Timeline for Is there an abstract logic satisfying the Löwenhein-Skolem property for single sentences but not for countable sets of sentences?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 7 at 1:27 | comment | added | Noah Schweber | Nothing better than applying Lindstrom's theorem itself, sadly. I'll think about it though, it's a fun problem! | |
| Jun 7 at 1:19 | comment | added | Rodrigo Freire | @NoahSchweber More positively, I would also be very interested in a direct proof that a compact regular logic extending FOL and satisfying dLS for sentences also satisfies dLS for theories. Do you know something about this? | |
| Jun 7 at 1:10 | comment | added | Noah Schweber | You might be able to get an example extending FOL by replacing "all uncountable structures" with something more well-behaved, but I don't immediately see it. | |
| Jun 7 at 1:09 | comment | added | Noah Schweber | This definitely doesn't extend FOL. I generally don't include "extends FOL" in my definition of "regular logic," but I think some texts do. | |
| Jun 7 at 1:06 | comment | added | Rodrigo Freire | @NoahSchweber Thanks! If this extends first order logic, then it is not regular. Take the conjunction of an $\omega$-categorical sentence $\varphi$ with a $\sigma_i$, for $i$ large enough. I should add that I am mostly interested in extensions of FOL. | |
| Jun 7 at 0:49 | comment | added | Noah Schweber | Sorry, that was a typo on my part - the countable models of $\sigma_i$ should be (up to isomorphism) the $\mathcal{A}_j$s with $j\ge i$. Unless I'm having a silly moment this does yield a regular logic; in particular, a conjunction of $\sigma$-formulas without a countable model must involve a negated $\sigma$-formula in which case it has no uncountable models either and so is unsatisfiable. | |
| Jun 6 at 23:11 | comment | added | Rodrigo Freire | @NoahSchweber Nice comment. If I understood correctly, this logic is not regular, for $\sigma_i\wedge\sigma_j$ has only uncountable models? | |
| Jun 6 at 22:25 | comment | added | Noah Schweber | Fix a sequence of nonisomorphic countable structures $\mathcal{A}_i$ ($i\in\omega$) and consider a logic generated by sentences $\sigma_i$ where for each $i$ the models of $\sigma_i$ are exactly the uncountable structures or the structures isomorphic to $\mathcal{A}_i$. This has dLS for single sentences but not for satisfiable countable theories. | |
| Jun 6 at 20:09 | history | edited | Rodrigo Freire | CC BY-SA 4.0 |
deleted 9 characters in body
|
| Jun 6 at 19:51 | history | asked | Rodrigo Freire | CC BY-SA 4.0 |