FOM: the cumulative hierarchy versus topos theory

[Colin McLarty]

Peter Aczel wrote:

>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).

        Yes. I was very happy to see this contribution.

>Let me call the two sides of the discussion SET and TOP.
>
> ........ passages deleted.........
>
>SET find the cumulative hierarchy picture very appealing.  By contrast
>TOP finds it ugly and irrelevent to core mathematics.

        It can serve as a foundation so it is "relevant" in a way. But I do
say its details are unnecessary for a foundation of mathematics. And I won't
call it quite "ugly". I prefer to do without it. Mac Lane likes to quote
Weyl saying the cumulative heirarchy is "too full of sand".    

>  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.

        I do think this is a nice feature of topos theory.
 
>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.

        This is true at the start. If you like, you can add assumptions to
restore any or all of classical logic, infinity, replacement, large
cardinals, whatever. But the point of topos foundations is that there are
good reasons for ALSO having foundations without these features of classical
set theory.

>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.  

        It seems to me that from a strict fom perspective we start from
axioms, not models. But I believe you are right, both sides are interested
in the things you describe.

>
>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. 

        I find that a fair description. Other categorists, such as Jim Lambek,
think there is a standard topos to start from, the "free topos" (see Lambek
and Scott INTRODUCTION TO HIGHER ORDER CATEGORICAL LOGIC). This is close to
what Peter calls for when he says

>Also there should be a
>standard model, based on something like Bishop style constructive
>mathematics. 

Unfortunately I cannot discuss it very sympathetically, it does not appeal
to me.

>
>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.

        This crystallizes my main concern with Peter's post. Peter defined
TOP as people "who believe that the notion of topos is important for
foundations". And indeed some of them just want foundations in toposes. But
more of us also think other categories have foundational roles. Certainly I
do. And, while there are toposes in core mathematics, there are many more of
other kinds of categories.

        I very much like Peter's distinction between "how much mathematics
can be relativized to" a given structure, and "how many examples there are
in core mathematics" of that structure. Well put. This is what I meant in an
earlier post when I said ZF has only a foundational role: You can relativize
much mathematics to it, but few models of it arise in core mathematics. 


>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

        I am sure that having some alternatives to SET and TOP will also
help SET and TOP communicate better.

Colin