jrs at math.duke.edu
Mon Feb 14 23:14:03 EST 2000
In response to some rather unfavorable remarks I made about MK
(Morse-Kelly set-class theory), Friedman has defended MK as natural
and important. Let me try to describe briefly what (in my opinion)
is the origin and purpose of NBG and MK.
For the moment, let us take a class to be a collection of sets
definable from set parameters in the language of ZFC. Classes in
this sense play an important role, even if one is working in ZFC.
For example, the Separation Scheme is most easily stated as: every
class which is included in a set is a set. In the language of ZFC,
we can only state this as a scheme. If we introduce variables for
classes, we can state it as a single axiom. Of course, we then
need axioms to insure us that all classes are values of the class
variables. This can be done by a simple scheme. As Friedmann and
others have pointed out, this scheme can be derived from a finite
number of its instances; but the proof of this is a bit tedious, and
is quite useless if one wishes to develop set theory in NBG.
The nice thing about NBG it that every model M of ZFC has a least
extension to a model of NBG; the classes in the extension are just
the classes in M. From this it follows that NBG is a conservative
extension of ZFC. Thus whether we do set theory in ZFC or NBG is
a matter of taste.
Now all of this naturally suggests an extended notion of class,
in which a class is an arbitrary collection of sets. We then extend
our class existence scheme to make every collection of sets
definable in our extended language a class. Of course not every
class (in the extended sense) is so definable; but these are the
only ones we can assert are classes in our extended language.
Unfortunately, it is no longer true that any model of ZFC can
be extended to a model of MK. We can prove Con(ZFC) in MK by
proving that the class V is a model of ZFC; and Con(ZFC) is a
statement in the language of ZFC not provable in ZFC. If the
model of ZFC has strong enough closure properties, we can extend
it. For example, if the model is closed under forming subsets, it
is clear that the Separation Scheme will hold independent of the
choice of the classes in the model. In this way we can show
(as Friedman observes) that a model V(k) where k is an inacessible
cardinal can be extended to a model of MK. The trouble with such
models is that they have strong absoluteness properties; most in-
teresting set theoretic statements are true in V(k) iff they are
true in V. This makes the models useless for most independence
Friedmann has given a sketch of an independence proof in MK by
forcing; but many of the details are unclear to me. He takes a
model M of MK, lets M' be the included model of ZFC and N' a generic
extension of M'. He then says N' canonically generates a model N of
MK. I do not understand how one selects the classes of N, nor how
one can prove the axioms of MK hold in N. I would be surprised
if the details wouls lead me to agree with Friedman that the question
he was considering is "not very much easier to solve for NBG than it
is for MK". In any case, there seems to be little reason to solve
it for MK.
Friedman concludes with some predictions about the future of
MK and similar systems, He says:
> We have only the bare beginnings of where the axioms of large
>cardinals come from or why they are canonincal or why they should
>be accepted or why they are consistent.
I agree whole-heartedly with this, and with the implied state-
ment that these are important questions. He then says:
> I have no doubt that further substantial progess on these
>crucial issues will at least partly depend on deep philosophical
>introspection, and I have no doubt concepts of both class and set
>and their "interaction" will play a crucial role in the future.
Here I strongly disagree. I think that if there is one thing
we can learn from the development of mathematical logic in the last
century, it is that all the major accomplishments of this subject
consist of mathematical theorems, which, in the most interesing
cases, have evident foundational consequences. I do not know of
any major result in the field which was largely achieved by means
of philosophical introspection, as I understand the term. I do
not see the the study of the interaction of sets and classes has
led to any very interesting results.
If the problems about large cardinals cannot be solved by philo-
sophical introspection, how can they be solved? Fortunately, I
have available an example of how to proceed, furnished by the recent
communication of John Steel. I think it says more about the prob-
lems of large cardinals then all the previous fom communications
combined. The idea is to examine all the results which have been
proved about large cardinals and related concepts, and see if they
give some hint of which large cardinals we should accept and what
further results we might prove to further justify these axioms.
We are still a long way from accomplishing the goal, but, as Steel
shows, we have advanced a great deal since large carinals first
appeared on the scene forty years ago.
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