[FOM] V does not exist

Richard Heck rgheck at brown.edu
Sat Oct 8 13:33:04 EDT 2005

>Both A.P.Hazen and Aatu Koskensilta have responded to an argument on my part (though not mine) to the effect that the standard interpretation of V in NBG is incoherent.
>Though I argued that calling V a class rather than a set would not escape the argument, Hazen felt that if V really were a different kind of thing:
>  "they are the (extensionalizations of) meanings
>   of predicates of our set-theoretic language, and they 
>   exist only by being definable."
>then my argument would fail.
Allen's language here is somewhat colorful, but I took his point to rest
upon the observation that quantification over classes NBG can be
understood as substitutional. Perhaps there is a problem here I'm not
remembering, one that is connected with the presence of parameters in
the comprehension axioms, but I don't think so. In any event, much the
same point could be made in a different way: NBG can be interpreted in
ZF(C) plus a weak truth-theory, one in which the truth-predicate is not
allowed to figure in instances of schemata. If you think of classes that
way, then I think it's clear enough what Allen's flourishes mean, and
there is no conflict between NBG and the definition:

>Defn:	A "pure well-founded set" is any definite collection of pure well-founded sets.
which I take to be equivalent to Boolos's insistence that set-theory is
supposed to be about /all/ collections.

As Allen mentions, these points all come from Charles Parsons's "Sets
and Classes". I believe that much of the technical work on which they
rest was also done by Charles, early in his career.

Richard Heck

Richard G Heck, Jr
Professor of Philosophy
Brown University
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