Timeline for Logics of proper class sized Kripke frames
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Sep 21, 2021 at 23:57 | comment | added | Andrew Bacon | Thank you both for tracking that paper down. I also found this paper by Marcus Kracht that seems to directly answer my question: he describes a normal modal logic $\Sigma$ that has a sentence that it satisfiable in a $\Sigma$-frame only if that frame has inaccessible cardinality. | |
| Sep 21, 2021 at 15:47 | comment | added | Emil Jeřábek | @NoahSchweber Wow, I completely forgot about that. I guess we are even, then. | |
| Sep 21, 2021 at 15:21 | comment | added | Noah Schweber | @EmilJeřábek I remembered you finding it for me some years ago! | |
| Sep 21, 2021 at 15:12 | comment | added | Emil Jeřábek | @NoahSchweber Yes, this is it. Thanks for finding it. | |
| Sep 20, 2021 at 22:55 | comment | added | Noah Schweber | I believe this is the Thomason paper @EmilJeřábek mentioned above. | |
| Sep 20, 2021 at 21:14 | comment | added | Andrew Bacon | @EmilJeřábek Thanks, yes that was my suspicion. I should read the Thomason now, but I was wondering in what sense frame validity is $\Pi_1^1$-complete, since there are even first-order conditions that cannot be expressed in terms of frame validity. | |
| Sep 20, 2021 at 21:03 | comment | added | Emil Jeřábek | IIRC frame validity is $\Pi^1_1$-complete by results of Thomason (?) from the 70s, thus it may well turn out that $\Pi^1_1$-reflection is actually necessary. | |
| Sep 20, 2021 at 20:16 | comment | added | Wojowu | @none It's not (necessarily) categorical, it has $V_\kappa$ as standard models for all inaccessible $\kappa$. | |
| Sep 20, 2021 at 20:14 | comment | added | Andrew Bacon | OK, I think the question of what counts as a "second order" set theory is distracting, so I've just reformulated the question in terms classes in Morse-Kelley. | |
| Sep 20, 2021 at 20:12 | history | edited | Andrew Bacon | CC BY-SA 4.0 |
I have restated the question in terms of MK set theory
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| Sep 20, 2021 at 6:37 | comment | added | none | I thought second order ZFC was categorical, analogously to the Peano axioms in second order logic. No first-order theory with infinite models can be like that, e.g. because of Löwenheim-Skolem. | |
| Sep 19, 2021 at 23:57 | comment | added | Andrew Bacon | I meant second order ZFC: an axiomatic system of second order logic with single universally quantified axioms instead of the replacement and separation schemas. I don't think there's much difference mathematically between it and Morse Kelley though, in the sense that you can translate straightforwardly between them. (I'll find a reference for that later.) | |
| Sep 19, 2021 at 19:08 | comment | added | Noah Schweber | What exactly is "second order $\mathsf{ZFC}$" here? Do you mean some appropriate class theory like $\mathsf{MK}$, or do you really mean the second-order-logic version of $\mathsf{ZFC}$? | |
| Sep 19, 2021 at 17:59 | history | asked | Andrew Bacon | CC BY-SA 4.0 |