i think the question of foundations needs to be considered together with the meta-question: why working mathematicians don't care for foundations? a trivial part of the answer is that it's a matter of taste: some people organize their diet following the pyramid of "so much fruit so much vegetables so much meat", other people smoke and drink coffee and eat chocolate. the less trivial part of the answer is that the world of working mathematics is not built on top of a static foundation. the questions and the meta-questions are asked together. categories are foundations of categories. russell's paradox and hilbert's idea that math should have a static foundation are old. a lot has happened. sets are not so rigid any more. starting from models of untyped lambda calculus, people built all kinds of reflective universes, even containing small complete categories. the category of small categories can probably be a small 2-category in such a universe. the set of all sets can hardly be a set because of the variance, but i think that the set of all sets of sets can be a set in some models. my 2p, -- dusko Vaughan Pratt wrote:
Encouraged by the lack of objections to my previous message about why Russell's Paradox should not be a big deal, I had a shot at shrinking the position I spelled out there down to one paragraph, as follows.
------------ We shall axiomatize certain 1-categories using 2-categories. We avoid Russell's paradox by treating any aggregation of $n$-categories as an $(n+1)$-category, and allowing for the possibility that the $(n+1)$-category $n$-$\CAT$ of all $n$-categories might be exponentially larger than any of its members. We impose no other size constraints besides the obvious one of keeping things small enough to remain consistent. Sets are defined as usual as 0-categories and categories as 1-categories. ------------
While I'm happy to field objections like "too flippant", I'm more concerned as to whether there are any technical flaws, and to a lesser extent philosophical or religious concerns. (I would not want to be held responsible for guns being brought to the next UACT meeting if ever there is one.)
Makkai and Pare address the same issue in AMS CM 104 (Accessible Categories) with a hierarchy of Grothendieck universes (three, since they like me stop at 2-categories for the application at hand).
Now the Grothendieck hierarchy is stepped through via ZF rather than Z, with Fraenkel's Replacement axiom doing the heavy hitting. This creates gaps mind-bogglingly larger than my teensy exponential gaps above. The general idea seems to be that these gaps ought to be large enough to take care of Russell while still not running headlong into inconsistency. However gaps this large do entail a certain amount of finger-crossing, and one might question the logic of hitting Russell with a nuclear weapon that might send some fallout your way when a harmless little tack-hammer will take him out.
One objection I can readily imagine to the above is that I've conflated the n-category hierarchy with Russell's proposal for a ramified types hierarchy. I would disagree with that. All I have done is to insist on two things that seem to me to be independent.
1. I have proposed to call aggregations of n-categories (n+1)-categories. Now morphisms between n-categories are n-functors, and where there are n-functors there are n-natural transformations, so this is hardly a bold proposal.
2. *Some* gap is needed between n-CAT and (n+1)-CAT, starting with the requirement that Set be bigger than any set. Russell's paradox is no respecter of n, applying just as effectively to an (n+1)-category of n-categories as it does to a 1-category of sets.
Certainly I have juxtaposed 1 and 2, but that is not the same thing as conflating them. Their mere juxtaposition provides sufficient armor against both Russell's paradox and the Icarus risk of flying too close to an inconsistently large cardinal.
The "prior art" for dealing with these issues has given rise to the adjectives "small", "large," "superlarge", etc. and the nouns "set" and "class." A good test for any revolution is the amount of blood it needs to shed. The following definitions are aimed at minimal upheaval through maximum compatibility with the status quo.
* An object is n-small when it belongs to an n-category.
* Small = 1-small, large = 2-small, superlarge = 3-small, etc.
* A set is a discrete 1-category.
* A class is a discrete n-category for unspecified n.
Hopefully Sol Feferman will give an even simpler solution in his talk tomorrow.
Vaughan Pratt