Timeline for Can superstability of a countable theory be characterized in terms of not 'weakly trace interpreting' a particular structure?
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| May 31, 2021 at 23:26 | comment | added | James E Hanson | @ErikWalsberg Thank you. I guess I didn't think about it hard enough. Perhaps the correct question really is how many structures you need to characterize superstability in terms of trace definability. Also, the structure on $2^\omega$ you can use to characterize $\omega$-stability (or more generally total transcendence) in terms of trace definability certainly has QE. | |
| May 31, 2021 at 23:25 | history | edited | James E Hanson | CC BY-SA 4.0 |
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| May 31, 2021 at 23:11 | history | edited | James E Hanson | CC BY-SA 4.0 |
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| May 31, 2021 at 22:02 | comment | added | Erik Walsberg | If I understand the definitions correctly, #1 is false because $(\mathbb{C},+,\times)$ weakly trace interprets $(\mathbb{Q},+,\times)$. Ok, this doesn't quite work because the language of fields is not relational, but you should be able to do something like add a relation symbol for every quantifier free definable set. The important fact is that every quantifier-free definable subset $X$ of $\mathbb{Q}^n$ is of the form $X' \cap \mathbb{Q}^n$ for a quantifier free definable $X' \subseteq \mathbb{C}^n$. (the last claim is trivial.) | |
| May 31, 2021 at 21:57 | comment | added | Erik Walsberg | I should also say that I don't know the answers to your questions. I would expect that #2 has a negative answer, but I also did not expect to be able to characterize in terms of trace definibility. (Ok, you get weak trace interpretability, but I think the structure you describe should have QE). I would be interested to see the result on omega-stable structures fully written down. | |
| May 31, 2021 at 21:56 | comment | added | Erik Walsberg | @EmilJeřábek As is common in abstract model theory, I only make these definitions for complete consistent theories, so there might not be as much confusion as you would think. | |
| May 31, 2021 at 21:50 | comment | added | Erik Walsberg | Two first comments: It should also be pointed out that this notion was also independently introduced by Vince Guingona. He only worked with algebraically trivial Fraisse limits, so I think my real contribution is to start thinking about this notion for arbitrary theories. Secondly, I do not think that I am good with coming up with names so I am not attached to any of my terminology. What I call trace definibility can be viewed as a weak form of interpretation, so maybe the terminology should reflect that. | |
| May 31, 2021 at 19:17 | history | edited | James E Hanson | CC BY-SA 4.0 |
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| May 31, 2021 at 17:57 | history | edited | James E Hanson | CC BY-SA 4.0 |
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| May 31, 2021 at 17:46 | comment | added | Emil Jeřábek | Weak interpretation is an already existing standard term: a theory $T$ weakly interprets a theory $S$ if there exists a consistent theory $T'\supseteq T$ (in the same language) such that $T'$ interprets $S$. You should choose a different name. | |
| May 31, 2021 at 17:27 | history | asked | James E Hanson | CC BY-SA 4.0 |