Timeline for Reference request: destroying saturation at an inaccessible?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 22, 2018 at 2:25 | comment | added | Sean Cox | In short, if you can force a saturated ideal on $\kappa$ with a $\kappa^+$-cc forcing, then you already have a saturated ideal in the ground model. | |
| Nov 22, 2018 at 2:19 | comment | added | Sean Cox | of an $I$-positive set $S$) to the condition $([ \check{S} \notin \dot{J} ], [\check{S}]_{\dot{J}})$; here $[\check{S} \notin \dot{J}]$ denotes the $\mathbb{P}$-boolean value of that statement, and $[\check{S}]_{\dot{J}}$ is the $\mathbb{P}$-name for the $\dot{J}$-equivalence class of $\check{S}$. This map is $\perp$-preserving, and so the $\kappa^+$-chain condition of the 2-step target poset carries down to $P(\kappa)/I$. | |
| Nov 22, 2018 at 2:17 | comment | added | Sean Cox | Regarding your question about recovering saturated ideals: this idea appeared (I think) in Kunen's paper on saturated ideals. Suppose $\mathbb{P}$ is $\kappa^+$-cc, and $\dot{J}$ is a $\mathbb{P}$-name for a $\kappa^+$-saturated ideal on $\kappa$. Let $I$ be the set of $A \subset \kappa$ such that $1_{\mathbb{P}}$ forces $\check{A} \in \dot{J}$. Then $I$ is an ideal on $\kappa$ (and is normal, if $\dot{J}$ is forced to be normal). Define a map from $P(\kappa)/I$ into $\text{ro}(\mathbb{P}) * \dot{P}(\kappa)/\dot{J}$ by sending $[S]_I$ (the $I$-equivalence class | |
| Nov 22, 2018 at 1:50 | comment | added | Sean Cox | My comments were confusing, you were not mistaken. Yes, I meant that I don't know how to destroy all saturated ideals on an inaccessible in a way that also preserves presaturation. | |
| Nov 21, 2018 at 23:32 | comment | added | Noah Schoem | Thanks for the info, Sean! I guess I was under the mistaken impression that there were known posets that destroyed the saturation of such ideals and the question only remained whether presaturation could simultaneously preserved. And what prevents recovering a $\kappa^+$-saturated ideal on $\kappa$ in further $\kappa^+$-cc extensions? | |
| Nov 21, 2018 at 2:24 | comment | added | Sean Cox | Not only would you have to destroy measurability of $\kappa$, but you'd have to ensure that $\kappa$'s measurability cannot be resurrected in any further $\kappa^+$-cc extension (and, more generally, that $\kappa$ does not gain a $\kappa^+$-saturated ideal in any further $\kappa^+$-cc extension). | |
| Nov 21, 2018 at 2:15 | comment | added | Sean Cox | Your question appears as Question 8.5 in our paper. It is an interesting problem, but I have no idea how to approach it. Note that any such forcing must destroy measurability of $\kappa$, since the dual of a normal measure on $\kappa$ is a trivial example of a saturated ideal. | |
| Nov 20, 2018 at 5:06 | history | edited | Noah Schoem | CC BY-SA 4.0 |
added 9 characters in body
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| Nov 20, 2018 at 5:05 | review | First posts | |||
| Nov 20, 2018 at 7:06 | |||||
| Nov 20, 2018 at 5:00 | history | asked | Noah Schoem | CC BY-SA 4.0 |