Timeline for Vopěnka's Principle for non-first-order logics
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 8, 2019 at 6:52 | comment | added | Keith Millar | Isn't this only the case for $L$ with set-sized occurrence number? For example, $\mathcal{L}_{\infty,\infty}$ can't have an LST number. | |
| Jul 19, 2015 at 4:11 | comment | added | Noah Schweber | How about first-order logic, assuming $\neg VP$? :P | |
| Jul 19, 2015 at 3:45 | comment | added | Thomas Benjamin | @NoahSchweber: Thanks for clarifying that. I will think on that for a while. Could you, perhaps, provide an example of a logic $L$ where $L$ has a LST number but $VP(L)$ doesn't hold? | |
| Jul 19, 2015 at 3:04 | comment | added | Noah Schweber | As usually when discussing large-cardinal-like principles, I mean distinct in terms of provable equivalence over ZFC (or related theories), or - even better! - in terms of consistency strength over ZFC (see OP paragraph beginning "in principle"). This is not directly addressed by the result you cite - in particular, it is not clear that "$L$ has a LST number" implies $VP(L)$ (consider non-$L$-elementary classes of structures; $LST(L)$ isn't directly useful here). | |
| Jul 19, 2015 at 1:08 | comment | added | Thomas Benjamin | @NoahSchweber: When you ask "whether the specific principles of the form $VP(L)$" are "distinct for reasonable natural $L$", what sort of 'distinctness' would you hope to find, if such 'distinctness' did, in fact, exist (perhaps Thm. 6 suggests that there might not be any versions of $VP$ stronger than the usual one)? | |
| Jul 18, 2015 at 18:25 | comment | added | Noah Schweber | By the way, a note about the proof of their Theorem 6: one direction is immediate. Suppose every logic has a LST number, and fix a proper class $C$ of structures. We can now build a silly logic $L_C$ containing first-order logic with a sentence which holds in exactly the structures in $C$. The existence of an LST number of $C$ then immediately implies the existence of lots of nontrivial elementary embeddings in $C$. The nontrivial direction is showing that VP is strong enough to produce LST numbers; for this, Magidor and Vaananen use a version of supercompactness. | |
| Jul 18, 2015 at 18:22 | comment | added | Noah Schweber | So, this is not directly related to my specific question - my question was not whether there were alternate characterizations of VP (of which there are lots), or even whether there were any in terms of abstract logics, but specifically whether the specific principles of the form $VP(L)$ were distinct for reasonable natural $L$. Still, this is interesting, so +1. | |
| Jul 18, 2015 at 15:57 | history | answered | Thomas Benjamin | CC BY-SA 3.0 |