Timeline for Stability and complete types (in Model Theory)
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 6, 2018 at 12:39 | vote | accept | THC | ||
| Jun 4, 2018 at 22:05 | answer | added | Alex Kruckman | timeline score: 3 | |
| May 29, 2018 at 15:48 | answer | added | Alex Kruckman | timeline score: 7 | |
| May 29, 2018 at 15:27 | comment | added | Alex Kruckman | Sure, but an abstract projective space in the incidence language is still a first-order structure, and you can take the usual definition of stability. All projective spaces of dimension at least $3$ satisfy Desargues's theorem, so there is at least a division ring around. You can google for "stable division ring" to get a sense for what these can look like. In the case $n = 2$, there are stable non-desarguesian projective planes, constructed using Hrushovski's method. For an exposition, see Section 10.4 of Tent and Ziegler's book A Course in Model Theory. | |
| May 29, 2018 at 15:14 | comment | added | THC | @Alex Kruckman: it is especially stability of that first-order structure via the incidence language that I want to understand. Suppose we replace $\mathbb{P}^n(k)$ by an axiomatic projective plane -- in that case, there isn't necessarily a field around -- could you explain what stability would mean ? | |
| May 29, 2018 at 15:02 | comment | added | Alex Kruckman | ... this is a perfectly good first-order structure, and you use the standard definition of stability from first-order model theory. In fact, $\mathbb{P}^n(k)$ (in the incidence language) is bi-interpretable with $k$ (in the field language), so it will be stable in the first-order model theory sense if and only if $k$ is stable (e.g. if $k$ is finite, algebraically closed, or separably closed). | |
| May 29, 2018 at 15:01 | comment | added | Alex Kruckman | Well, the right way to make the definition depends on what properties you are interested in. Also, there are a number of definitions of stability which are equivalent in the first-order setting, but not necessarily in other settings. So for example, when people study stable classes of finite graphs, they usually define stability by a bound on the size of instances of the order property for the edge relation, which has nothing to do (in this context) with counting complete first-order types. In the case of a projective space in the incidence language (points, lines, incidence relation), ... | |
| May 29, 2018 at 14:49 | comment | added | THC | @Alex Kruckman: you should read it as "far away (at first sight)." I know that stability of certain types of graphs has been studied, so I could have asked the question about the point graph or incidence graph of $\mathbb{P}^n(k)$ instead. My question remains the same: what would the statement mean for this type of graph ? For instance, what are the "complete types" over such a graph ? | |
| May 29, 2018 at 14:36 | comment | added | Alex Kruckman | What makes you think this statement makes sense in a context "far away from Model Theory"? Shelah is only making the definition for elementary classes. In fact, it is possible to make sense of the definition in more general contexts (e.g. metric structures or abstract elementary classes), but my view is that making sense of this definition in some context would actually be evidence that that context is actually not so far away from classical model theory. | |
| May 29, 2018 at 11:50 | history | asked | THC | CC BY-SA 4.0 |