Timeline for Cohen/Random reals over intermediate models in countable support iterations
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 14, 2018 at 16:04 | comment | added | Monroe Eskew | @Ashutosh, isn't is possible to adjust each $r_n$ on finitely many coordinates, to make the sum $r$ converge to any chosen value? | |
| Jul 8, 2018 at 18:52 | comment | added | Not Mike | @Ashutosh It's possible that $r$ is contained in some meager set coded in the ground-model, in which case $r$ is not Cohen-generic over $V$ (and consequently isn't Cohen-generic over any of the $V^{P_n}$.) | |
| Jul 8, 2018 at 17:37 | comment | added | Ashutosh | Take $r = \sum_n 2^{-n} r_n$ where $r_n \in (0, 1) \cap V^{P_n}$ is Cohen over $V^{P_{n-1}}$. | |
| Jul 8, 2018 at 17:01 | comment | added | Stefan Hoffelner | I should probably add that the motivation for this question has nothing to do with questions concerning cardinal characteristics, so I am not worried about $\aleph_2$ being collapsed. An idea how to solve the problem under CH would be helpful as well. | |
| Jul 8, 2018 at 15:15 | comment | added | Goldstern | It is a bit unusual to start with $\mathfrak c=\aleph_2$. There is a perfect antichain in $P_\omega$, so the forcing $P_\omega$ will not have the $\aleph_2$-cc. (And iterating it for $\omega_1$ many steps will collapse $\aleph_2$. But apparently you do not want to continue the iteration beyond $\omega$., so perhaps this is not a problem.) | |
| Jul 5, 2018 at 5:27 | comment | added | Stefan Hoffelner | Yes, I recall learning that from one your answers here. I picked $Add(\omega, \omega_1)$ as it is closer to the actual problem I am facing. Plus it rules out some tricks involving rearranging of the factors. | |
| Jul 4, 2018 at 23:38 | comment | added | Joel David Hamkins | Do you mean specifically that the forcing at stage $n$ is the forcing $\text{Add}(\omega,\omega_1)$? (After all, even adding one Cohen real will add continuum many Cohen reals to the ground model.) | |
| Jul 4, 2018 at 21:23 | history | asked | Stefan Hoffelner | CC BY-SA 4.0 |