Timeline for Does an ultrapower of an Aronszajn tree have an $\omega_{1}$-branch?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 23, 2014 at 20:09 | vote | accept | Joseph Van Name | ||
| Oct 31, 2013 at 22:06 | comment | added | Joseph Van Name | Here the partition relation $\kappa\rightarrow[\alpha]^{2}_{\lambda,<\mu}$ means that if $f:[\kappa]^{2}\rightarrow\lambda$, then there is some subset $K\subseteq\kappa$ with $|K|=\alpha$ where $|f[[K]^{2}]|<\mu$. I am unsure if the condition that these conditions are necessary though. | |
| Oct 31, 2013 at 22:06 | comment | added | Joseph Van Name | More generally, assume that if $\kappa,\mu,\lambda$ are cardinals with $\kappa$ regular, $\mu$ measurable or $\mu=\aleph_{0}$, and $\kappa\rightarrow[\kappa]_{\lambda,<\mu}^{2}$. Furthermore, assume that $\mathcal{U}$ is a $\mu$-complete ultrafilter on a set $I$ generated by a set of $\lambda$ many elements and $T_{i}$ is a tree on $\kappa$ of height $\kappa$ for $i\in I$. Then the only $\kappa$ branches on the trimmed ultraproduct of trees $\prod_{i\in I}T_{i}/(\mathcal{U})$ are the ones induced by the ultraproducts of the $\kappa$-branches of the $T_{i}'s$. | |
| Oct 31, 2013 at 21:43 | comment | added | Joseph Van Name | Contrasting the case of trees of height $\omega_{1}$, if $\kappa$ is a weakly compact cardinal, $|I|<\kappa$, $\mathcal{U}$ is an ultrafilter on $I$ and $T_{i}$ is a tree of height $\kappa$ for $i\in I$, then the $\kappa$-branches of the trimmed ultraproduct of trees $\prod_{i\in I}T_{i}/(\mathcal{U})$ are simply the ultraproducts of the $\kappa$-branches of the trees $T_{i}$ (i.e. taking an ultraproduct does not add any more branches then we have to). The proof of this fact is a straightforward application of the partition relation for weakly compact cardinals. | |
| Oct 31, 2013 at 21:17 | answer | added | Ali Enayat | timeline score: 7 | |
| Oct 31, 2013 at 1:32 | answer | added | Andreas Blass | timeline score: 10 | |
| Oct 31, 2013 at 0:37 | answer | added | Goldstern | timeline score: 16 | |
| Oct 30, 2013 at 19:55 | comment | added | saf | of interest: matwbn.icm.edu.pl/ksiazki/fm/fm118/fm118113.pdf | |
| Oct 30, 2013 at 19:37 | history | asked | Joseph Van Name | CC BY-SA 3.0 |