From: Steve Vickers <sjv@doc.ic.ac.uk> 1) The "element of" relation is absolutely _un_fundamental - this is part of the force of Freyd's example about simple groups. What are the elements of a real number? Yes, when you need to kick a set theorist, the real number line is a particularly sensitive part. Excellent kick. Ouch. In reality there is no single universal "element of" relation that describes the nature of everything, including the reals; instead, the reals are described by various relations with other specific objects. I'm with you 100% philosophically on this. We move and rest, and need vocabulary for both activities. Sets are great for lolling around in (I wonder how Cantor was at sports in school), but terrible for zipping perfectly smoothly along the real line. Categories give a much smoother ride than sets because they're *built* to. But they also have stationary parts, that show up in two places. (i) The objects. This is where one rests between morphisms. Having your objects form a set is a Good Thing, it fits their punctual style: "Be there on the dot" says the set. (ii) Homsets. Sets are dual to their elements. This seems to be because when dealing with powersets you have a higher probability of meeting the contravariant power set functor herself than one of her covariant brothers. (Not a theorem, just a feeling.) Homsets are no exception. When you pass from moving along one morphism at a time in a category to contemplating the morphisms lying there in profusion in the homsets, you dualize your viewpoint and see a morphism as merely a punctual dot of a homset. This explains the paradox of the very unpunctual morphisms able to make themselves the punctual members of a homset, we just view them dually "from the other direction" (not one parallel to the morphisms though). I agree that it is a type error to try to understand a smooth line as made of points when it is just trying to do its job and get you from point A to point B uninterrupted (poor gap, I interrupted it). Interruptions are incompatible with smooth travel. But travel with no rest at all is pretty grim and hard to sustain when you're trying to get some real work done, whence line *segments* rather than entire lines for category theory. Hell is having to move forever along the real line! But it is equally a type error to try to turn your rest stops into motion. You *need* to rest from time to time. For one thing it gives your maintenance department a turn at doing stuff, maintenance has to shut down when you're on the move, on-the-fly maintenance is *much* harder. And if you're the type of person that has opponents, it's all in the game to give them a turn. But if I'm with you 100% philosophically on this example, I'm only with you about 50% mathematically. It seems to me that set theory supports the go part of stop-and-go traffic *pretty* well. Not perfectly, as we agree. But when you consider how much of the continuum set theory *is* able to comprehend, I'm not terribly sold on the inadequacy of sets for modeling go almost as well as stop. 2) Topology: Normally one thinks of open sets as being sets of points, but localic topology views points as being sets of opens (e.g. reals as neighbourhood filters above). There is obviously a fundamental relation of points being "in" opens, and localic arguments can be expressed quite reasonably using it. Set theoretic expression using "element of" completely obscures this. In other words, set theory prevents you from adequately expressing reasonable arguments. I'm with you <10% on this one. I can't tell what limitation of Set you're talking about here, but unlike your philosophically excellent 1) it doesn't seem to match up to any of the well-known limitations. It sounds like you're saying that set theory doesn't let you talk about the converse of the membership relation. That's certainly not the case. Or maybe you mean that Set\op is not a concrete category. What's wrong with its concrete representation as CABA? If that seems too complicated, how about its coconcrete representation as the category of sets and their *converse* functions (binary relations whose converse is a function)? Maybe you're saying that Hom:Set\op*Set -> Set is outside set theory. That too is implausible. None of these issues hold the sort of terrors for set theorists that the continuum does. At least not a terror that has them terribly bothered mathematically the way the continuum problem does. 3) Generic objects: 0%, needle wrapped around the post. You seem to be saying that set theorists are scratching their heads over what a variable is while the category theorists have it all sorted out. News to set theorists. You can't kick a set theorist in the variables, they're very well protected there. 4) What about theories such as that of accessible categories, that, for set theoretic reasons, have to be liberally sprinkled with infinite cardinals? Doesn't this make you think that perhaps set theory is somehow obscuring simple ideas? I don't know how to define "accessible category" without bringing cardinals into the picture, but maybe I'm the last one to find out how, as usual. What's the trick? To my mind, the evidence suggests that despite its undoubted successes, set theory is not right for mathematical foundations, and we should be looking for its replacement. Whoa, needle went negative there. But I would have the same reaction to a set theorist who claimed that categories were nothing but an alternative language for the mathematics ordinarily and satisfactorily treated by set theory. ("Alternative language for" is the polite code word some people use for "weird way of talking about") Set theory itself will only go away when the natural numbers go away. Let's get real here. The natural numbers are the very oxygen of mathematics, and they are not going away in *any* foreseeable future. Hence neither is the (internally) bicomplete topos of finite sets. This topos is a very nice concrete way of working with natural numbers---it makes numbers *more* categorical, not less, by letting you transform them. But sets *do* have to transform via arbitrary functions, there's no way to wriggle out of *that* one! Transforming sets with binary relations neuters them, neutered sets are only good for the side lines. *Converse* functions on the other hand are fine, if you're not Bill Thurston, who eloquently expressed his inability to relate to Set\op at UACT. I don't care how the infinite sets are organized, just so long as there's a way of doing it that doesn't make the logic that bears on my life inconsistent. I'm not planning on rubbing up against any infinite sets personally without a sturdy layer of math and logic between me and them. You can seriously injure yourself with an infinite set. Mathematics founded on set theory stays close to the air supply. Any religious upstart of a foundations claiming to offer a viable *replacement* for set theory is going to have to argue real hard about the disadvantages of breathing! Whether we should work with sets and categories in exactly equal proportions is an interesting question. I tend to be slightly more settish than cattish in my thinking, if only because I'm lazy and sit around a lot, on the dime if not on the dot. But there's not a big gap. Categories: can't live with sets, can't live without them. Sets: can't live with categories, can't live without them. Vaughan Pratt PS. No reactions yet to my ordinal-based definition of Set. Main thing I want to know is, is it old hat? Second thing, is it good for anything else besides what I made it up for, namely to shorten the passage from function composition in Set to the membership relation on ob(Set)? I don't claim any more *significance* to the membership relation in my version of Set than Mike Barr et al were claiming for other versions of Set. Well, maybe a little more: it is undeniably the membership relation for ordinals, which is all my Set claims to contain *explicitly*. But is pi essentially an ordinal? Of course not. Can pi exist in my Set? No more or less than in anyone else's Set. Is pi made any more real by installing it in a category? Rubbish.