Timeline for Potentiality classes and Borel reductions
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 8, 2015 at 16:52 | vote | accept | Richard Rast | ||
| Sep 7, 2015 at 17:15 | answer | added | Burak | timeline score: 2 | |
| Sep 7, 2015 at 6:50 | comment | added | Burak | @RichardRast: If by "Borel equivalent" you mean "Borel bireducible", then the answer should be no. I recently had to quote this theorem, which should be somewhere in the paper you linked (but I did not bother checking). Given this fact, every essentially countable Borel equivalence relation which is the orbit equivalence relation of a Borel action of $S_{\infty}$ is potentially $\Sigma^0_2$. You can easily find three non-equivalent such orbit equivalence relations. (For example, isomorphism of torsion-free abelian groups of rank 1,2,3) | |
| Mar 5, 2015 at 6:01 | comment | added | tomasz | Fair enough. However, I think saying that these are the only interesting potentiality classes from perspective of model theory is a bit too general, Borel equivalence relations in model theory are not limited to isomorphism relations of countable models. | |
| Mar 4, 2015 at 15:00 | comment | added | Richard Rast | From a model theory perspective, those are the only potentiality classes of interest (that are Borel) - the isomorphism relation for the set of countable models of an $L_{\omega_1,\omega}$-sentence is always such a class. The really interesting thing here would be if this showed there are only $\omega_1$ classes of Borel equivalence for such situations. I agree it sounds a bit dubious. | |
| Mar 3, 2015 at 19:03 | comment | added | tomasz | Maybe I'm misunderstanding something. As far as I can tell, the main theorem is not that there are only a small number of potentiality classes, but rather potentiality classes of relations induced by closed subgroups of $S_\infty$! This seems like quite a difference. I don't know about this relation, but in general, there are no more than $\omega_1$ potentiality classes, while there are $\mathfrak c$ many classes of Borel equivalence, so without CH at the very least, the two can't be the same (and even with CH it sounds dubious). | |
| Mar 2, 2015 at 14:35 | history | asked | Richard Rast | CC BY-SA 3.0 |