Re: Let's you & him fight


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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