Timeline for Proof (or reference) about the cc-ness of termspace forcing
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Feb 28, 2023 at 17:40 | vote | accept | Hannes Jakob | ||
| Feb 28, 2023 at 14:23 | answer | added | Yair Hayut | timeline score: 7 | |
| Feb 28, 2023 at 9:42 | comment | added | Yair Hayut | It seems like it's going to be very difficult to get below a weakly compact, even consistently: Let $\mathbb{Q}$ be a $\kappa$-c.c. forcing notion such that $\mathbb{Q}^2$ is not $\kappa$-c.c. Take $\mathbb P$ be the atomic forcing with 2 atoms, $a_0, a_1$. Let $\{(q_i, q_i') | i < \kappa\}$ be an antichain in $\mathbb{Q}^2$. Take the sequence of terms $\dot t_i$, such that $\dot{t}_i = q_i$ is $\min G_P = a_0$ and $\dot{t}_i = q_i'$ otherwise. This is an antichain in the termspace forcing. | |
| Feb 28, 2023 at 9:25 | comment | added | Rahman. M | @HannesJakob You did the right thing! :-) | |
| Feb 28, 2023 at 9:24 | history | edited | Rahman. M | CC BY-SA 4.0 |
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| Feb 28, 2023 at 9:17 | comment | added | Hannes Jakob | I am not sure what you mean by "the right question"... | |
| Feb 28, 2023 at 9:16 | history | edited | Hannes Jakob | CC BY-SA 4.0 |
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| Feb 28, 2023 at 7:05 | comment | added | Rahman. M | You may want to edit your question by mentioning Philipp's comment and stating the right question! | |
| Feb 26, 2023 at 7:45 | comment | added | Hannes Jakob | It certainly seems to. Thanks. I additionally misspoke in the question: He only proves it in the case when $P<\kappa$. What do you suggest i do with the question? Would you mind adding your comment as an answer? | |
| Feb 25, 2023 at 21:16 | comment | added | Philipp Lücke | Let $\mathbb{P}$ be the forcing that adds $\kappa$-many Cohen reals, let $\dot{\mathbb{Q}}$ be the $\mathbb{P}$-name for the forcing that adds a single Cohen real and for each $\alpha<\kappa$, let $\dot{q}_\alpha$ denote the canonical $\mathbb{P}$-name for a Cohen condition that is equal to the first digit of the $\alpha$-th Cohen real added by $\mathbb{P}$. Doesn't $\langle\dot{q}_\alpha\vert\alpha<\kappa\rangle$ enumerate an antichain in $T(\mathbb{P},\dot{\mathbb{Q}})$? | |
| Feb 25, 2023 at 14:21 | comment | added | Rahman. M | The best thing I am aware of is that it holds if $\kappa$ is weakly compact, and $\mathbb P$ is of size less than $\kappa$. Maybe it is true for a poset with the $\kappa$-chain condition. But just with inaccessibility...! | |
| Feb 25, 2023 at 8:16 | history | asked | Hannes Jakob | CC BY-SA 4.0 |