Re: SUO: Echos
> >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.
Chris Menzel
--
Christopher Menzel # web: philebus.tamu.edu/~cmenzel
Philosophy, Texas A&M University # net: chris.menzel@tamu.edu
College Station, TX 77843-4237 # vox: (979) 845-8764