Timeline for What is the "iterated definability" limit of first-order logic?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 18, 2023 at 6:29 | history | bounty ended | Noah Schweber | ||
| Jan 16, 2023 at 21:47 | comment | added | Noah Schweber | The set of all finite-arity relations. (It was a typo, but it's too late to edit now.) | |
| Jan 16, 2023 at 21:46 | comment | added | Farmer S | What do you mean by $\mathcal{P}^{<\omega}$? | |
| Jan 16, 2023 at 21:36 | comment | added | Noah Schweber | As an upper bound, I think this works for all structures: if we define an analogue $L_{\mathfrak{A},-}$ of the $L$-hierarchy starting with a copy of a structure $\mathfrak{A}$ as urelements, the $\mathsf{FOL^{(\infty)}}$-definable relations on $\mathfrak{A}$ should all be in $L_{\mathfrak{A},\theta}$ for $\theta$ least such that $L_{\mathfrak{A},\theta}\cap\mathcal{P}^{<\omega}(\mathfrak{A})=L_{\mathfrak{A},\theta+1}\cap\mathcal{P}^{<\omega}(\mathfrak{A})$. It's not clear to me, though, that this is always sharp. | |
| Jan 14, 2023 at 18:51 | vote | accept | Noah Schweber | ||
| Jan 14, 2023 at 18:51 | comment | added | Noah Schweber | I think the only thing that needs to be added is that when we look at models coded by reals it's enough to look at reals already in the relevant layer of $N$, but this is immediate. Thanks! (I'm going to hold off on awarding the bounty for now since the other component of the question is still open.) Meanwhile re: your first sentence, is there something in particular that I could clarify in the question? I'm generally interested in this sort of topic and I'd like to make my post maximally understandable. | |
| Jan 13, 2023 at 17:55 | comment | added | Noah Schweber | I think this is right, but let me read through it more carefully later today. | |
| Jan 13, 2023 at 14:33 | history | answered | Farmer S | CC BY-SA 4.0 |