Timeline for Finite properties of finite models of first order theories
Current License: CC BY-SA 3.0
7 events
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| Oct 17, 2017 at 10:34 | comment | added | Ehud Meir | Thank you both for your answers. It seems that the examples I find for omega categorical structure rely in a strict way on the fact that the domain has infinite cardinality. For example, dense linear orders are omega categorical, but they do not really have a finite counterpart. Do you have an indication about where can one find also finite examples? | |
| Oct 13, 2017 at 11:59 | comment | added | Peter Heinig | [...] roughly speaking, the 'place' where to look for examples is: highly symmetrical structures on $\omega$. | |
| Oct 13, 2017 at 11:58 | comment | added | Peter Heinig | @EhudMeir: keywords which will lead you to examples are "$\aleph_0$-categorical theory" and "oligomorphic permutation group". In particular, countable-infinity of the domain of the structures in question seems to play an essential role in this topic. If a structure has coutable domain, then it's $\aleph_0$-categorical iff its automorphism group is oligomorphic. Very very roughly: for examples, you'll have to look into very symmetrical structures. And in particular: the theorem is vacuously true for finite cardinality, and false for uncountable cardinality. So, [...] | |
| Oct 13, 2017 at 10:01 | comment | added | Ehud Meir | I was not aware of that theorem. Thank you for that. Is there any place where I can find some examples for theories satisfying that? | |
| Oct 13, 2017 at 9:59 | comment | added | Emil Jeřábek | That is, for theories $T$ in a finite language, your condition holds iff the theory of pseudofinite models of $T$ has finitely many completions, each of which is $\omega$-categorical. | |
| Oct 13, 2017 at 9:54 | comment | added | Emil Jeřábek | Are you aware of the Ryll-Nardzewski/Engeler/Svenonius theorem? | |
| Oct 13, 2017 at 9:04 | history | asked | Ehud Meir | CC BY-SA 3.0 |