Re: SUO: Echos
Chris,
Interesting points. Please see my comment below.
At 06:56 PM 2/15/01 -0600, Chris Menzel wrote:
> > >There's the set of things that are abstract.
> > >...
> >
> > No there isn't. You are talking about something that is equinumerous
> > with, e.g., the collection of all sets, namely a "proper class" (von
> > Neumann), "ultimate class" (Quine), "megacollection" (Langendoen and
> > Postal), or, perhaps best of all, an "inconsistent multiplicity"
> > (Cantor's 1899 letter to Dedekind). According to Fraenkel, Bar-Hillel,
> > and Levy, "The only difference between proper classes and sets is
> > that, because of the antinomies, the proper classes cannot be members
> > of classes whereas sets can." (_Foundations of Set Theory_,
> > North-Holland Publishing Co., Amsterdam, 1973, p. 137.)
>
>That is a difference, but it is a bit disingenuous of F, B-H, and L to
>say it is the *only* difference. That makes it sound as if the
>distinction is utterly ad hoc, retrofitted onto the universe of set
>theory simply to avoid the paradoxes. In fact, it is reasonable to
>argue that the paradoxes in fact brought to light a distinction that had
>been obscured. Reflection on Zermelo's original axioms and subsequent
>work in set theory eventually led to the development (by Mirimanoff, von
>Neumann, and Zermelo himself, among others) of the so-called "iterative"
>conception of set on which sets fall into a natural hierarchy based upon
>the membership relation: start with some "urelements" (concrete objects,
>say), then the first level consists of sets of urelements, and the n+1th
>level consists of all the sets that can be formed out of the objects in
>the first n levels together with the union of those levels. Unions are
>taken at limit stages. The members of earlier levels thus accumulate in
>later levels, and so this conception is often also called the
>"cumulative hierarchy". It is the "intended" model of ZF for most
>working set theorists.
>
>Now, say that the *rank* of a set is the ordinal number that indexes the
>level of the hierarchy in which that set first appears. Then we can
>draw the distinction between sets and proper classes simply and cleanly
>as follows: sets are those collections that have a rank. Proper
>classes, by contrast, are collections that contain sets of arbitrarily
>high rank; their members occur arbitrarily high up in the hierarchy.
>There is thus no level of the hierarchy at which a proper class ever
>appears, no level at which it is "formed" from objects in lower levels,
>and hence there are no collections that are themselves formed out of
>*them* -- i.e., unlike sets, they can't be members of other classes.
>
>The set theoretic paradoxes all arise by taking some proper class or
>other to be a set. By providing a substantial, structural account of
>the difference between the two types of collection that is independent
>of the set theoretic paradoxes, it is reasonable to claim that the iterative
>conception provides a genuine *explanation* of those paradoxes.
JT: The appeal of the iterative conception lies in its safe means of
"forming" collections -- so safe, in fact, as to prohibit the formation of
any collection that is not a set. That is why some people regard the
iterative conception as better suited to denying the existence of proper
classes than to (affirming their existence and) distinguishing them from
sets. I'm not sure whether Chris really believes in proper classes, but I
should mention, for what it's worth, that some people who believe in them
will favor an approach, like that of von Neumann, Bernays, and Godel, in
which "class" is treated as a primitive notion corresponding to the
intuitive notion of a collection, a set is defined as a class that is a
member of some class, and not all classes are sets. Whatever its defects,
the appeal of such an approach is that it permits the most straightforward
interpretation of Chris's claim that "The set theoretic paradoxes all arise
by taking some proper class or other to be a set", i.e. there really are
proper classes, and the paradoxes arise when they are mistaken for sets.