🚀 go-pugleaf

In article <24509@optima.cs.arizona.edu> gudeman@cs.arizona.edu (David Gudeman) writes:
>...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.  

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.  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).  

>What I take issue with is the apparent claim by realists that this
>syntactical peculiarity tells us something fundamental about real
>sets.  

The failure of comprehension--roughly, that there is always a set {x :
A(x)} for any condition A(x)--is neither syntactic nor a peculiarity.
It is a deep and indisputable ontological fact.  Ok, it *is* syntactic
insofar as we quantify over conditions; but what we're interested in
is what it says about sets.  If it turned out that the denotation of
`George Bush' knew all about the Iran-Contra affair from the
beginning, it would still be a fact about Bush, despite the (in this
case, unnecessary) syntactic trappings.

>Specifically, that the paradox tells us (1) that not all
>properties describe sets,...

Understanding "describe" to mean "have as extensions"...

>...(2) that the idea of self-containing sets is nonsensical,...

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.  The paradox implies nothing
about the incoherence of self-membership.  Type theory, of course,
takes this extra step and implicitly declares the idea of
self-membership incoherent and encodes this in a syntax that doesn't
allow its expression.  This avoids the paradox, but isn't terribly
satisfying as an explanation of it.

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

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.

>I don't believe any of these things.

Unbelief noted.

>Take the (naive) axiom of comprehension
>
>  y elem {x : p(x)} <=> p(y)
>
>as a realist statement rather than as a rule in an inductive system.

You're losing me.

>Clearly, this can have no other meaning than as a definition of set
>membership (for until we posit this axiom, we have no idea what it
>means to be a member of a set).  

Hmm.  ZF (or, for Randall Holmes and his ilk, NFU) sure *seems* to
give us a pretty good idea.  The above is of course missing from both.

>Also clearly, the definition is not
>eliminable, and so is suspect for that reason alone, without ever
>talking about what it means.  In particular, the definition contains
>an ill-founded recursion which can be seen if you rewrite it (in a
>higher-order logic) as
>
>  "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.  

Recursion isn't quite the notion you're looking for.  I think what
you're after here is the idea of "impredicativity".  An impredicative
definition, roughly, is one in which the range of a quantifier in the
definition includes the object being defined.  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.  Poincare and, early on,
Russell traced the paradox to the impredicativity of the definition of
the Russell class R: R is that class such that, for ANY class x (R
included), x \in R iff x \not\in x.

*However*, if you're a realist, and you don't buy into the
constructivist line, then an impredicative definition is nothing but a
way of picking out something that already exists (sets, after all,
exist independently of our ability to conceive them or pick them out)
and hence is perfectly legitimate.  Quine has a very nice discussion
of this in Set Theory and Its Logic, and strict predicativity and its
dizzying, full blown manifestation in ramified type theory is
discussed at length in Chihara's very nice book Ontology and the
Vicious Circle Principle.

  Chris Menzel
  Philosophy Dept.
  Texas A&M University