From: zeleny@husc10.harvard.edu (Michael Zeleny)
Newsgroups: sci.philosophy.meta,sci.logic
Subject: Russell's Paradox (was re: Of apples and their skins)
Date: 10 Oct 92 05:12:42 GMT
Article-I.D.: husc3.1992Oct10.011243.16219
Organization: The Phallogocentric Cabal

In article <1992Oct9.155935.18205@guinness.idbsu.edu>
holmes@garnet.idbsu.edu (Randall Holmes) writes:

RH:
>The Liar (This statement is false) was originally proposed in a spirit
>of deliberate paradox (in ancient Greece, not by Russell).  Thomas's
>criticism may apply to the original proposer.
>
>It certainly does not apply to Russell.  Russell was interested in the
>Liar because he had produced it unintentionally in the course of
>reasoning which seemed perfectly sensible.  If reason produces the
>Liar, reason is in deep trouble; thus Russell's genuine, legitimate
>concern.
>
>Here's how he did it.  It seems prefectly acceptable that for any
>property P (say, redness) there is a set of all things with property P
>(the set of all red things).  Now, if P is the property "is a set",
>the set we get (the set of all sets) contains itself as a member; it
>is a set.  If P is "is red", the set we get does not contain itself;
>the set of all red things is not a red thing (it is an abstraction).
>Let P be "is a set which does not contain itself as a member".  The
>collection we get is the "Russell class", the set of all sets which
>are not members of themselves.  If R is a member of itself, then it
>fails to have its defining property and is not a member of itself.  If
>R is not a member of itself, then it has its defining property and
>should be a member of itself.  This is a contradiction.
>
>The conclusion of this argument is not "there is something wrong with
>reason"; it is that the "perfectly acceptable principle" above is
>false; it is not the case that every property corresponds to a
>collection in this way.  But notice that the sentence "R is a member
>of itself" is basically the Liar; it asserts its own falsehood.
>
>The definition of R is not artificial, either.  It results from
>applying Cantor's proof that the collection of all subsets of a set A
>is "larger" than the set A to the largest possible set, the universe.
>
>Russell's motive was not to show off or to confuse anyone; it was
>serious alarm about basic concepts being applied in mathematics at the
>time.  When Russell discussed the Liar, his aim was to look for
>correct principles of reasoning which would avoid the paradox above
>and other similar problems.

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.  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.  His philosophical righteousness, augmented by
lucid interpretive skills of Zermelo and G\"odel, clearly gives lie to your
oft-repeated claim concerning the "rightness" of NFU.

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.  Other reasons can be readily found in Hallett's book on the
limitation of size.

And to retort, as you have been known to do on occasion, that the universal
set is perfectly conceivable in type-free theories like Jensen's NFU, would
obviously commit you to the onerous, and, as far as I know, hitherto
uncompleted task of describing the intended model for such a theory, at
least to the extent such models have been described in the Cantor-Zermelo
tradition.

>-- 
>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
"Nous donnons la mort, nous saurons la subir."
Newsgroups: sci.philosophy.meta