[FOM] The defence of well-founded set theory

[Roger Bishop Jones]

On Tuesday 04 October 2005  4:39 am, Aatu Koskensilta wrote:
> On Oct 3, 2005, at 11:16 AM, Roger Bishop Jones wrote:
> > On Thursday 29 September 2005  9:54 pm, Stanislav Barov 
wrote:
> >> H. Wayl
> >> influented intuitionistic criticue formulated his own
> >> version of set theory which was precisly described and
> >> improved by S. Faffermann. As sone in ZF was obvious lack
> >> for big sets Geodel and Bernais make defenitions for notion
> >> of class which isn't set and not contained in other class.
> >
> > Its not clear to me what cogent critique of the iterative
> > conception these respond to.
> > (indeed NBG might be, perhaps usually is, considered
> > consistent with that conception)
>
> NBG is consistent with the usual conception of set theoretic
> hierarchy in a rather trivial sense: one can consider the
> totality of classes to consist of properties or collections
> definable in the language of set theory. The intelligibility
> of the language of set theory itself seems to imply the
> acceptability of such a totality of classes.

I find this hard to swallow.

The iterative conception of set describes the pure
well-founded sets, which we may accept as the domain
of discourse of first order set theory (ZFC) at the expense
only of some difficulty with the metatheory.

However, as soon as we try to make the totality of
pure well-founded sets into a collection we run into
trouble, because of course it must be and cannot be
a pure well-founded set.  Calling this totality a
class rather than a set doesn't really help, because
if there were a class which collected ALL pure well-founded
sets then it would be a pure well-founded collection
and there would have to be a set with the same extension.

If first order set theory is interpreted with quantifiers
ranging over all pure well-founded sets then so far we
have not hit this problem, because there is no claim
that there is a collection containing all the sets.
It is still possible to talk about formulae of set
theory defining classes so long as we regard this
as a "virtual theory", of the kind discussed by Quine,
where no ontological claims are made.
However, this interpretation cannot be used to
support a set theory which quantifies over classes.

If indeed "The intelligibility of the language of
set theory itself seems to imply the acceptability
of such a totality of classes" we would have to
conclude that set theory is unintelligible.

Of course, all this changes if we compromise by
interpreting set theory in one or more V(alpha)
rather than in V as a whole.
Then NBG (or even set theory in w-order logic)
becomes just another language for talking about
(interpretable in) part of V.

> The only
> controversial thing in NBG not motivable in this fashion is
> the global axiom of choice for which you need to use forcing
> to prove conservativity. 

But surely choice is (notwithstanding Boolos' reservations)
entailed by the requirement in the iterative conception that
at each stage ALL sets are formed whose members have been
formed at previous stages.
Even if you interpret NBG in a V(alpha) rather than V
(as I argued above one must) surely the choice sets must
be there?

> I'm not sure how commonly the global
> axiom of choice is actually considered to belong to NBG, but
> it is included at least in the original von Neumann
> formulation (or, rather, is an immediate consequence of the
> limitation of size principle).
>
> NBG is of course only one first step in the sequence of
> "predicative" class extensions of ZFC arguably as acceptable
> as ZFC and the notion of set theoretic truth. These theories
> are all weaker than ZFC+"There is an inaccessible" (in the
> sense that ZFC+IA proves every set theoretic sentence provable
> in any of these extensions of ZFC).

Is there a rationale for sticking here to "predicative" class
extensions?
It seems (as discussed above) only to work if the interpretation
falls short of encompassing all sets, and in that case it works
also for impredicative theories.
This is just to keep the theory no more controversial than ZFC?
So someone who accepts large cardinals would be rational to
drop the predicativity?

Roger Jones