Timeline for What is the general opinion on the Generalized Continuum Hypothesis?
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12 events
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| Oct 19, 2017 at 3:14 | comment | added | David Roberts♦ | @goblin a topos is meant to have sets for homsets <-- toposes are described by some first-order theory, and don't have to rely on any kind of ambient sets. They of course are cartesian closed, so have an "internal" homsets, but the "real" collection of arrows $a\to b$ doesn't have to be anything like a set... | |
| Jun 27, 2015 at 17:19 | comment | added | goblin GONE | The problem with this viewpoint in my opinion, Tom, is that a topos is meant to have sets for homsets, and those sets are meant to live in a universe of sets, and that universe either satisfies $\mathrm{GCH}$, or it doesn't. But perhaps this can be gotten around somewhat by considering each topos as canonically self-enriched. | |
| Feb 6, 2010 at 16:54 | comment | added | Mike Shulman | Large cardinals do turn up in accessible categories and even occasionally some other places in topos theory, like whether there can be an inaccessible lex endofunctor of Set. I don't know to what extent these sorts of things happen without AC or even intuitionistically. But I'm not sure whether any of that is very related to GCH. No large cardinal hypotheses have been found that decide GCH one way or the other, have they? | |
| Feb 6, 2010 at 16:52 | comment | added | Mike Shulman | In a general topos I don't think the question of GCH is particularly meaningful. Without the axiom of choice, there's no reason for 2^c to be an aleph, so it doesn't even make sense to ask which aleph it is. I'd be surprised to find many toposes that violate AC but in which GCH is "true" in the sense that P(N) is isomorphic to the set of well-orderings on N. | |
| Feb 6, 2010 at 5:00 | comment | added | Tom Leinster | Yes, the accessible category stuff does seem to reach deep into issues about large cardinals. I don't know much about it. Maybe Mike Shulman will turn up and shed some light. Anyway, I think this is an exception. | |
| Feb 6, 2010 at 4:23 | comment | added | François G. Dorais | I think we misunderstood each other. My use of finite includes all finite cardinals, but I see how that was unclear in my original question. I then understood from your answer that combinatorial species were also used to count infinite objects. Anyway, I have seen infinite cardinals (even large cardinals!) appear in works on accessible categories. I don't know if this is very exceptional or not. | |
| Feb 6, 2010 at 4:13 | comment | added | Tom Leinster | Sorry, François, maybe I misunderstood you. Were you interested in the relevance of counting infinite sets in category theory? | |
| Feb 6, 2010 at 3:10 | comment | added | François G. Dorais | I've only seen combinatorial species used to count finite things; I would love to see them in action in the infinite. | |
| Feb 6, 2010 at 2:46 | comment | added | Tom Leinster | Francois: no, I don't think it would be fair. For instance, the theory of combinatorial species (especes de structures) is all about counting, and it's a fundamentally categorical theory, created by Joyal - even though when combinatorialists use it, the category theory often gets hidden. | |
| Feb 6, 2010 at 2:29 | comment | added | Harry Gindi | Well, Universe successors are important. | |
| Feb 6, 2010 at 2:20 | comment | added | François G. Dorais | Would it be fair to say that counting, beyond large/small/countable/finite, is mostly irrelevant in category theory? | |
| Feb 6, 2010 at 1:46 | history | answered | Tom Leinster | CC BY-SA 2.5 |