Timeline for How can we control the cardinality of $j(\kappa)$ for $\kappa$ an $\aleph_1$-strongly compact cardinal?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 1, 2023 at 14:58 | comment | added | Gabe Goldberg | In fact, it is an open question whether given a model containing a strongly compact cardinal, it is possible to force GCH and preserve the strongly compact (see Apter-Dimopoulos-Usuba's "Strongly compact cardinals and the continuum function," Question 5.1). I think it is an interesting question whether one can start with an $\omega_1$-strongly compact and force to make the first $\omega_1$-strongly compact a strong limit cardinal; maybe this is possible just by "pushing up" the first $\omega_1$-strongly compact. More information in Gitik's "On $\sigma$-complete uniform ultrafilters." | |
| Nov 1, 2023 at 10:11 | vote | accept | Calliope Ryan-Smith | ||
| Nov 1, 2023 at 10:11 | comment | added | Calliope Ryan-Smith | GCH seems quite a strong additional assumption on top of the existence of an $\aleph_1$-strongly compact cardinal; do you know if it is forcable from merely 'there exists an $\aleph_1$-strongly compact cardinal'? | |
| Oct 31, 2023 at 23:40 | history | edited | Asaf Karagila♦ | CC BY-SA 4.0 |
Elementary, my dear Watson.
|
| Oct 31, 2023 at 14:46 | history | answered | Gabe Goldberg | CC BY-SA 4.0 |