FOM: "the static nature of sets"
cxm7 at po.cwru.edu
Wed Feb 4 14:00:25 EST 1998
I like Carsten's post a lot. While I would not say everything just
the same way, I have no objections to make. There is one point where it may
appear he disagrees with me, given some confusions that have appeared on the
list before, and I want to make it clear. This is also a point Bill Lawvere
often insists on.
>In what follows I use "we" to speak about category theorists, and
>"you" to speak about set-theorists. Although I can only speak for me.
>We like "sets" almost as much as you like.
This is true of even the most radical categorical foundationalists,
on the understanding that "sets" means sets in categorical set theory. We
also have a broader usage of "sets" where it includes "variable sets" which
are the objects of any topos. But here let us focus on classical sets, with
Boolean logic, the singleton 1 has just two subsets (the empty one and the
whole), there is no difference between internal and external logic, and so
on. The theory of these sets is naively intertranslatable with Zermelo style
set theory based on the membership relation.
Math textbooks rarely go into axiomatic technicalities. Some
calculus texts go so far as to define a function as a set of ordered
pairs--but very few bother to define an ordered pair as a set! Apart from
such technicalities, categorical set theory works exactly like
membership-based set theory. Mathematicians in general are completely
indifferent to the differences between categorical set theory and ZF or
Now, if you insist on the particulars of ZF, many category theorists
will object. So will Feferman, as he already has on the list and in
publications. He thinks a somewhat different theory is better--indeed a
theory that gives more attention to functions, though it is based on set
membership. We category theorists begin with functions and get by entirely
without a "global membership" relation--a relation making it meaningful to
ask of any two sets X and Y whether X is a member of Y.
I like sets "almost as much as you", but not quite. I also explore
independent, "set-free", foundations for many parts of math.
The NYT obituary for Samuel Eilenberg yesterday (in the Final
edition yesterday) quoted Columbia Prof. John W. Morgan, saying "The theme
that runs through Sammy's mathematics is always to find the absolutely
essential ingredients in any problem and work only with those ingredients
and nothing else -- in other words, to get rid of all the superfluous
I don't know what Sammy thought of categorical foundations (let
alone Prof. Morgan). I guess he wasn't much interested in them or I would
have heard more. Anyway, I like this attitude not only in working math but
in foundations. I want to know what minimal apparatus I can use to organize
differential geometry for example. And I find (by reading Lawvere and
others--I have worked in this but hardly invented it) that the topos axioms
plus a simple assumption about the line gives me a good start. To get more,
I need more assumptions.
Two uncontroversial claims: I like this kind of research and
consider it foundational. Nobody yet knows any other framework for this as
flexible and as close to practice as category theory.
Sets are great. But now, what kind of axioms should we use for
transfinite set theory? There we disagree.
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