# FOM: Quine's NF

Matt Insall montez at rollanet.org
Wed Mar 8 16:26:17 EST 2000

 Several years ago, I read several books on different set theories.  Quine's
NF was mentioned, and I seem to recall that in at least one reference, it
was claimed that NF is inconsistent.  I have been unable to find such a
reference lately, but in the text Foundations of Set Theory'' by
Fraenkel,
Bar-Hillel and Levy, the following facts are mentioned:
1.  Quine's system presented in his Mathematical Logic'', called ML'',
was shown to be inconsistent by Rosser in 1942, because it implied the
Burali-Forti Paradox.  (Fraenkel, Bar-Hillel and Levy, footnote on page
168)
2.  NF contradicts certain simple and obvious facts of classical set
theory'' (Fraenkel, Bar-Hillel and Levy, page 163, line 2):
a)  In NF, some sets X are such that if Y = {{x}|x \in X}, then X and Y
are not equinumerous.
b)  In NF, some sets X are such that X is equinumerous with the power
set of X.

Perhaps I misremembered the quote about ML as a quote about NF.  In any
case, item 2 seems a good deal more serious to me (as it seems to have
appeared to Fraenkel, Bar-Hillel and Levy, cf. page 163, line 1).  To me,
as
I expect is the case for most mathematicians, a) and b) are just
counterintuitive, especially since no sets of real numbers satisfy either
a)
or b).  It is true that in the NBG extension of ZF to include classes,
there
are *classes* X  of *sets* such that the class P_S(X) of all *subsets* of X
is equinumerous with X, but all such classes are proper classes and cannot
be members of P_S(X) for that reason.  In a technical (or formalistic)
sense, these objections may be mere linguistic problems'' (and my gut
reaction is to say they are just that), but that is quite an unsatisfying
answer, in my opinion.  The fact that one must always be careful which kind
of set one is working with in NF in order to use such fundamental intuitive
notions as the equinumerosity of a set with its set of singletons would
automatically cause me to prefer ZF, or, even better, NBG.  (Note that in
NBG, every class is equinumerous with the class of its singletons, and the
linguistic distinction'' between classes and sets is actually not just a
technical advantage, but an intuitive one as well.  Witness all the
following topics in Mathematics:  the use of varieties of algebraic
structures in universal algebra, universal spaces in topology, category
theory, algebraic K-theory (I think), etc., etc., etc.)

Steve asked whether NF (or some suitably related system) can be interpreted
in ZF (or some extension, such as NBG), or vice versa.  Apparently, this
was
asked also years ago, for the similar questions appear on page 166 of
Foundations of Set Theory''.  I guess this hasn't been answered yet?  Who
might be working on this?

Matt Insall
Associate Professor
Mathematics and Statistics Department
University of Missouri - Rolla
insall at umr.edu
montez at rollanet.org
http:/www.umr.edu/~insall
http:/www.rollanet.org/~montez