Timeline for Large Cardinal Principles that Imply $\Sigma_3^1$-Generic Absoluteness
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 2 at 11:01 | comment | added | Ember Edison | Why is $\textbf{Ord}^\sharp$ required for $Σ^1_3$ generic absoluteness to hold, but not $\mathbb{R}^\sharp$? How should I understand the conclusion in link.springer.com/article/10.1007/BF02897063 ? | |
| Sep 1, 2015 at 2:42 | comment | added | Andrés E. Caicedo | @William Yes.${}$ | |
| Sep 1, 2015 at 2:42 | comment | added | William | @AndresCaicedo In your paper "Projective Well-Ordering of the Reals" Theorem 3 seems to be what you are referring to. But the definition of generic absoluteness is stronger something like a two-step generic absoluteness. Also what is the sharp of an arbitary set. Should the existence of $A^\sharp$ be equivalent to there is an elementary embedding of $L[A]$ to $L[A]$? Can it be some kind of mouse? | |
| Sep 1, 2015 at 2:31 | comment | added | William | @AndresCaicedo Is sharps really necessary? Bagaria and Friedman showed that it consistent with $\mathsf{ZFC}$ that (lightface) $\Sigma_3^1$-generic absoluteness holds, for example in the $\text{Coll}(\omega, <\lambda)$ extension where $\lambda$ is a $\Sigma_2$-correct cardinal (which is not a large cardinal). | |
| Aug 28, 2015 at 7:52 | comment | added | Asaf Karagila♦ | @Andres: One might even say that the consistency bound is... sharp :-P | |
| Aug 26, 2015 at 21:42 | comment | added | Andrés E. Caicedo | The existence of sharps (as in Philip's second paragraph) is also necessary, so this assumption is optimal. | |
| Aug 26, 2015 at 15:48 | history | answered | Philip Welch | CC BY-SA 3.0 |