Timeline for symmetric models and HOD
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2017 at 21:40 | comment | added | Asaf Karagila♦ | The forcing which adds a generic bijection from $\omega$ to $A$ (with finite injections as conditions) restores the full Cohen extension. Or rather, there is a generic filter doing just that. It is indeed a homogeneous forcing. | |
| Mar 26, 2017 at 21:39 | comment | added | Asaf Karagila♦ | Well. I think that would be easy if you prove that every set in Cohen's model is definable from ordinals and parameters in $A$ there. This should be doable, since you can identify a minimal support within Cohen's model, (i.e. you can find, for each set, a finite sequence of reals which serve as its minimal support). You can also have a partial interpretation function, which will then serve as a definition, and the ordinal would give you the name (which lies in $L$). Again, this is all very long and very technical, but that's how things tend to be when working in specific models by hand. | |
| Mar 26, 2017 at 21:36 | comment | added | Vladimir Kanovei | One of key things which I am interested to fix is to prove that every set $x\in\text{HOD}(A)$ is a member of $\text{OD}(A)$ inside $\text{HOD}(A)$. What if we extend $\text{HOD}(A)$ by a generic bijection $f:\omega$ onto $A$ (finite conditions)? This looks like a homogeneous forcing | |
| Mar 26, 2017 at 19:53 | comment | added | Asaf Karagila♦ | Well, Cohen's original forcing was based on his definition of forcing, which is now known as "weak forcing" (or is it "strong forcing"? I always forget), and things were a bit weirder there. As far as I know, it was Halpern and Levy who formalized the construction as $L(A)$. | |
| Mar 26, 2017 at 19:22 | comment | added | Vladimir Kanovei | By the way you are right, Cohen's original technique, which is essentially $L(A)$, is perhaps most instrumental for major facts including the ultrafilter existence and all sets being HOD($A$). | |
| Mar 26, 2017 at 15:41 | comment | added | Asaf Karagila♦ | Yes, these things are often complicated. I'd be happy to help, feel free to drop me an email with some details if you're interested. | |
| Mar 26, 2017 at 15:37 | comment | added | Vladimir Kanovei | Well, unfortunately it takes 6 or so pages in Jech, deeply contaminated with the boolean algebra stuff. Minimally I'd like to prove self-containedly that the symmetric version of the construction satisfies: all sets belong to HOD($A$). I need a really self-contained proof since I want to prove that this model $L(A)$, even after extending by another real $x$ (say random or Mathias($U$), to begin with) keeps the D-finiteness and the existence of a free ultrafilter. It needs an OD($A,x$) wellordering inside, so the same problem arizes in a much more difficult setting. But thank you for the help. | |
| Mar 26, 2017 at 13:35 | history | edited | Asaf Karagila♦ | CC BY-SA 3.0 |
added 309 characters in body
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| Mar 26, 2017 at 12:31 | history | answered | Asaf Karagila♦ | CC BY-SA 3.0 |