Strong chain conditions and preservation of choice principles
There are 2 comments on this post.I recently returned from a wonderful week in Italy, where I attended the Young Set Theorist 2013 conference. I met a lot of new people, some old acquaintances, baffled people with oversized pickles, and most importantly shared and learned some great ideas.
One of the nicer things I'd done was to work with Thomas Johnstone on some preservation theorem related to forcing and choice principles (see also this announcement by Victoria Gitman). In order to clean up a bit the proof, I'll introduce a new definition which is going to slightly extend the ideas originally discussed in Italy. So without further jibber jabber, let's talk mathematics.
Canonical names. Given a forcing
This notation makes it slightly simpler to talk about names which are generated by collecting some names in the ground model. Next we discuss antichains. Recall that in the lack of choice it is consistent that there is a forcing extension which does not have maximal antichains, or that not all antichains can be extended to maximal antichains. It could also be the case that no antichain is well-ordered. In those cases it is possible to have the usual
Definition. We say that a notion of forcing
In particular every strong
Theorem. Letbe a model of . If is a notion of forcing which has strong .c.c, and is -generic, then .
Proof. Let
Since we have that
This was an observation made by both myself and Thomas in the case where
Theorem. (Johnstone) Letbe a model of . If is a -strategically closed forcing and is -generic, then .
There are 2 comments on this post.
(Jun 28 2013, 20:59)
Hi Asaf, Question: Is it known that there are posets that have the strong -cc but are not well-orderable?
I believe that the result about preserving extends to preserving because we use the well-orderability of the poset for mixing, which I guess we can also do with strong -cc. Also, I don't think we have the preservation of for -Baire posets, but only for -strategically closed posets. We actually don't know whether -Baire suffices. Do you think it does? I sent Tom the link to your post.
(Jun 28 2013, 22:51 In reply to Victoria Gitman)
Hi Victoria,
I have to admit that I have nothing of the top of my head, but I also feel that there is no reason for strong .c.c to imply well-orderability. I am certain that one could concoct something up using all sort of strange pathologies. I'll think about it some more, I'm sure that if I get my brain to overdrive I'll be able to find such example.
As for the second issue, I think that you're right. He did show me the proof for strategically closed forcings, which explains why I had such a difficult time to finish the proof for the Baire case. I feel suddenly very confused about my memory (and unfortunately all I had was the drafts we worked on, which mostly included the Baire case, rather than the strategically closed case). I'll correct this in this post, and perhaps for the next post I'll actually show more than that! (And thank you for sending the link to Tom.)