Timeline for Set-theoretic forcing over sites?
Current License: CC BY-SA 2.5
17 events
| when toggle format | what | by | license | comment | |
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| Jan 22, 2015 at 7:43 | comment | added | Simon Henry | If I remember correctly, "yes", the double negation topology is indeed essentially the only one that gives a boolean topos. More precesely, for a topos T, the only boolean subtoposes of T are obtained by taking the double negation topology on an open subtopos of T. I can't remember where I saw this result, but I think it was somewhere in Johnstone's elephant... | |
| Dec 22, 2010 at 5:24 | vote | accept | Mike Shulman | ||
| Feb 1, 2010 at 4:40 | comment | added | François G. Dorais | @Mike: I wasn't sure how to handle the Blass and Scedrov reference. I struck out the previous statement to make the edit more obvious, but I didn't think about the notification issue. That's a very good point. | |
| Feb 1, 2010 at 2:15 | comment | added | Mike Shulman | BTW, my inclination would be that instead of editing a "no this doesn't exist" answer to change it to "yes it exists in this paper," it would be better to post the later discovered paper as a separate answer. I think edits should be more for minor changes that leave the central point intact. In particular, the asker doesn't get notified when you edit an old answer, so I wouldn't have noticed your added reference to the Blass+Scedrov paper if I hadn't come back here to follow the comment discussion. | |
| Feb 1, 2010 at 2:11 | comment | added | Mike Shulman | The Blass-Scedrov paper looks very interesting, but the introduction suggests to me that it isn't quite what I was looking for. They construct "for almost any category C... a notion of forcing and a topological group of automorphisms of it" representing the double-negation sheaves. But I was thinking rather of a direct construction of a model of set theory from a site, rather than by going through a combination of classical poset-based forcing and symmetric models. | |
| Feb 1, 2010 at 2:06 | comment | added | Mike Shulman | So, from an opposingly biased perspective, one might say that the reason for set-theorists' focus on poset-based forcing is their addiction to the axiom of choice? (-:O | |
| Jan 30, 2010 at 22:51 | comment | added | Joel David Hamkins | Very good answer, François! | |
| Jan 30, 2010 at 22:36 | history | edited | François G. Dorais | CC BY-SA 2.5 |
precision
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| Jan 30, 2010 at 22:18 | history | edited | François G. Dorais | CC BY-SA 2.5 |
addendum
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| Jan 30, 2010 at 22:02 | comment | added | François G. Dorais | @Joel: I think you're right. Grigorieff's results still apply to general sites. | |
| Jan 30, 2010 at 22:00 | history | edited | François G. Dorais | CC BY-SA 2.5 |
correction
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| Jan 30, 2010 at 19:27 | comment | added | Joel David Hamkins | My point instead is that we are simply less interested in the models that symmetry/permutation give rise to, as by design Ac fails in them. There are such deep applications of forcing, which seem to reveal fundamental aspects of the nature of sets, which don't require us to give up AC or classical logic. (e.g. cardinal invariants, proper forcing axiom, large cardinals and forcing, etc. etc.). | |
| Jan 30, 2010 at 18:58 | comment | added | François G. Dorais | @Joel: There is more to the site approach than just the unification of forcing models and symmetric models, it is a natural and much more general framework which happens to contain both as special cases. To transpose the chicken and the egg, wouldn't it be possible that the fact that there are so few symmetric models compared to forcing models is a result of the division of the two? I think the fact that symmetrizing is commonly viewed as an extra step after forcing makes it less accessible as a tool. | |
| Jan 30, 2010 at 18:16 | comment | added | Joel David Hamkins | Isn't it also true even that the work on symmetric and permutation models is a relatively minor part of the applications of forcing? Indeed, I class the symmetric model/permutation model ideas along with the many methods of describing inner models (which are numerous and highly developed), rather than as particularly connected with forcing. Most of the interest in forcing among set theorists has been in a fully classical, full ZFC context, exploring the ubiquity of ZFC independence. I would guess that more than 95% of the forcing arguments in the literature use the axiom of choice. | |
| Jan 30, 2010 at 18:15 | comment | added | François G. Dorais | @Mike: That's why I added "to a certain extent." It is my impression that the double-negation topology is the only one which is "of general purpose," but I would be very happy to be proven wrong! I very much agree with your last point. | |
| Jan 30, 2010 at 17:54 | comment | added | Mike Shulman | Thanks! I agree that such crossovers are sorely lacking. Is it really true, though, that the double-negation topology is the only one that makes sense for classical set theory? I mean, there are other topologies which still give rise to Boolean topoi, aren't there? Anyway, even double-negation topologies on arbitrary categories are a larger common generalization of those on posets and groups which seem to mostly pervade the set-theoretic literature. | |
| Jan 30, 2010 at 17:44 | history | answered | François G. Dorais | CC BY-SA 2.5 |