Re: Sets vs. categories as a foundation

 : > How exactly one would one even DEFINE "the category of sets" if one
 : > were NOT STARTING with ZFC or some other rich set theory as a
 : > foundation? My point is simply that if you have ZFC, what do you
 : > need categories for?  Sets are already adequate as a foundation; you
 : > can do EVERYthing, INCLUDING categories, AS sets.  "The category of
 : > sets" starts to get viciously circular.  But if you don't have a set
 : > theory, if you are using categories as a foundation instead, then
 : > "the category of sets" is simply nowhere in evidence: how do you
 : > even DEFINE "set"?

[EMAIL PROTECTED] (Jesse F. Hughes) writes:
 : And here, I think George goes too far (or I don't get his point).
 : Some folks want category theory as an alternate foundation, it's true.
 : Others just like it for its unifying qualities and ability to make
 : apparently disparate phenomena particular instances of a common
 : structure.  I don't see why there's any particular issue for talking
 : about the category of sets as a particular category for *either*
 : group.

Well, until individuals assert their claim to represent their respective
camps, anything said by the rest of us risks mis-characterizing their actual
positions.

A core point is that there is more than one way to axiomatize either
of category theory OR set theory.  For some of these issues, it does
actually matter which axiomatization you pick.  Any treatment that
alleges that a category needs to have a set of objects is going to
get accused of begging the question.

There is a simple (in classical FOL with equality) axiom-set for category 
theory in which every element of the domain is an arrow, the objects
are all&only the subclass of arrows that go from themselves to 
themselves, and every arrow has to go from an object to an object.
That treatment identifies each category with a whole model of
the axiom-set, which pushes some of the interesting questions into
your preferred model theory, which might not be category-theoretic.
It might not be set-theoretic either, but then again it might be.
If it is then there is a sense in which you haven't really picked
a side, in which you are simultaneously demanding the necessity of
both foundations.

Ensuring that a category defined by THAT axiom-set grows into a
category of sets basically requires adding enough axioms to restrict 
the models to topoi.  The necessity of that MUCH work IS "an issue
for talking about the category of sets", because intuitively,
sets seem simpler than that -- so MUCH simpler that you could
in fact have defined categories in terms of them.  There are textbooks
that introduce the category of sets (and functions) in the
first chapter, and don't get around to topoi until the last.

On the other side, if you are using set-theoretic foundations,
then one obvious issue in talking about the category of all sets
is that in such axiomatizations of set theory as ZFC, it is too
large to exist at all.  That IS DEFINITELY "an issue in talking
about" it.
-- 
 ---
  "It's difficult ... you need to be united to have any 
   strength, but internal issues have to  be addressed."
 --- E. Ray Lewis, on  liberalism in America