Timeline for Ultrainfinitism, or a step beyond the transfinite
Current License: CC BY-SA 3.0
8 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 4, 2012 at 23:29 | comment | added | David Roberts♦ | Or how about this for a question: is it legitimate to ask 'is a = b?' where a and b are sets of different universes? If you don't have a global equality predicate like this, then one has to consider that these things form themselves into categories as I described above. | |
| Jul 4, 2012 at 18:11 | comment | added | Mirco A. Mannucci | they has to go outside ZFC and classical logic, to find "sets" which are iso with their function space. Now, I suspect that higher category theory will also become unmanageable from the point of view of ZF + large cardinals, but I am not sure it is there yet. | |
| Jul 4, 2012 at 18:06 | comment | added | Mirco A. Mannucci | I could not agree more on your last assertion. To me ALL thosecategories are indeed "universes of discouse", no more no less than Set (in fact, it is enough to think of quantum mechanics, where 'sets" are Hilbert spaces and subspaces thereof). But I reiterate my point: what is at stake here is -Is ZFC + large cardinals enough?- Most set theorists (see Joel's answer) would argue yes, and to be sure, fact is that so far there is some evidence they are right. Now, there is also some evidence they are wrong. For instance, when folks tried to find natural models of the untyped lambda calculus | |
| Jul 2, 2012 at 11:06 | comment | added | David Roberts♦ | ... internal to some background category of sets, just as one can talk about small categories. But in practice, you don't go around assuming that the collection of all vector spaces forms a set (unless you want to use Grothendieck universes to avoid size issues). There really are as many (or more) vector spaces than sets. And then there is the categories of groups, of modules, algebras, fields, ... all of which are just as big as $Set$. So each of these can be considered as a sort of 'universe of discourse' on par with $Set$, and there are many more of these than plain old universes of sets... | |
| Jul 2, 2012 at 11:01 | comment | added | David Roberts♦ | Let me put a question back to you, then. Given the collection of all universes, what sort of structure does this form? Or consider the collection of all proper classes which satisfy the axioms of your favourite set theory (let us say not NBG here). Is it legitimate to have a predicate $V_1 = V_2$ where $V_i$ is a universe? One can pretend that really one is working inside a large cardinal, and that all these proper classes are really just inaccessibles, but that is really not how set theorists think of the universe. Indeed it is possible to talk about models of higher categories ... | |
| Jul 2, 2012 at 10:08 | comment | added | Mirco A. Mannucci | nice answer David! Need some time to process it. The chief question though is this: can higher order categories be simulated (ie modeled ) within ZFC + some huge cardinal principle? If yes, they are within the scope of Joel's multiverse, as it stands now. If they cannot, then they could be the answer I am looking for. I have so say, though, that I tend to think they can. | |
| Jul 2, 2012 at 6:05 | history | answered | David Roberts♦ | CC BY-SA 3.0 |