Timeline for Complexity of the statement 'P is proper'
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Nov 5, 2010 at 20:16 | comment | added | Amit Kumar Gupta | Andres, sorry if I'm being dense, but I've looked through those three sources and can't seem to find this result or something obviously equivalent to it in them. Any chance you can specify where in one of those books/articles this result can be found? Thanks. | |
| Nov 5, 2010 at 16:04 | comment | added | Amit Kumar Gupta | Which formulation of properness are you using that says $P$ is proper iff "$\exists \alpha V _{\alpha} \vDash \psi$" | |
| Nov 5, 2010 at 12:49 | comment | added | Joel David Hamkins | Every statement of the form "$\exists\alpha V_\alpha\models\psi$" is equivalent to a $\Sigma_2$ statement, regardless of the complexity of $\psi$, and conversely all $\Sigma_2$ statements can be expressed in this form. One direction is easy. For the other direction, you do a little Lowenheim-Skolem argument. (You can use $H_\kappa$ instead of $V_\alpha$ if you prefer, and the argument may be more attractive that way.) I can post the argument here later, if you like. | |
| Nov 5, 2010 at 2:45 | comment | added | Amit Kumar Gupta | Could you expand on that? Any property of anything is observable in a sufficiently large $V _{\alpha}$ (if I'm understanding you correctly), that's just the Reflection Theorem. But it's not the case that any property is $\Sigma _2$. | |
| Nov 3, 2010 at 14:28 | comment | added | Andrés E. Caicedo | Shelah's "proper and improper forcing", Jech's "multiple forcing" and the chapter on proper forcing in the Handbook all have proofs of this result. | |
| Nov 3, 2010 at 11:33 | vote | accept | Stefan Hoffelner | ||
| Nov 3, 2010 at 11:29 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |