FOM: iterative conception of set
Charles Silver
csilver at sophia.smith.edu
Sat Feb 28 07:32:09 EST 1998
I have a question about NF (or NFU) that may relate somewhat to
the SET v. CAT controversy. Does NF seem to you to be based on some
fundamental conception of what a collection is? One reason I ask is that
I vaguely remember someone criticizing one or more of Quine's set theories
for having no underlying conception. But, I don't recall who said this
(was it Wang?) and I don't know the scope of the criticism. That is,
supposing this was a fair criticism, which set theories of Quine's would
it apply to?
My second reason for asking is that about a million years ago I
took a set theory course that used Quine's _Set Theory and Its Logic_. In
that book, the set theory laid out toward the beginning (there are some
alternatives towards the end, if I remember correctly) was presented as a
clever system built up step by step from first-order logic. At each step
in his formulation, Quine proposed various technical alternatives, arguing
in each case why the formulation he favored was to be preferred over all
others. So, his formulation of set theory proceeded by philosophical
argument, based on various concerns, like what would be the spiffiest way
to pretend, using the epsilon symbol, that you had a class when you really
didn't. For example, I (vaguely) remember "virtual classes" and I sort-of
half remember some extremely clever way that Quine had of explaining that
the epsilon symbol could precede an individual, in which case the epsilon
symbol didn't really mean "belong to", but meant "equal to"--I think.
Maybe that's wrong, I don't remember exactly. I also have the impression
that he identified individuals with their unit classes. The result was
that he had virtual classes and real ones, and then he showed--again with
much cleverness--how virtual classes and real ones could be meshed. (The
title "The Virtual Amid The Real" stands out in my mind.)
In short, I think it is clear that Quine had no underlying
conception pertaining to "collection" whatsoever in _Set Theory and Its
Logic_. Rather, he wanted to develop a set theory that would maximize
various philosophical principles that he endorsed, so he tinkered and
fiddled and fussed with alternative definitions in order to arrive at a
version of set theory that was least objectionable to him.
Finally my question: Is NF a "real" set theory in the sense of
embodying a "real" conception of the underlying notion of a "collection"?
That is, is its development quite different from the development of set
theory in _Set Theory and Its Logic_?
I think the fact that this question can be raised shows that not
any set theory can be claimed to be foundationally superior to, say, CAT,
in being firmly based on an underlying conception. One can *say* that set
theory is based on "collection", but not have it turn out that the
resultant set theory is a satisfactory explication of this concept. To my
mind, this raises further questions about how to judge whether an
intuitive conception is successfully embodied within a technical
formalism.
Charlie Silver
Smith College
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