FOM: Quine's NF

Matt Insall montez at
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
 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
 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,
 I expect is the case for most mathematicians, a) and b) are just
 counterintuitive, especially since no sets of real numbers satisfy either
 or b).  It is true that in the NBG extension of ZF to include classes,
 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
 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
 montez at

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