Re: proper classes in ZF

On 2007-05-19, in sci.math, MoeBlee wrote:
Just as a matter of personal taste, I kind of don't like speaking of
these things in this manner, so I'm somewhat uncomfortable here. But I
think the point here is that there are only a denumerable number of
formulas, yet Bernays set theory proves the existence of more than
just a denumerable number of classes, and since ZFC and Bernays set
theory are "in certain resepect essentially the same" (there are
formal ways of saying that, but, for now I demur from a formulation
since there are a couple of points regarding it that I feel I haven't
worked out properly in my own understanding), then we have to
recognize that there are (even in a "ZFC context", which again I've
not made formal) classes that don't correspond to a ZFC formula.

You seem to be disregarding the qualifier "with parameters" in my text. In
NBG the existence of any class definable by a formula with set parameters
(and free class variables) is provable. There are certainly more than
denumerable number of them -- in fact there are at least as many proper
classes as there are sets.

It was some time ago I wrote the PlanetMath article, and today I would
probably phrase it differently. There is nothing seriously wrong it, though.

P.S. One of the reasons I'm uncomfortable talking about this is that I
haven't worked out a problem I discussed with Aatu. See, I'd rather
not use a two-sorted language, since, at least for me, that
complicates comparing among first order theories. So I started working
on a strict formalization of Bernays set theory using relativization
rather than two-sorted logic. But what I found out is that when you
use relativization, you don't get that Bernays set theory is truly,
technically, a conservative extension of ZFC, but rather there is yet
another technical qualification that has to be made to get a rigorous
formulation of the statement that Bernays is "essentially conservative
over ZFC". However, that is a concern that is perhaps mainly one of my
fussiness as to technicals, while for the working purposes of most
people it is enough to understand the basic sense in which Bernays set
theory is convservative over ZFC.

As said, it's a mere technicality.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.