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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*. 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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