Re: proper classes in ZF

On May 17, 7:34 pm, 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. 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 - - -".

Quotation finished.

One thing that realy perplex me about this notion, is that if proper
classes in ZF are the open formulas, then we know that MK quantify
over these proper classes, this mean that MK which is a first order
logic set theory is actually quantiying over open formulas in ZF.

See, this an example of the kind of problem I find with this kind of
loose way of talking. Are ZFC open formulas ACTUALLY proper classes;
or rather is there a RELATION BETWEEN ZFC formulas and proper classes?
I would be very wary of committing to the propostion that ZFC open
formulas are ACTUALLY proper classes. In fact, in my meta-theory, ZFC
open formuas are definitely NOT themselves proper classes (in my meta-
theory, which is itself a ZFC "one step up from object level ZFC", ZFC
open formulas are sequences of symbols, and sequences are a certain
kind of function, and functions are a certain kind of set, so ZFC
formulas are sets, and not proper classes). However, I do understand
that we might formalize a certain relation between open formulas in
the language of ZFC and objects that fulfill the "role" of Bernays
proper classes. But that's just my own way of looking at it.

This
provok me to ask the following question: can a theory in first order
logic be equi-interpretable with a theory in second order logic? since
it appears to me that MK which is a first order logic thoery is
working like second order ZF, and I think they would be equi-
interpretable.

I don't know about equi-interpretability of Morse and ZFC (my guess is
that they are not equi-interpretable or at least not proven to be equi-
interpretable, since, as I understand, there's not a known relative
consistency proof from ZFC to Morse), but, as I understand, there are
instances of equi-interpretability between theories in languages of
different order. But, dealing with this precisely at least begins with
a definition of 'interpretable'. There are definitions in mathematical
logic of these things; so we should learn them and be very clear about
them if we are to really understand what is involved in such questions
as yours.

Anyway, as you get further into this you will discover relations among
first order theories, first order theories in multi-sorted languages,
relativizations of theories, first order set theories, second order
theories of arithmetic, hierarchies in set theories, etc.

MoeBlee

.

Relevant Pages

  • Re: The set of All sets
    ... > In a conservative extension of ZFC called NBG ... > are not sets are called proper classes). ... type of ordinals would be an ordinal is called the Burali-Forti paradox ...
    (sci.math)
  • Re: proper classes in ZF
    ... 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. ... The pure formulas of MK correspond to the formulas of ZFC. ...
    (sci.math)
  • Re: The collection of all sets and proper classes.
    ... hmmm...., such an absolute statement. ... and NBG is consevative over ZFC, while Kelley-Morse is stronger than ... ZFC, and in both of these there is a proper class of all sets, without ... the collection of all sets and proper classes well defined ...
    (sci.math)
  • Re: Cardinality and Proper classes.
    ... NBG does, which can be thought of as the equivalent of ZFC ... without the proper classes. ... The statement "Every nonempty class of ordinals has a least element" becomes the scheme, for any formula fwith one free variable, ...
    (sci.math)
  • Re: proper classes in ZF
    ... be translated as proper classes in NBG, ... of 'translate a proper class to a formula' and what do you mean by ... that cannot be reduced to open formulas in ZFC and the alike theories. ...
    (sci.math)