Re: proper classes in ZF

On May 17, 8:32 am, zuhair <zaljo...@xxxxxxxxx> wrote:

I will quote the following from Planet math.

"- - - In many set theories there are formally no proper classes; ZFC
is an example of just such a set theory. In these theories one usually
means by a proper class an open formula , possibly with set
parameters . Notice, however, that these do not exhaust all possible
proper classes that should "really" exist for the universe, as it only
allows us to deal with proper classes that can be defined by means of
an open formula with parameters.

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.

The theory NBG formalises this usage:
it's conservative over ZFC (as clearly speaking about open formulae
with parameters must be!).

There is a set theory known as Morse-Kelley set theory which allows us
to speak about and to quantify over an extended class of
impredicatively defined proper classes that can't be reduced to simply
speaking about open formulae - - -".

This probably has to do with the fact that Morse allows the 'P' in the
comprehension schema to have general class variables (i.e., not
restricted to set variables). In other words, IF I understand all of
this correctly: Z (thus ZFC too) is impredicative since the 'P' in the
separation schema allows variables whose range includes the very set
that is being asserted to exist; then Bernays set theory is
impredicative in that way too, but it is not impredicative over proper
classes too, since the variables in the 'P' in the comprehension
schema are restricted to set variables; but Morse is impredicative
over everything (including proper classes) since, for asserting the
existence of a proper class, the 'P' in the comprehension schema
allows variables whose range includes the very proper class that is
being asserted to exist.

(If that is not a correct summary, then hopefully someone will correct
or sharpen it.)

So, perhaps that is the sense in which the article is describing Morse
as providing an "extended" ("more"?) proper classes even than Bernays
set theory. However, I don't quite catch what difference is being
claimed between Bernays and Morse with regards to proper classes
"reduced to" open formulas.

So what this quotation is telling us, is that all proper classes in
NBG can be reduced in ZFC (or any thoery that doesn't have proper
classes as objects in its universe of discourse) to open formulas.

That's not my impression (though I could be wrong). I'm not so sure
that the article is saying that it is not the case that Bernays set
theory proves the existence of more proper classes than can be
specified in ZFC by a ZFC formula for each proper class.

However the source is mentioning that in MK there are proper classes
that cannot be reduced to open formulas in ZFC and the alike theories.

To the extent you quoted the article, I'm not so sure that it is
saying, as it seems to me that you are thinking, that ZFC can catch
all the Bernays proper classes but not all of the Morse proper classes
(maybe it is saying that, but it's just not clear to me that it is).
Anyway, again, the main difference has to do with the differences in
what variables are allowed in the respective comprehension schemata.
But I'm not clear how that plays out toward what proper classes can be
"captured" by a ZFC formula.

I want to know which proper classes in MK that cannot be reduced to
open formulas in ZFC, and why?

I don't know enough about Morse in particular. But at least on general
grounds, again, as I mentioned, all of these theories prove the
existence of more than can specified by a formula for each class (or
even for each set). There are only denumerably many formulas. But
there are more than just a denumerable number of sets (and, I would
think, in Bernays set theory, even more than just a denumerable number
of proper classes; and, I would think, even more proper classes in
Morse than in Bernays).

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.

MoeBlee



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