maximum, nor, _a fortiori_, to any coherent presentation as a completed
>whole.

Having registered my sympathy for the Cantorian view, one still has to
ask, Mikhail, is it that easy?  I mean, for the platonist, the sets
are all *there*.  How can it not be possible to consider them a
completed whole, just as we can consider \omega a complete whole?
What are we talking about in VNBG, for example, when we refer to ON =
{x | Ord(x)}?  Why is this not a "multiplicity presented to the mind
as a whole?"  You might respond that we are not *really* talking about
a completed whole, since we could in principle eliminate such terms in
favor of formulas involving only "Ord(x)".  But this sort of
reductionist reply (to which a good realist shouldn't be terribly
sympathetic anyway!) doesn't do away with the fact that it sure
*seems* like we're not in principle doing anything different in the
case of ON than in the case of \omega.  Seems to me there is a deep
tension here between platonism and the very "quasi-constructivist"
intuitions that get set theory off the ground in the first place.

As a platonist myself, I am *not happy* about this!

>cordially,
>mikhail zeleny@husc.harvard.edu

--Chris Menzel
  Department of Philosophy