Dmytro Taranovsky dmytro at mit.edu
Tue Jul 20 15:31:31 EDT 2004

Jesse Alama wrote:
> Some more work would be needed to conclude that most mathematicians 
> really do use classes only as figures of speech.  There is evidence for 
> the contrary position: substantial use of proper classes is made in the 
> study of universal algebra, and it seems that there classes are treated 
> as honest mathematical objects.

I do not know enough about category theory or about minds of mathematicians to
give a definitive answer, but it is clear that mathematicians generally use
proper classes without asking, "Metaphysically speaking, what exactly is a
proper class?"  The view that proper classes are simply large collections of
objects is (in my opinion) naive and generally rejected.  However, there are
three alternatives.

One can treat proper classes as syntactical objects and sets as sets, and work
in a theory conservative over set theory.  For example, NBG includes axiom of
global choice and is conservative over ZFC.

Alternatively, one can consider only sets whose rank is smaller than a certain
ordinal kappa, and treat collections of these sets as classes.  The ordinal
should be sufficiently large to include all objects under consideration, and
V(kappa) should satisfy enough correctness so that relevant statements, if true
in V(kappa), are true in V.

Finally, one can treat classes semantically as descriptions, and for a set x and
a proper class X, treat x \in X as "set x satisfies description X".

In each of these alternatives, however, something more than ZFC is desirable...

Dmytro Taranovsky

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