[FOM] V does not exist

[Roger Bishop Jones]

On Saturday 08 October 2005  6:33 pm, Richard Heck wrote:
> >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,

However, as I pointed out in my message, my argument is
independent of the nature of V, speaking only to its intended
extension, and has nothing to say about quantification over
classes (though I could easily offer relevent corollories).

I did not argue that NBG cannot be interpreted.

> 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.

Well it certainly is not intended to be equivalent to it.

First of all, I don't see how a definition can be equivalcnt
to an "insistance"!

If I were to take this alleged insistance as a definition
then I guess it would read "a set is any collection".

The difference between this and my own definition, which
I will paraphrase for comparison as "a set is any
definite collection of sets", seems to me very considerable.

My definition contains so much information that it runs
very close to inconsistency (its a reductio absurdum
on the possibility that the iterative conception
could be completed).
The one you attribute to Boolos contains so little information
that it runs close to vacuity.

My definition is a well-founded recursive definition.
It is a definition by transfinite induction, and should
be understood as involving the tacit codicil: nothing
is a set unless its sethood is entailed by the definition.

>From the definition we are can derive a principle
of transfinite induction asserting that sets have every
"hereditary" property, where, in this context, a property
is hereditary iff it is posessed by set whenever it is
posessed by all its members.

Using this induction principle we can then prove that:

1.  All sets are pure.
2.  All sets are well-founded.
and hence
3.  All sets are "heteronymous"
	(i.e. do not contain themselves) 

None of these conclusions flows from the insistance
which you attribute to Boolos.

More controversially perhaps, it is plain from my
definition that:

4.  All definite collections of sets are sets.

and hence that there are no proper classes, unless
something containing things other than sets or lacking
a definite extension might be said to be a class.

Boolos's alleged insistance, would have the additional
disadvantage that, taken out of context but with some
knowledge of Boolos's metaphysics, we might reasonably
interpret it as referring to all "actual" collections,
where the meaning of "actual" if any, can only be
discovered by probing Boolos's metaphysical intuitions.

By contrast, my definition may be understood as a
definition, not of all the sets which "really exist"
but as a definition of all the sets which might possibly
exist, of which the sets intuited by Boolos are
an infinitesimally small part.

A final but important difference between the "definitions"
is the occurence in mine of the concept "definite".
Without this the argument yields a contradiction without
consideration of classes, suggesting that the concept of
"set" is incoherent.
With it, it appears to demonstrate that there must be some
characteristic of the extesions which yield sets which
is not shared by the extensions which yield classes.
In my view it is best to read "definite" as a feature
implicit in the first order formalisation of set theory,
viz: that for any set s and any putative member x either
x is in s or x is not in s.
This is of course, an instance of excluded middle.
For a theory to emcompass collections which are not definite
in this sense one would have to represent membership by
something more complicated than a relation.
Possibly this motivates attempts to interpret
classes as rules or formulae.   However this won't
help if the rule or formula or whatever, is still
supposed to have a definite extension.

Since NBG is a first order language of set theory
which includes "classes" such as V, this kind of
"definiteness" of extension is possessed both by
the sets and the classes, and the argument shows
that the supposition that the extension of V is
all the sets encompassed by the iterative conception
(rather than all the sets in some other
interpretation of NBG) is incoherent.

I guess that, even with this additional explanation
you will not be convinced by this argument, and in
that case I would be interested to know where you
find the argument to be faulty.

For my part, coming across this particular definition
of "set", (even though its an obvious definition of
pure well-founded set and seems, obviously, to say the
same thing as the iterative conception),
has made a significant change to my beliefs
about classes and about what kinds of accounts of
the semantics for set theory are coherent.

I used to be suspicious about V, doubting whether
the iterative conception of set could coherently
be considered completeable.
But I knew of no argument for or against which I
considered wholly convincing.
I now believe not only that the intended interpretation
of NBG is incoherent, but also that formal set
theories which do not mention classes cannot coherently
be considered to be interpreted in the complete domain
described by the iterative conception of set.

Of course, these are philosophical matters, so
I don't imagine that these arguments are conclusive.

Roger Jones