Dear Neil, This is not an answer to your question, so please ignore it if you're not so interested in these broader issues. My broad question is this. When you look at making set theory more categorical, are you just looking for a categorical way to do essentially the same thing, or are you trying more deeply to expose possible limitations of set theory? One thing shared by ETCS and ZFC is the well-pointedness: that the object is determined by its global elements (morphisms from 1). That can seem obvious if what you're trying to capture is some idea of collection, but in fact it breaks down when the collection has topological structure. The cohesion between points goes beyond what can be explained in terms of the global points themselves, and in point-free topology we see non-trivial spaces with no global points at all. This is not necessarily a pathology of point-free topology but can be related to topological facts such as the existence of principal bundles with no continuous global sections. It also feeds back into "sets" as discrete spaces, with non-well-pointed toposes of sheaves (= local homeomorphisms = fibrewise discrete bundles). In these terms, many questions about cardinalities - such as that of the real line - become artifacts of the way in which you strip off the topology to get a set of global points. Abandoning well-pointedness also allows us to be more relaxed about whether we need to use principles such as choice to assert the existence of elements that we know we can never construct. Regards, Steve Vickers p.s. -
2. How would you state that {{}} and {\beth_\omega} are very different objects?
I would say the answer is that you find an object X where {} and \beth_\omega are very different elements, and treat {{}} and {\beth_\omega} as subobjects of X. In other words, you elucidate their different structures by working over X.
On 24 Nov 2017, at 22:36, bartonna@gmail.com wrote:
Dear All,
I'm very interested in how categorial and material set theories interact, and in particular the advantages of each.
It's well-known that categorial viewpoints are good for isolating schematic structural relationships. We can look at sets through this lens, by considering a categorial set theory like ETCS (possibly augmented, e.g. with replacement). A remark one sometimes finds is that once you have defined membership via arrows from terminal objects, you could use ETCS for all the purposes to which ZFC is normally put.
My question is the following:
(Q) To what extent can you ``do almost the same work'' with a categorial set theory like ETCS vs. a material set theory like ZFC?
Just to give a bit more detail concerning what I'm thinking of: Something material set theory is reasonably good at is building models (say to analyse relative consistency), or comparing cardinality. However, there's no denying that for representing abstract relationships the language is somewhat clunky, since the same abstract schematic type can be multiply instantiated by structures with very different set-theoretic properties. So, to what extent can a categorial set theory like ETCS supply the good bits of the fineness of grain associated with material set theories, whilst modding out the `noise'?
For example, the following are easily stated in material set theory:
1. \aleph_17 is an accessible cardinal.
In material set theory, it's easy to define the aleph function and then state that the 17th position in this function can be reached by iterating powerset and replacement. But I wouldn't even know how to talk about specific sets of different cardinalities categorially. I suppose you could say something in terms of isomorphism between subobjects, and then exponentials, but it's quite unclear to me how the specifcs would go. Is that an easily claim to state (and prove) in ETCS?
2. How would you state that {{}} and {\beth_\omega} are very different objects? Set-theoretically, these look very different (just consider their transitive closures, for instance). But category-theoretically they should look the same---since they are both singletons they are isomorphic. So is this a case where their different set-theoretic propeties are considered just `noise', or where ETCS just wouldn't see a relationship, or where ETCS can in fact see some of these properties (and I'm just missing something)?
3. How would ETCS deal with model theory and cardinality ascriptions? (This links to a question asked earlier on this mailing list concerning syntactic theories in category theory, and whether from the categorial viewpoint we should be taking notice of them at all.) For instance, it's an interesting theorem (for characterising structure) that a first-order theory categorical in one uncountable power is categorical in every uncountable power (Morley's Theorem). But I have no idea how one might formalise this in something like ETCS---I know of Makkai and Reyes textbook (which I am currently reading) on categorial logic (where theories are represented by categories and models by functors), but I don't see how you could get categoricity-in-power claims out of the set up there. Can this be done?
Any help and/or discussion would be greatly appreciated!
Best Wishes,
Neil
-- Dr. Neil Barton Postdoctoral Research Fellow Kurt Gödel Research Center for Mathematical Logic University of Vienna Web: https://neilbarton.net/
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