FOM: the cumulative hierarchy versus topos theory

[Peter Aczel (Guest Valentini)]

The cumulative hierarchy versus topos theory
--------------------------------------------

There seems to be a great deal of difficulty in having a good
discussion on foundations between set theorists (and other logicians
who are persuaded to some extent by the cumulative hierarchy picture) 
and category theorists (who believe that the notion of topos is
important for foundations).

I do not feel myself to be entirely located on one side or the other
of the discussion.  But I would like to understand the issues better.
So here are my thoughts, written rather hastily.  I try to describe
the discussion, as it appears to me.  I may have got it seriously
wrong.  I am aware that there are aspects that I have not covered
properly.

Let me call the two sides of the discussion SET and TOP.

SET uses the picture of the cumulative hierarchy to justify a focus on
something like the ZFC axiom system, possibly augmented with large
cardinal axioms.  There is the feeling that there is something
categorical about the picture, which leads to thinking of it as giving
us the standard model of say ZFC.  Over the last 30 years or so there
has developed a good theory of models of ZFC and this has been a very
useful tool in set theory.  But, alongside the mathematical focus on
this theory of models of ZFC there has continued to be a foundational
interest in the standard model.  So for example the problem of CH is
whether or not it is true in the standard model given by the
cumulative hierarchy.

SET find the cumulative hierarchy picture very appealing.  By contrast
TOP finds it ugly and irrelevent to core mathematics.  Moreover the
model theory of ZFC is too non-algebraic.  It was discovered that by
changing the picture suitably one gets a very good algebraic notion,
that of an elementary topos, with many examples coming from core
mathematics. 

The changes to the axiomatisation and picture that are needed are (i)
the weakening of the logic to intuitionistic logic, (ii) staying at
something like Zermelo set theory (without infinity) rather than
jumping to ZFC, and (iii) focusing on the category of sets rather than
the membership structure of limit ordinal segments of the cumulative
hierarchy.

What is common to both sides of the discussion is an interest in the
study of the models of an axiom system (and variants of it) that can
play the role of a framework within which an interesting and
significant amount of mathematics can be represented.  On both sides
one is interested in the interplay between doing mathematics inside a
model and looking at what one gets from outside the model.  Also one
is interested in constructions which can construct a new model from an
old one, the new one being interestingly related to the old one.  

A difference between SET and TOP comes when SET considers the standard
model.  This has some special status, at least on a first
consideration.  It is the starting model out of which all other models
must arise.  But TOP seems only to be concerned to consider a relative
form of this situation.  Start with any topos and treat it as the base
topos for the construction of other topoi over it.  And this very
activity, which is of course a piece of mathematics, can be considered
as taking place inside a topos, any base topos.  It seems that for TOP
there need be no starting model, such as the standard model that SET
uses.  

Traditional set theoretic foundations lays stress on the standard
model of set theory.  As it appears to me, category theorists who
consider the notion of topos of foundational importance, seem to have
no need for a standard topos.  It would appear that TOP has a
different concept of foundations than the one modelled on the
traditional set theoretic foundations.  

I do not recall having seen a clear statement of the sense of
foundations in which topos theory is providing a foundations.  Here is
my attempt:

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
The notion of elementary topos (with natural numbers object?) is a
very good algebraic notion of a mathematical structure, with many
examples in core mathematics, which is such that a significant body of
mathematics can be relativised to each such structure.  For that
reason alone it is of foundational interest.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

On this conception of foundations the notion of model of ZFC is not so
good, because of its lack of a good algebraic formulation and lack of
examples coming from core mathematics.

I have tried to not get involved in taking sides in this debate, but
only want to try to clarify the issues.  My interest here is to ask:

Is the above attempt at characterising the sense of foundations, in
which topos theory has been considered of foundational interest, along
the right lines?

I would hope to try to express my own views another time.  Briefly,
here, I think that the notion of topos is in some sense the right kind
of thing for a good foundations, but is not actually quite right, for
more than one reason (it is too impredicative and it should probably
be a theory of classes, not of sets and ...) Also there should be a
standard model, based on something like Bishop style constructive
mathematics.

I only started reading the fom mail in late December.  I have found
some of it highly stimulating and all of it impossibly time consuming
- something like a very tempting drug that should probably be
resisted.

Anyway I am away on a trip for the next week or so and no doubt will
have severe withdrawal symptoms.

Peter Aczel