Timeline for Perfectly transversable theories
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2017 at 18:42 | history | edited | Noah Schweber | CC BY-SA 3.0 |
added 64 characters in body
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| Mar 26, 2017 at 18:41 | vote | accept | Noah Schweber | ||
| Mar 26, 2017 at 18:37 | answer | added | Danielle Ulrich | timeline score: 4 | |
| Mar 26, 2017 at 5:57 | comment | added | Gerhard Paseman | Indeed it may not help. I just remember working through it and using topology and something like perfect sets to solve it. I don't have any other suggestions. Gerhard "Lives In An Imperfect World" Paseman, 2017.03.25. | |
| Mar 26, 2017 at 5:50 | comment | added | Noah Schweber | @GerhardPaseman I don't think that actually helps - the sizes of the automorphism groups or related of the individual models of the theory doesn't obviously let me build a perfect set of reals coding each countable model exactly once . . . In fact, if there is a counterexample $T$ to Vaught's conjecture, then there is one whose countable models each have continuum-many automorphisms: just consider the theory whose models consist of a model of $T$ and an unrelated dense linear order (since $\mathbb{Q}$ has continuum-many automorphisms and is $\aleph_0$-categorical). | |
| Mar 26, 2017 at 5:47 | comment | added | Gerhard Paseman | Let A be a countable (infinite or finite) universal algebra of countable (i. or f.) type. Show each of four associated structures is either at most countably infinite or else has size precisely that of the continuum: automorphism group, sub algebra lattice, endomorphism monoid, and congruence lattice of A. It's on page 35 of my copy, but I don't know if there is an online copy. Gerhard "Copy Not Meaning Isomorphism Type" Paseman, 2017.03.25. | |
| Mar 26, 2017 at 5:38 | comment | added | Noah Schweber | @GerhardPaseman One of every isomorphism type. Sorry, by "copy" I meant "isomorphic copy of", so "representative of isomorphism type of." (What's the exercise?) | |
| Mar 26, 2017 at 5:36 | comment | added | Gerhard Paseman | I guess I am confused. Did you want P to be a set that captures one of every model, or one of every isomorphism type? I am seeing ambiguity in the post. Also, there is an exercise in Chapter I of Algebras Lattices and Varieties which suggests (but does not assert) to me a yes answer. Gerhard "Burris And Kwatinetz, I Think" Paseman, 2017.03.25. | |
| Mar 26, 2017 at 5:28 | comment | added | Noah Schweber | @GerhardPaseman No, of course in general they won't be rigid and each model will be (in all probability) represented continuum-many times. So? | |
| Mar 26, 2017 at 5:18 | comment | added | Gerhard Paseman | Do you expect all the models to be rigid (have trivial automorphism group)? Unless you have an infinite language, I would expect several reals to encode different but otherwise isomorphic models. Gerhard "Maybe Permutation Structures Help Here?" Paseman, 2017.03.25. | |
| Mar 26, 2017 at 5:08 | history | asked | Noah Schweber | CC BY-SA 3.0 |