Timeline for Elementary chains in forcing extensions of $M_1$
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 28, 2015 at 9:49 | answer | added | Grigor | timeline score: 3 | |
| Dec 28, 2015 at 9:30 | comment | added | Grigor | M_1 is definable in this extension. it is K up to the woodin and then L above it. | |
| Nov 3, 2015 at 21:30 | review | Close votes | |||
| Nov 7, 2015 at 10:36 | |||||
| Nov 3, 2015 at 21:20 | comment | added | Stefan Hoffelner | Ok, I see, then I had a misconception of how $M_1$ behaves under generic extensions. What I would want is an inner model with a Woodin cardinal, which has a projectively definable wellorder on the reals, and which is definable in small forcing extensions. I will rethink the exact formulation of the question. | |
| Nov 3, 2015 at 21:01 | comment | added | Andrés E. Caicedo | It is not clear to me. The point is that $M_1^\sharp$ is not on the extension, so you cannot define $M_1$ in the extension by a formula saying that you are defining $M_1$. As I said, presumably what you want is to define $M_1$ as $K$, but then you need to clarify what this means precisely in the presence of Woodin cardinals. Or perhaps you have another idea in mind? | |
| Nov 3, 2015 at 20:59 | comment | added | Stefan Hoffelner | @Yizheng@Andres: Thanks for pointing that out. If I assume the existence of $M_1^{\sharp}$, then it should be definable right? | |
| Nov 3, 2015 at 20:57 | history | edited | Stefan Hoffelner | CC BY-SA 3.0 |
added 33 characters in body
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| Nov 3, 2015 at 20:06 | comment | added | Andrés E. Caicedo | @AsafKaragila Yes, the latter. The point is that to identify $M_1$ as such you need its sharp, which of course is not in $M_1[G]$, so you need to define $M_1$ some other way. (Presumably as a version of $K$, though this would also need to be clarified.) | |
| Nov 3, 2015 at 17:50 | comment | added | Asaf Karagila♦ | @Yizheng: Ground model definability. Or do you mean definable without parameters? | |
| Nov 3, 2015 at 17:03 | comment | added | Yizheng Zhu | Why is $M_1$ definable in $M_1[G]$? | |
| Nov 3, 2015 at 16:21 | history | asked | Stefan Hoffelner | CC BY-SA 3.0 |