Timeline for Is there a complete uncountable theory with two countable models?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 6 at 15:15 | vote | accept | Noah Schweber | ||
| Jan 6 at 14:29 | answer | added | Colin McQuillan | timeline score: 4 | |
| Nov 16, 2023 at 14:51 | comment | added | Alex Kruckman | An answer to the question has been posted on MSE. link | |
| Sep 6, 2023 at 18:44 | comment | added | Noah Schweber | @AlexKruckman 'Tis done. | |
| Sep 6, 2023 at 2:23 | comment | added | Alex Kruckman | @NoahSchweber I believe it is computable - if you ask this as a separate question, I'll write an answer. | |
| Sep 6, 2023 at 0:50 | comment | added | Noah Schweber | @AlexKruckman That's wild! Is (say) the set of $(m,n)$ such that there is a countable complete theory with exactly $m$ isomorphism classes of size-$\aleph_n$ models known to be computable? | |
| Sep 5, 2023 at 16:52 | comment | added | Alex Kruckman | Similarly, the only numbers less than $20$ that occur as the number of models of size $\aleph_3$ for some countable theory are: $1$, $4$, $7$, $10$, and $13$. | |
| Sep 5, 2023 at 16:51 | comment | added | Alex Kruckman | After some offline discussion, James and I realized that for any finite $k$, there is a countable theory with exactly $k$ models of size $\aleph_1$ up to isomorphism: take the theory of an equivalence relation with exactly $k$ classes, each of which is infinite. Then a model of size $\aleph_1$ is determined up to isomorphism by how many of the classes are uncountable. However, there is no countable theory with exactly $4$ models of size $\aleph_2$ up to isomorphism. Other numbers which cannot be realized in $\aleph_2$ include $2$, $8$, and $9$ (assuming I've done my computations correctly). | |
| Sep 3, 2023 at 18:50 | comment | added | Alex Kruckman | @JamesHanson No, I haven't thought about it. You're right that the Lachlan result reduces it to a pure finite combinatorics / group theory problem. I don't know how tricky it is. | |
| Sep 3, 2023 at 18:44 | comment | added | James E Hanson | @AlexKruckman Do you know a complete classification of which finite numbers can occur as the number of models of size $\aleph_1$? This (and the analogous question for $\aleph_n$ in general) seems like it might be a kind of tough combinatorics problem. | |
| Sep 3, 2023 at 18:25 | comment | added | Alex Kruckman | @JamesHanson and Noah: the original reference is Theories with a finite number of models in an uncountable power are categorical by Lachlan. I don't think he makes the remark about $4$ there, but it follows from the proof. | |
| Sep 3, 2023 at 4:55 | comment | added | James E Hanson | @NoahSchweber This is a modern reference (with a full classification of the spectra of countable theories). I was under the impression that the result was originally due to Lachlan, but I can't find it right now. | |
| Sep 3, 2023 at 2:12 | comment | added | Noah Schweber | @JamesHanson The fact in your second comment is neat, do you have a citation for that? I'm not familiar with it. | |
| Sep 3, 2023 at 2:01 | comment | added | James E Hanson | Also, fun fact: A countable complete theory cannot have precisely $4$ models (up to isomorphism) of cardinality $\aleph_1$. | |
| Sep 3, 2023 at 1:59 | comment | added | James E Hanson | Gabe Conant pointed out that MSE question to me a while ago and in thinking about it, I started to think that a more tractable related question would be this: Is there an expansion of $(\mathbb{N},+,\cdot)$ that has a unique countable elementary extension up to isomorphism? (There are some results about uncountable language expansion of $\mathbb{N}$ that might be useful here.) I wasn't able to resolve it and I was considering asking it on MO, but I never got around to it. | |
| Sep 3, 2023 at 0:07 | comment | added | Joel David Hamkins | Nice question . | |
| Sep 2, 2023 at 23:47 | history | asked | Noah Schweber | CC BY-SA 4.0 |