FOM: Re: class theory
Harvey Friedman
friedman at math.ohio-state.edu
Sun Jan 30 21:26:52 EST 2000
Reply to Shoenfield Sun, 30 Jan 2000 17:24.
I want to clarify a point about local/global choice, and also reply to
Shoenfield about some remarks concerning class theory and MK.
In Sat, 29 Jan 2000 15:50, Shoenfield mentions the well known result that
"global choice is not provable from
local choice in NBG." Here local choice is AxC = the axiom of choice for
sets, and glocal choice is C = the axiom of choice for classes. Shoenfield
attributes this to Solovay, 1963.
In my posting of Sat, 29 Jan 2000 17:09, I said that this well known result
was proved by "adding a generic class using set conditions."
My phrase "adding a generic class using set conditions" is actually the
obvious proof (often credited to Solovay) of a different result "every
model of NBG + AxC can be extended to a model of NBGC with the same sets"
which immediately implies that "global choice is a conservative extension
of local choice over NBG."
For "global choice is not provable from local choice in NBG," most people
probably think of Easton. However, Shoenfield says, "Solovay did this, I
think in 1963, although it has never (so far as I know) been published or
even announced".
In any case, additional arguments are needed - see my posting of 9:27AM
1/27/00 - to answer the very specific question of Charles Parsons stated in
his posting of 6:04PM 1/26/00. This is a question that comes directly out
of a consideration of some of Frege's work, as indicated in the Parsons
6:04PM 1/26/00 posting.
Parsons' question is a very sensible question about class theory, and the
answer I gave is most sharply stated in terms of MK = Morse/Kelley theory
of classes, which is an extension of NBG. This important theory of classes
has the very natural models
V(theta + 1)
where theta is a strongly inaccessible cardinal.
Furthermore, the Parsons question is not very much easier to solve for NBG
+ AxC than it is for the stronger theory MK + AxC.
Parsons' question and theories of classes like MK reflect a concept of
class of status independent from the concept of set, in that the class
concept is not considered to be defined from the set concept.
There can be no question that the theory of classes is a fundamental theory
in its own right that is to be distinguished from the theory of sets. This
is apparent through the history of logic, and nothing that has been done in
modern times should change one's mind about this.
In particular, we have only the bare beginnings of - but still a relatively
poor understanding of - where the axioms for large cardinals come from
and/or why (if?) they are canoncial/inevitable and/or why they should be
accepted and/or why they are consistent. The same can be said, perhaps not
as strongly, for the axioms of ZFC, or even for much weaker systems going
all the way down to fragements of arithmetic.
I have no doubt that further substantial progress on these crucial issues
will at least partly depend on deep philosophical introspection. And I have
no doubt that the concepts of *both* class and set, and their
*interaction*, will play a crucial role in the future.
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