Timeline for What goes wrong in Easton forcing if we don't just use regular cardinals?
Current License: CC BY-SA 3.0
20 events
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| Jun 21 at 0:10 | answer | added | Arvid Samuelsson | timeline score: 3 | |
| Nov 20, 2012 at 19:55 | comment | added | David Roberts♦ | Aha, a straight answer at last. This is what I meant when I mentioned tameness. | |
| Nov 20, 2012 at 14:24 | comment | added | François G. Dorais | The basic problem is that if you add a subset of $\kappa$ without being very careful then you will probably add a subset of $\mathrm{cf}(\kappa)$ too. If you do that at every $\kappa$ then every power set will blow up to a proper class. One solution is to add sets only at regular cardinals and then lift sets from $\mathrm{cf}(\kappa)$ to $\kappa$ when you need to. | |
| Nov 20, 2012 at 11:28 | comment | added | David Roberts♦ | I'm not saying it doesn't work, I'm just curious to know if I can do it. The poset of conditions I was after needed to have a certain structure, and by imitating Easton but removing the restriction to regular cardinals seemed to be the easiest way to do it. Also I'm not intending to do this forcing in ZFC, but with sheaves :-) | |
| Nov 20, 2012 at 9:12 | comment | added | François G. Dorais | I still don't see where you're going. Easton forcing still adds subsets to singular cardinals. For example, if $x$ is a new subset of $\omega$ then $\lbrace\omega_n:n\in x\rbrace$ is a new subset of $\omega_\omega$. Why doesn't this work for you? | |
| Nov 20, 2012 at 7:37 | comment | added | David Roberts♦ | Ah, so you can do the forcing, but it's just that it will have no effect at limit stages. I think this is the answer to my question. | |
| Nov 20, 2012 at 6:50 | comment | added | Asaf Karagila♦ | Sure. But just note how these sets are defined. Unions at limit stages. You don't add sets, you just accumulate what you already created. | |
| Nov 20, 2012 at 5:57 | comment | added | David Roberts♦ | Ah, that's interesting. You know my email, could you elaborate? | |
| Nov 20, 2012 at 5:50 | comment | added | Asaf Karagila♦ | So you want to add new sets for every $V_\alpha$? Note that if $\alpha$ is limit then no new sets are added at $V_\alpha$. | |
| Nov 20, 2012 at 4:11 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
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| Nov 20, 2012 at 1:14 | comment | added | François G. Dorais | I think your statement that Friedman's examples are all variants of Easton forcing is not quite accurate. Nevertheless, if you want to add set-many sets at each rank then the first thing to try is iterating with Easton or reverse Easton support depending on what you're trying to do. If these both fail then you can try something else... | |
| Nov 18, 2012 at 23:59 | comment | added | David Roberts♦ | -> "Planning to add set-many sets at each rank, for all ranks" | |
| Nov 18, 2012 at 19:57 | comment | added | David Roberts♦ | @JDH not necessarily in ZF :-) @Andreas that's good to know. I was planning to add sets at all ranks, so that's one objection out of the way. | |
| Nov 18, 2012 at 13:33 | comment | added | Andreas Blass | A Jech-Sochor theorem for a proper class of atoms will need some additional nontrivial structure on the atoms. The point is that the pure sets that play, in the ZF model, the role of the atoms can't all have the same rank. So the set of these surrogate atoms comes with a certain structure, namely (at least) a partition according to rank. If the original set of atoms didn't have this structure, then you won't get the sort of transfer theorem that Jech and Sochor got. | |
| Nov 18, 2012 at 10:48 | comment | added | Joel David Hamkins | David, given your purpose and since there are a proper class of regular cardinals, why does it matter to do something at singular cardinals? For example, you can view Easton's iteration as adding a subset to $\kappa^+$ for every infinite cardinal $\kappa$, including singular $\kappa$, since all $\kappa^+$ are regular. | |
| Nov 18, 2012 at 7:45 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
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| Nov 18, 2012 at 7:14 | comment | added | David Roberts♦ | Yes, it's to do a class-of-atoms version of Jech-Sochor. All I need in the end is a model of ZF. | |
| Nov 18, 2012 at 5:28 | comment | added | François G. Dorais | Presumably, you still want to add a proper class of sets with a purpose? | |
| Nov 18, 2012 at 1:51 | answer | added | Andreas Blass | timeline score: 12 | |
| Nov 18, 2012 at 1:25 | history | asked | David Roberts♦ | CC BY-SA 3.0 |