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In article <24780@optima.cs.arizona.edu> gudeman@cs.arizona.edu (David Gudeman) writes:
>In article  <1992Oct29.012959.12245@tamsun.tamu.edu> Chris Menzel writes:
>]In article <24509@optima.cs.arizona.edu> gudeman@cs.arizona.edu (David Gudeman) writes:

DG:
>]>...I have always thought that when people said things like
>]>"Russel's paradox proves that not all properties describe sets" that
>]>they meant _semantic_ properties.  But in reading through Quine's _Set
>]>Theory and its Logic_ I see that he writes things like {x : alpha}
>]>where alpha is specifically a syntactic object.  
>]

CM:
>]Sure, a syntactic object whose semantic value (intuitively, anyway) is
>]a property.  So one way of expressing the lesson of Russell's paradox
>]is that there are first-order formulas (e.g., x \not\in x) such that
>]the properties they express do not have sets as their extensions.  One
>]could, of course, dispense with property talk altogether and talk
>]instead about formulas.

DG:
>It looks like you are assuming that all first order formulas express
>properties.  

As clearly noted, the property talk is informal and intuitive, and can
be dispensed with--though I do believe that every first-order formulas
expresses a property, relation, or proposition (but would want to be
clear about what the primitives of the language are, and what it's
logic is).

DG:
>This is clearly false since Russel's Paradox can be
>expressed in such formulas without reference to sets.

Not so clear; there are of course property theoretic versions of the
paradox, but don't forget that it takes more to generate the paradox
than the assumption that every FO formula with a single free variable
expresses a property.  In particular, one needs the property theoretic
equivalent of lambda conversion, i.e., that where [x:A] is any
property denoting term, b EX [x:A] iff A(b), where EX is the (2-place)
exemplification relation.  One approach to the paradox is to restrict
this principle in one way or another.  See, e.g., Ray Turner's "A
Theory of Properties" that appeared in the JSL a few years back.
Turner defines a language with truth and exemplification predicates in
which every formula can be used to form a property (or n-place
relation) denoting term, with a consistent logic.  The semantics for
the language uses a variant of Gupta and Herzberger's fixed point
constructions.

CM:
>]Say that a *condition* is a first-order
>]formula A(x) (in an interpreted language) with at most a single free
>]variable x, and that an object m *satisifies* A(x) just in case A(x)
>]is true when x is assigned m.  Then one can say that RP showed that
>]there are conditions A(x) such that there is no set consisting of
>]exactly the things that satisfy A(x).  

DG:
>If A(x) is not a property at all, but merely a syntactic well-formed
>formula that does not express a proposition, this is merely a
>syntactic curiosity.


Your assertion is groundless, and, as far as I can see, confused.
This is very straightforward, standard stuff.  Is it also a syntactic
curiousity that in ZF set theory theory, every condition of the form x
\in a & A(x) *does* pick out a set?  What is the difference?

CM:
>]The source of the paradox is sometimes mistakenly traced to the idea
>]of self-membership, but that's a mistake, as, e.g., Peter Aczel's
>]recent work on nonwellfounded sets shows.

DG:
>Is this generally acknowledged now or still controversial?

There are no grounds for controversy; nonwellfounded set theory (on
which someone recently gave a nice overview in this group) is provably
consistent relative to ZF.

DG:
>]>(3) that there is some mysterious distinction between
>]>sets and classes.  

CM:
>]A distinction yes, a mystery not at all, at least on the iterative
>]picture.  (Proper) classes are collections that are unbounded in rank,
>]i.e., collections containing objects that occur arbitrarily high in
>]the iterative hierarchy.  On this picture, the real error behind
>]Russell's paradox is the failure to make this distinction.

DG:
>Sorry, I find that quite mysterious...

Well, you may not like that as an explanation of RP, but mysterious it
is not; this is very basic set theory.  The iterative hierarchy, the
notion of rank, the set/class distinction, these are all perfectly
well defined and (modulo the intrinsic complexities of the hierarchy)
well understood.

DG:
>In the first place, I have no
>reason to supose that there is any metaphysical significance to the
>iterative hierarchy.  

What ARE you talking about?

DG:
>In the second place, the paradox applies to classes as well as it
>does to sets.

Not according to ZF (supplemented with class terms) or VNBG.  You may
not like that, but then go invent your own set theory.

DG:
>All I see in the class/set
>distinction is a primitive type theory that doesn't even solve the
>problem it was meant to solve.

Cloudy vision noted.

DG:
>]>Take the (naive) axiom of comprehension
>]>
>]>  y elem {x : p(x)} <=> p(y)...

This is not comprehension; naive comprehension is the principle that
every condition determines a set.

DG:
>ZF and
>NFU both exclude the existence of arbitrary self-containing sets, and
>I think this is a weakness.

Ok, but why?  In the iterative hierarchy, which is the natural model
of ZF (I defer to Randall on NFU), it is plain why there are no
self-membered sets.  Hence, ZF excludes them.  But if you don't like
that, adopt Aczel's theory which replaces foundation with an
anti-foundation axiom that generates a huge array of self-membered and
other varieties of nonwellfounded sets.  Does that suit you?

DG:
>]>  "elem" is that relation satisfying
>]>    Forall p,y . y elem {x : p(x)} <=> p(y)
>]>
>]>where p is taken as quantifying over all relations _including set
>]>membership_.  Since set membership is not yet defined, this is a
>]>recursive definition and so you cannot assume it is consistent, you
>]>have to prove it.  

CM:
>]...  If you think that sets
>]are things we "construct" somehow, then there does seem to be
>]something illicit about impredicative definitions: if the object being
>]defined in some sense exists in virtue of our having defined it, then
>]we jump the ontological gun if we attempt to define it in terms of a
>]range of objects that already includes it...
>]*However*, if you're a realist,...
>]then an impredicative definition is nothing but a
>]way of picking out something that already exists...

DG:
>No, the problem with recursive (a more general term than
>"impredicative") definitions...

Impredicative definitions are not in general recursive.  Again, this
is pretty fundamental material.  And are you seriously suggesting
there is something wrong with the notion of a recursive definition?

DG:
>[Lengthy musings on definitions deleted]
>
>This (or so I claim) is the single, simple, obvious cause of Russel's
>paradox.  

Two l's in "Russell".

DG:
>In particular, R = {x : x not elem x} is a definition...
>that contains an implicit reference to R because x quantifies
>over a set that contains R.  

So, after all that, your problem is with the impredicativity of the
definition of the Russell class, just like Poincare and the early
Russell.  Why didn't you say so in the first place?


>					David Gudeman
>gudeman@cs.arizona.edu

  Chris Menzel
  Philosophy Deparment
  Texas A&M University