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Rupert wrote:
> george wrote:
> > "you" being Aatu Koskensilta and "him" being Roger Bishop Jones.
> > AK wrote:
> > > 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.
> >
> > RBJ replied,
> > >> I find this hard to swallow.
> > >>...[because]
> > >> 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
> >
> > so far so good
> >
> > >> and there would have to be a set with the same extension.
> >
> > This is surely false. "Class" just MEANS "collection" in this
> > context. But I would appreciate it if somebody (like AK) would
> > give a more cogent rebuttal.
> >
> > Calling this totality a class rather than a set PROVABLY
> > helps (you get a contradiction one way and you don't get
> > one the other way). And it is NOT the case that there "would have
> > to be" a set with the same extension; in models of NBG, THERE ISN'T
> > a set with the same extension, so clearly there doesn't have to be.
> >
> > The problem with "calling this totality a class rather than a set"
> > is NOT that it doesn't help; rather, the problem is that the fact
> > that it helps is THE ONLY available JUSTIFICATION for it; it does
> > not seem PRIORly analytic or obvious-in-definition that something's
> > being "too big" should prohibit it from being the SAME KIND of
> > collection.
> > The problem with the set/class distinction is that it seems like a
> > dodge
> > specifically concocted for coping with this problem; it does not seem
> > to have inherent viability.
> >
> > Though of course things are not always as they seem.
>
> Never mind all this talk about sets and classes. Suppose I adjoin to
> the first-order language of set theory a satisfaction predicate, the
> idea being that Sat(n,x_1,x_2,...,x_k) holds when n is the Goedel
> number of a formula in the first-order language of set theory with k
> free variables and it is true when these variables are interpreted to
> be x_1, x_2, ... x_k. I adjoin all the obvious axioms concerning the
> satisfaction predicate and I assume that the axiom of separation and
> replacement hold for formulas involving Sat as well. The resulting
> theory, call it ZFC*, is not conservative over ZFC, it has much
> stronger consistency strength. But surely if we accept ZFC we should
> accept ZFC*. And NBG can be interpreted in ZFC*. That is, there's a
> birecursive map from the theorems of NBG to a recursive subset of the
> theorems of ZFC*.
Sorry, I should have said "there's a birecursive map from the sentences
in the first-order language of set theory to a recursive set of
sentences in the language of ZFC* which maps theorems of NBG to
theorems of ZFC*."
> If we accept that talk about semantics of the
> first-order language of set theory is meaningful, then we can give a
> meaning to the sentences of the second-order language of set theory on
> which the theorems of NBG come out true. On the other hand, the same
> probably couldn't be said of Kelly-Morse set theory.
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