FOM: NBG and ZFC
Joseph Shoenfield
jrs at math.duke.edu
Sat Jan 29 15:50:45 EST 2000
This is (mostly) a comment on Friedman's communication on cardi-
nality operations.
I am not exactly sure what Friedman had in mind, but one of his
aims seems to be to show that global choice is not provable from
local choice in NBG. This was proved by Solovay, I think in 1963,
although it has never (so far as I know) been published or even
announced. This was the first independence proof by class forcing;
the ideas eventually found their way into Easton's thesis. I do
not know exacly what forcing class Solovay used; I'm sure he'd be
glad to tell you if you send him an e-mail.
As to the digression on whether NBG or VB is more appropriate,
one thing is sure; if an initial is used for von Neumann, it must
by N, not V. As to Godel's role, his changes in Bernays' system
were fairly minor; they are listed in a footnote to his book (on the
consistency of GH). I think his main contribution was to demon-
strate that NBG is a very suitable system for doing some kinds of
axiomatic set theory.
One thing is important to understand; each of N,B, and G
intended a class to be not an arbitrary collection of sets but a
collection which is definable (from set parameters) in the language
of set theory. Thus a model of NBG is best regarded as obtained
by taking a model M of ZFC and defining the classes of the model to
be the collections of individals of the model which are definable
from parameters. This is the smallest extension of the notion of
a class which will make M into a model of NBG (with local choice);
and there is no reason to consider any other. (Failure to under-
stand this has led some to think that Morse-Kelley is a simpler or
better axiomatic system. However, not every model of ZFC can be
extended to a model of MK; so there are many new problems in the
semantics of MK.)
Why did Godel use NBG? In his original announcement in the
PNAS, he used the much more familiar ZFC. To define the construc-
tible sets in ZFC, he needs to define the operator O, where O(x) is
the set of subsets of x definable from parameters in the structure
with universe x and the usual membership relation restricted to x.
This requires formalizing some syntax and semantics in ZFC. This
is no big problem for anyone with a command of elementary logic.
However, if classes (in the above sense) are available, things are
more direct; one can simply talk about sets and classes without
bringing in notions like "formulas" and "validity of formulas".
I think Godel felt this would make his work easier to understand by
those unfamiliar with logic.
I think it is sad that neither writers of texts on axiomatic
set theory nor researchers in the field have tried using NBG in
place of ZFC. I think it would often promote clarity and under-
standing without any apparent disadvantage.
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