Timeline for Can we find CH in the analytical hierarchy?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 18, 2015 at 21:08 | comment | added | Andrés E. Caicedo | It may be worth adding that, in a sense, $\mathsf{CH}$ is the $\Sigma^2_1$ statement. Woodin's $\Sigma^2_1$-absoluteness theorem can be stated as follows: Assuming enough large cardinals (a proper class of Woodin cardinals suffices) and that $\mathsf{CH}$ holds, if $\phi$ is a $\Sigma^2_1$ statement, and $\mathbb P$ is a poset forcing $\mathsf{CH}$, then $\mathbb P$ forces $\phi$ iff $\phi$ already holds. That is, informally, any $\Sigma^2_1$ statement compatible with $\mathsf{CH}$ already follows from $\mathsf{CH}$. | |
| Sep 18, 2015 at 18:21 | vote | accept | Wojowu | ||
| Sep 18, 2015 at 18:13 | comment | added | Asaf Karagila♦ | @Wojowu: It means $\kappa$-closed, for $\kappa=\omega_1$. It sounds like a snide comment, but it's not. The term $\kappa$-closed forcing is a lot easier to find. What Emil says is correct, for $\kappa=\omega_1$ it is often called $\sigma$-closed instead. | |
| Sep 18, 2015 at 17:43 | comment | added | Emil Jeřábek | It means that every countable chain of forcing conditions has a lower bound (in the present case, this amounts to the observation that the union of countably many compatible countable partial functions is again a countable partial function). This condition is also called “countably closed” or “$\sigma$-closed”. | |
| Sep 18, 2015 at 17:37 | comment | added | Wojowu | I suppose it's standard terminology, but what does "$\omega_1$-closed" mean? It turned out to be unsuccessful to search for this on Wikipedia. | |
| Sep 18, 2015 at 17:31 | history | answered | Emil Jeřábek | CC BY-SA 3.0 |