Timeline for Amoeba collapse
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 26, 2023 at 9:21 | vote | accept | Monroe Eskew | ||
| Feb 19, 2023 at 13:30 | answer | added | Andreas Lietz | timeline score: 5 | |
| Feb 18, 2023 at 12:10 | history | edited | Monroe Eskew | CC BY-SA 4.0 |
added 64 characters in body
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| Feb 18, 2023 at 12:02 | answer | added | Monroe Eskew | timeline score: 8 | |
| Feb 18, 2023 at 5:29 | comment | added | François G. Dorais | Thank you! This is a great forcing! | |
| Feb 17, 2023 at 18:09 | comment | added | Monroe Eskew | In my answer, my translation of names from $\sigma'$ to $\sigma''$ was flawed. It should have been a substitution of conditions under an isomorphism, but I was assuming conditions brought in from the smaller partial order played the same forcing role when moved over. I will delete my answer for now. So the question is open. | |
| Feb 17, 2023 at 15:05 | comment | added | Monroe Eskew | Rather, I conjecture it doesn't add generics for $\kappa$-c.c. posets of size $\kappa$. | |
| Feb 17, 2023 at 14:57 | comment | added | Monroe Eskew | continued: This is because, if you take two of them for $\alpha<\beta$, then $f_\alpha,f_\beta \in Add(\beta^+)$, so a regular superorder of both would have to respect that they are incompatible in $Add(\beta^+)$. This may seem silly but it's the nature of the ordering of Amoeba which is designed to "freeze" how the smaller partial orders appear by having the larger ones be literal super-orders. Now I suspect something deeper is true; it doesn't add any generics for $\kappa$-c.c. posets. | |
| Feb 17, 2023 at 14:54 | comment | added | Monroe Eskew | @FrançoisG.Dorais It's not $\kappa$-c.c. Let $\{ f_\alpha : \alpha < \kappa \}$ be a sequence of binary functions, where $f_\alpha$ has domain $\alpha+1$, takes value 0 below $\alpha$ and value 1 at $\alpha$. Each is a member of $Add(\lambda)$, where $\lambda$ is any regular cardinal between $\alpha$ and $\kappa$. So consider the conditions $(Add(\alpha^+),f_\alpha)$ in Amoeba. I claim these are incompatible in Amoeba. | |
| Feb 17, 2023 at 14:18 | comment | added | François G. Dorais | Is there an easy way to see that this is different from the plain Levy collapse? | |
| Feb 16, 2023 at 15:19 | comment | added | Joel David Hamkins | Very nice. Woodin told me once that in his dissertation, he had used a "generic Boolean algebra" construction, which has a certain similarity to this (but without the inaccessible). One expands the Boolean algebra generically. But I think his forcing wasn't fully amoebic like yours. | |
| Feb 16, 2023 at 14:43 | history | asked | Monroe Eskew | CC BY-SA 4.0 |