[FOM] V does not exist
Roger Bishop Jones
rbj01 at rbjones.com
Fri Oct 7 03:52:27 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.
Koskensilta's response I didn't entirely understand,
but seemed to be directed toward justifying
quantification over classes, whereas my objection
was not to quantification over classes. It was to
the possibility of one particular class, V, being
what it is supposed to be.
I provide below a new presentation of the argument
which I think makes the argument more general and
precise, and clarifies the character of the result.
The argument, as now presented is an argument about
the concept "pure well-founded set" (which is what
I take the iterative conception of set to be describing).
It is to the effect that this concept does not have
a "definite" extension.
The meaning of "definite" here is not crucial to the
argument. In classical set theory as described in
the iterative conception of set "definite" means
something very weak (much weaker than the notion
of "definite property" used in defining separation).
It just means something like that the predicate or
membership relation is boolean.
I offer the following definition:
Defn: A "pure well-founded set" is any definite
collection of pure well-founded sets.
>From which I allege follows:
Lemma: Pure well-founded sets are pure and well-founded
(in the usual sense of these terms).
My thesis is:
The extension of the concept "pure well-founded set"
is not definite.
Proof: By reductio. Assume that it is definite and conclude
that it both is and is not heteronymous.
Since the argument is about the concept of set itself, any
object which purports to have a definite extension which
coincides with that concept, however different that object
may be from a set, must be tainted with the incoherence
of supposing that the concept set has a definite extension.
For anyone who finds this argument too tenuously connected
to the iterative conception of set, it can be reduced
to something closer to that account via a similar argument
to the effect that the extension of the concept ordinal
(which corresponds of course to the stages in the iterative
conception) cannot have a definite extension, and hence
that the conception cannot describe a definite collection
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