Timeline for Is this compactness property for "satisfiability on $\mathbb{R}$" consistent?
Current License: CC BY-SA 4.0
10 events
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| Jun 17, 2021 at 1:29 | comment | added | Farmer S | @NoachSchweber I added Update 2, relating to your questions on $(\kappa,\kappa^+)$-compactness with large continuum. The position of my statement "I think $(\omega_4,\omega_5)$ will be more subtle" in the text might have been misleading: I meant I thought it would be more subtle to do an argument analogous to those for $(\omega_2,\omega_3)$ and $(\omega_3,\omega_4)$, not the $L[G]$ question. This is because (i) $u_{n+3}$ is singular, so $u_{n+3}\neq\omega_{n+3}$, and (ii) if there was something like it, it should lead into larger cardinals. But this is moot now given the recent answers on MSE. | |
| Jun 17, 2021 at 1:12 | history | edited | Farmer S | CC BY-SA 4.0 |
Added Update 2.
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| Jun 9, 2021 at 3:13 | history | bounty ended | Noah Schweber | ||
| Jun 9, 2021 at 3:12 | comment | added | Noah Schweber | Separately, could you say a bit about why you think the $(\omega_4,\omega_5)$-case may be particularly subtle? | |
| Jun 9, 2021 at 3:12 | comment | added | Noah Schweber | Re: your final addition, for the "gap-two" stuff I'm most interested in what can be done in $\mathsf{ZFC}$ alone (or very mild strengthenings thereof), so while supercompactness works it's not what I'm looking for there. | |
| Jun 8, 2021 at 15:59 | comment | added | Farmer S | @NoahSchweber I added some more on these questions. | |
| Jun 8, 2021 at 15:58 | history | edited | Farmer S | CC BY-SA 4.0 |
Added some answers to further questions from comments.
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| Jun 8, 2021 at 2:50 | vote | accept | Noah Schweber | ||
| Jun 8, 2021 at 2:50 | comment | added | Noah Schweber | Oh I like this a lot! Two quick questions. First, this does not obviously help with questions like "Is it consistent that $2^{\omega}\ge\omega_4$ and $\mathcal{R}$-satisfiability is $(\omega_3,\omega_4)$-compact?," right? I think this is fairly special to the $(\omega_2,\omega_3)$-case, but I just want to check that I'm not missing something. Second, do you think it's plausible that there is some $\kappa$ such that $\mathcal{R}$-satisfiability is $(\kappa,\kappa^{++})$-compact? | |
| Jun 8, 2021 at 2:23 | history | answered | Farmer S | CC BY-SA 4.0 |