Timeline for What is the descriptive complexity of a set added by Cohen forcing?
Current License: CC BY-SA 3.0
8 events
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| Dec 1, 2016 at 14:46 | history | edited | Will Brian | CC BY-SA 3.0 |
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| Dec 1, 2016 at 14:36 | comment | added | Will Brian | @fhyve: It is impossible to add $\Sigma^1_2$ or $\Pi^1_2$ reals by forcing, so $\Delta^1_3$ is the best possible. I'll add this to my answer, and edit out an erroneous statement that I made about Jensen's real: it is not true that $\{a\}$ is $\Delta^1_3$ in $L[a]$ (it is actually $\Pi^1_2$; this follows from the proof found in chapter 28 in Jech's book). | |
| Dec 1, 2016 at 7:12 | comment | added | fhyve | Maybe I shouldn't have said Cohen in particular. Cohen forcing is what I'm (vaguely) familiar with. Can you go any lower than $/Delta_3^1$? | |
| Nov 30, 2016 at 22:05 | comment | added | Noah Schweber | Oh, indeed, that wasn't meant as criticism, just elaboration for the OP - I wanted to clarify the relation between your answer and the Cohen-specific case. (I +1'ed, by the way.) | |
| Nov 30, 2016 at 22:03 | comment | added | Will Brian | @NoahSchweber: Yes, that's right. I'm just pointing out that, even though Cohen forcing won't add projective reals, there are other forcings that will (at least with the right ground model). (Notice that, even though the title just refers to Cohen forcing, the OP does ask about arbitrary forcings in the first paragraph.) | |
| Nov 30, 2016 at 22:00 | comment | added | Noah Schweber | Note, however, that this isn't Cohen forcing. And indeed, Cohen forcing can do no such thing, as Andreas states. | |
| Nov 30, 2016 at 20:56 | comment | added | Will Brian | Also, in case you can't access Jensen's paper, Jech discusses this result in chapter 28 of his book (which is how I know about it). | |
| Nov 30, 2016 at 20:55 | history | answered | Will Brian | CC BY-SA 3.0 |