Timeline for CH and automorphisms of ultrapowers of $\mathbb{Z}$ and $\mathbb{R}$
Current License: CC BY-SA 3.0
8 events
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| Mar 30, 2018 at 14:17 | comment | added | Ali Enayat | @NoahSchweber Yes, with the proviso provided by Joel Hamkins in his comment about the cardinality of the structure, one can ask the more general question you posed. On the other hand, as pointed out by Douglas Ulrich in his comment, the interesting case is when the structure is rigid. | |
| Mar 30, 2018 at 14:13 | comment | added | Ali Enayat | @DouglasUlrich Thanks for your comments and for the pointers, I will take a look. | |
| Mar 30, 2018 at 13:50 | comment | added | Danielle Ulrich | The "Viva/Vive la difference" series by Shelah (arxiv.org/abs/math/9201245, arxiv.org/pdf/math/9304207.pdf, arxiv.org/pdf/math/0112237.pdf) and "Automorphism Groups of Ultraproducts of Finite Symmetric Groups" (by Lucke and Thomas) seem relevant. In particular, in "Vive la difference III" Shelah seems to prove that consistently, there is an ultrafilter $\mathcal{U}$ on the set of primes such that $\prod_\omega \mathbb{F}_p/\mathcal{U}$ is rigid. Perhaps the proof concept can be modified... | |
| Mar 30, 2018 at 13:28 | comment | added | Danielle Ulrich | Generalizing that $\mathbb{C}^*$ has many automorphisms: for any structure $M$ and any nonprincipal ultrafiflter $\mathcal{U}$, every automorphism of $M$ lifts canonically to an automorphism of $M^\omega/\mathcal{U}$, i.e. we can view $\mbox{Aut}(M)$ as a subgroup of $\mbox{Aut}(M^\omega/\mathcal{U})$; this is because ultraproducts commute with expansions. So the interesting case is when $M$ is rigid (e.g. $\mathbb{Z}, \mathbb{N}$ as asked). | |
| Mar 30, 2018 at 11:58 | comment | added | Joel David Hamkins | @NoahSchweber You should add the adjective "infinite". | |
| Mar 30, 2018 at 4:30 | comment | added | Noah Schweber | This might be a silly question, but: can an ultrapower (of a structure in a countable language of size at most continuum with respect to a nonprincipal ultrafilter on $\omega$) ever be rigid? | |
| Mar 30, 2018 at 3:20 | history | edited | Ali Enayat | CC BY-SA 3.0 |
added 9 characters in body
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| Mar 30, 2018 at 2:33 | history | asked | Ali Enayat | CC BY-SA 3.0 |