Timeline for A question about ordinal definable real numbers
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Jul 30, 2014 at 22:45 | history | edited | François G. Dorais | CC BY-SA 3.0 | corrected spelling |
| Jun 23, 2011 at 18:17 | history | edited | Andreas Blass | CC BY-SA 3.0 | appended what used to be a comment |
| Jun 23, 2011 at 16:05 | comment | added | Andrés E. Caicedo | @Andreas: Thanks! Why don't you add this to the body of the answer? There is more chance people will see it that way. | |
| Jun 23, 2011 at 15:30 | comment | added | Andreas Blass | Andres, I don't think this works as it stands. If you take a Mathias-generic $r\subset\omega$ and shift it to the right (or left) one unit, the result is still Mathias-generic and generates the same model. Instead of a simple shift, you could apply any strictly monotone function from $L$. But suppose you did Mathias-forcing with respect to, say, the constructibly-first non-principal ultrafilter in $L$. That would avoid this problem. (Note that Joel's comment also depends on the fact that Prikry forcing is with respect to an ultrafilter in the ground model.) | |
| Feb 10, 2011 at 0:30 | comment | added | Joel David Hamkins | Andres, the analogue of your conjecture is true for Prikry forcing. If you start, say, in $L[\mu]$, and do Prikry forcing to add the Prikry sequence $s$, then the only Prikry sequences in $L[\mu][s]$ that still generate all of $L[\mu][s]$ are the sequences that differ finitely from $s$. | |
| Feb 9, 2011 at 22:33 | history | undeleted | Andrés E. Caicedo | ||
| Feb 9, 2011 at 22:33 | history | edited | Andrés E. Caicedo | CC BY-SA 2.5 | New attempt; Post Made Community Wiki |
| Jun 7, 2010 at 4:50 | history | deleted | Andrés E. Caicedo | ||
| Jun 6, 2010 at 19:08 | comment | added | Joel David Hamkins | No problem, Andres! I had tried the same argument myself, and I think it is a subtle point... | |
| Jun 6, 2010 at 19:03 | comment | added | Andrés E. Caicedo | Hi Joel. This is what happens when I think while playing outside with my baby. Sorry. | |
| Jun 6, 2010 at 19:02 | history | edited | Andrés E. Caicedo | CC BY-SA 2.5 | deleted 1633 characters in body |
| Jun 6, 2010 at 18:46 | comment | added | Joel David Hamkins | Justin, yes. The $V_{\delta+1}$ you need is from $V[c]$, and the extension can't tell whether to use $V[c]$ or some other $V[c']$ . | |
| Jun 6, 2010 at 18:43 | comment | added | Joel David Hamkins | You can see that $HOD^{V[G]}$ can differ from $HOD^V$ by first adding a Cohen real $c$, then coding it into the continuum function of $V[c][H]$, and then collapsing cardinals to $V[c][H][g]$. The combined forcing $H+g$ is equivalent to collapsing forcing, as is the whole forcing $c+H+g$, so HOD of the final model is contained in $V$. Thus, $c$ went from definable in $V[c][H]$ to nondefinable in $V[c][H][g]$, even though it was homogeneous forcing. | |
| Jun 6, 2010 at 18:41 | comment | added | Justin Palumbo | Is the problem then that this $V_{\delta+1}$ isn't OD in the extension? If it were you could define the ground model, and thus the levels of the cumulative hierarchy of the ground model, and thus the ground model OD sets. Is that right? | |
| Jun 6, 2010 at 18:33 | comment | added | Joel David Hamkins | To explain, for homogeneous forcing what you have is $HOD^{V[G]}\subset HOD^V$, rather than equality. If your model is $V[c][G]$, where $c$ is the Cohen real and $G$ is the collapse forcing, then we can reconstitute it as $V[c][G]=V[c'][G']$, where $c'=c\oplus d$ is built from an additional Cohen real $d$ added by $G$, and the $g'$ is the corresponding quotient forcing, which is still collapse forcing. Any proposed definition of $X$ cannot distinguish between the reals of $V[c]$ and those of $V[c']$. So $X$ is not ordinal definable in $V[c][G]$. | |
| Jun 6, 2010 at 18:22 | comment | added | Joel David Hamkins | About the Laver theorem: the history is that Laver proved this theorem, and Woodin later observed a similar fact independently. But this theorem doesn't give you OD preservation. It is possible to have models of V=HOD whose HOD's drop in further extensions by homogeneous forcing. | |
| Jun 6, 2010 at 18:18 | comment | added | Joel David Hamkins | Andres, homogeneous forcing doesn't necessarily preserve OD. (Although you are likely thinking of the fact that it doesn't enlarge OD.) So I don't see why X should be OD-definable after the collapse. Indeed, it cannot be, since the same extension could have arisen by absorbing additional Cohen forcing into the collapse forcing. | |
| Jun 6, 2010 at 18:08 | history | answered | Andrés E. Caicedo | CC BY-SA 2.5 |