[FOM] Sets and Proper Classes

[Dmytro Taranovsky]

The purpose of this posting is to correct some misunderstandings about
the term "proper class".

A set is an arbitrary collection of objects.  A property of sets is a
formula with parameters and one free variable such that a set either
satisfies the formula or it does not.  It is convenient to view
properties as collections, where x in {y: phi(y)} means phi(x) and {x:
phi(x)}={y:psi(y)} means for all x (phi(x) iff psi(x)).  However, there
is no collection of all collections, so for not every formula is there a
collection of all objects that satisfy the formula.  A property of sets
for which there is no set of all sets that satisfy the property is known
as a proper class.

We know that for every set s, there is the set of all subsets of s.  The
same reasoning should lead to that for every collection s there is the
collection of all subcollections of s.  Therefore, if proper classes did
exist as collections, then we could form a superclass of all classes of
ordinals, proceed to form a super-superclass of all sub-superclasses of
the superclass of all classes of ordinals, and so on. In that case, by a
set we meant a set in the initial segment of the cumulative hierarchy,
perhaps, a member of V(kappa) where kappa is the least inaccessible
cardinal such that V is an elementary extension of V(kappa) with respect
to first order formulas about sets, and by a class, we meant a subset of
V(kappa).


Dmytro Taranovsky