I guess simplicity is in the eye of the beholder. For example, I do not consider the categorical version of either choice (epis split) or well-pointed (1 is a generator) to be translations of set theory, but perfectly natural categorical axioms. The point is that Harvey is a set-theorist, so he thinks comprehension and all that stuff (which at least 95% of all mathematicians could not state properly if their lives depended on it) is perfectly natural and I don't. But my actual criticism of ZF(C) is much simpler. I have taught these courses in set theory and we spend a lot of time developing these epsilon trees and then totally ignore the structure. In other words, the epsilon tree structure of sets is totally irrelevant to what you do with them. There are a number of definitions of pairs, but they are irrelevant. The only thing we need to know about pairs (and the only thing a categoriest does know) is when two pairs are equal. All the defintions of pairs have that property of course, but they also have irrelevant properties. Mike

I remember to have seen a paper on frameworks for measuring complexity of mathematical concepts by two autors earlier. (I don't exactly recall where) but I also remeber that one of the authors was H.Friedman! P.S.Subramanian Tata Institute.

How does Harvey Friedman know that the formulation of real analysis as carried out in set theory will do all that real analysis *should* do? My favourite example is that of partial functions. Most teachers of real analysis (calculus) rightly impress on students the importance of the domain of a function, and the domain of f+g, etc. So a student might think that the algebra and analysis of partial functions would occupy a good part of the literature. Solutions of ODEs (such as dy/dx=exp(-y) ) are often given by partial functions with domain involving a parameter, and the solution (including its domain) seems to vary smoothly with this parameter. In fact there is very little in the literature on such matters. I had a small go with 29. (with A.M. ABD-ALLAH), ``A compact-open topology on partial maps with open domain'', {\em J. London Math Soc.} (2) 21 (1980) 480-486. It is not clear that the most general case of the functional analysis of partial functions with domain neither open nor closed can be successfully handled within classical set theory. There is a chance it can be handled within topos theory. (Try functions defined on the leaves of foliations. Any answers?) Another point of topos theory is to handle categories such as that of directed graphs in a similar manner to the category of sets, and to make comparisons between such categories. (Bill Lawvere has of course written a lot on this.) Ronnie Brown

According to Harvey Friedman:

I challenge anyone to write down their favorite fully formal axioms for topoi that are sufficient to do real analysis, so we can compare them side by side with the fully formal axioms I write down here.

The papers I announced here the other day do not offer any such "dramatically simple" or "tremendously powerful" axioms, comparable with Friedman's, nor indeed anything as politically engaged --- but I think they may have to do with the issue. Given a field in any category with enough limits, we implement analytic functions, solve differential equations --- do quite a bit of real analysis. It turns out that most of it can be captured as guarded induction on final coalgebras. We did not try to use this to embed real analysis in any fully formal foundational theory, but to provide a uniform way of implementing it on a computer, together with infinitary constructions and all. The result may not be as "amazingly simple" as Friedman's axioms, but it is fairly simple, and conceptually clear. Check it out (the last two papers at http://www.cogs.susx.ac.uk/users/duskop/). Sorry for the plug, but I actually want to make a point. I don't think category theory should spend time trying to beat set theory on its own territory of fully formal systems. Sets were invented in the time of doubt about the consistency of mathematics, when foundations were really sought for. Nowadays people go and prove Fermat Theorem. Set theory wants to be something like a formal grammar of mathematics. That is fine, can be very interesting in itself, but great stories are usually told without thinking of grammar. Category theory might perhaps do better to try to be some kind of a programming language or mathematics, a set of object-(or better: method-)oriented tools, conceptualizing large "software" projects of the day. How about that? -- Dusko Pavlovic PS On the other hand, us toposophers competing against them setologists who can do analysis --- aren't we a bit like A Frenchman and an Englishman making the bet who can faster translate a sentence from German. Little Gretschen comes by and says: (fg)' = f'g + fg'... I mean, didn't analysts tell *everyone* by now: THEY DO NOT CARE FOR FOUNDATIONS. They do not think of their manifolds neither as sets of little elements, nor hanging on a bunch of morphisms among other manifolds. Most of the time, they are quite happy with their manifolds as manifolds. Next time they need foundations, they'll invent them again, like they invented sets and sheaves.

Well the first question has an easy answer. It is just as possible to talk about a category without knowing what a set is as it is to talk about a set without knowing what a set is. Of course, you cannot talk about homsets. It is interesting to read the very first Eilenberg-Mac Lane paper, which did not talk about homsets. A set is an undefined notion and there is a relation, epsilon that may hold between one set and another, subject to certain axioms, one version of which Friedman listed. A category consists of undefined things called arrows and three relations, two functional and the third partially functional (actually better than that, but leave that aside). Friedman's axioms are not coherent, as has been pointed out, while the categorical axioms are. On the other hand, one can state Friedman's axioms, in all their glorious incomprehensibility (I think I could stare at the 8th one from now until the middle of next year without understanding what it says, and the 6th, asserted to be the axiom of infinity is not much clearer) in a couple hundred words, while it is pretty much necessary to interrupt the topos axioms for some definitions (at least monic and subobject) to do the topos axioms. Thus each one looks simpler to its devotees and there is really no point in arguing about it. Michael On Thu, 29 Jan 1998, categories wrote:

Date: Wed, 28 Jan 1998 22:15:27 +0000 From: Carlos Simpson <carlos@picard.ups-tlse.fr>

Being a newcomer to the category list, I have a really naive and stupid question (concerning H. Friedman's challenge). Namely, I was always under the impression that you had to know what a set was before you could talk about what a category was (in particular a topos). Is it possible to talk about toposes without knowing what a set is?

This seems somewhat related to a question that has been bugging me for some time, namely how to talk about a ``category'' which is enhanced over itself, but not necessarily having any functor to or from Sets. The very first part of the structure would be a class of objects O together with a function (x,y)\mapsto H(x,y) from O\times O to O, but I can't get beyond that.

---Carlos Simpson