From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Russell's Paradox (was re: Of apples and their skins)
Organization: Boise State University
Date: Tue, 13 Oct 1992 17:55:07 GMT

In article <1992Oct10.011243.16219@husc3.harvard.edu> zeleny@husc10.harvard.edu (Michael Zeleny) writes:
>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.

I know all of this.  Where did I say that Cantor believed that there
was a universe?

  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.

Of course, he was wrong, since the set of all ordinals _does_ exist.
But anyone gullible enough to believe the inductive proof that each
ordinal is the order type of the corresponding initial segment of the
ordinals in the natural order... :-)  [in joke for NF(U) fans] 

  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.

I know this, too.  I don't understand how Cantor's adherence to his
(consistent, I agree) set theory gives the lie to my adherence to my
(equiconsistent) set theory.

>
>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.

It is certainly a ludicrous viewpoint, as I proceeded to prove!  But a
lot of people interpreted the situation exactly as I described it.  I
do not interpret it in this way; I think that there are _a priori_
reasons why the Russell "property" does not define a class (and the
trivial condition on x that x = x does define the universe
successfully), prior to any consideration of the paradox.

  Recall that a set is "defined" by
>Cantor as a multiplicity presented to the mind as a whole.

Now you know that I would not agree that the mind has anything to do
with it!

  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.

Sure.  The set of all strongly Cantorian ordinals does not exist; the
absolute infinite is there in Quine-Jensen set theory, too.

  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.

"Limitation of size" proves to be a workable but excessively strong
criterion.


>
>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 obvious model is all around you :-) But, strictly speaking, there
is no model of Cantorian set theory as a whole; a model would be "too
big".  Similarly, there is no model of Quine-Jensen set theory "as a
whole"; membership is not a relation.  A model of Cantorian set theory
is defective, in that it is too small.  A model of Quine-Jensen set
theory is also defective; since the membership on the model becomes a
relation, the ordinals of the model are not well-ordered by the
natural order (externally).  You can't talk about the intended model
of Cantorian set theory _as a whole_ without skirting the absolute
infinite, an incoherent notion.  Quine-Jensen set theory has very nice
models; they give a perfectly coherent picture of what is going on.
Both on the one-liner level (the intended model of NFU is the actual
logical universe) and on the serious level (read the account of models
of NFU in Forster's book) consider yourself answered.  Do you really
not know that it is easy to model NFU, and the modelling scheme is
just as extensible as the natural scheme for modelling of Zermelo set
theory?

It is true that a model of NFU in Zermelo style set theory will have
certain defects (and so will a model of NFU in NFU!), but this is not
a fair test; the model is NOT the universe.  Note that models of
Zermelo set theory in Zermelo set theory (or NFU) are also defective;
they fail to contain all ordinals.

>
>>-- 
>>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."

Mikhail, are you trying to pick a fight with me?  You know I can't
resist!  :-)



-- 
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
Xref: sparky sci.philosophy.meta:2134 alt.postmodern:2493