Timeline for Large cardinals in ZF + DC + AD
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 9, 2023 at 12:49 | comment | added | Alex O. | $ZF+AD$ consistent with choiseless cardinals, example Berkeley cardinals? | |
| Mar 25, 2023 at 7:50 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
the tag (determinacy) might be suitable here - since it is about AD
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| Mar 23, 2023 at 23:47 | answer | added | Gabe Goldberg | timeline score: 9 | |
| Mar 22, 2023 at 14:05 | comment | added | Andrés E. Caicedo | The known results are much stronger. Kanamori's book on large cardinals is a good introduction. For Jónsson cardinals and the like, see for instance "Determinacy and Jónsson cardinals in $L(\mathbb R)$", MR3343535. Work by Sargsyan and others has pushed significantly what we know about large cardinals below $\Theta$ in models of $\mathsf{AD}$ under various assumptions. | |
| Mar 22, 2023 at 9:58 | comment | added | Holo | Furthermore, in $ZF+AD$ every regular $\kappa<\aleph_{\omega_1}$ is Jonsson | |
| Mar 22, 2023 at 9:49 | comment | added | Holo | It is consistent that every regular cardinal bellow $\Theta$ is measurable (specifically, $L(\mathbb R)$ will satisfy this). IIRC, there is a similar result about supercomapcts | |
| Mar 22, 2023 at 7:11 | comment | added | Hanul Jeon | Also, it is known that if we have enough large cardinals (like a proper class of Woodin cardinals) then $L(\mathbb{R})$ is a model of ZF + AD + DC. I guess $L(\mathbb{R})$ also possesses moderate large cardinals like inaccessibles. (A cardinal $\kappa$ is inaccessible if $V_\kappa$ is a model of second-order $\mathsf{ZF}$. This definition is equivalent to a usual definition if we have Choice.) | |
| Mar 22, 2023 at 7:08 | comment | added | Hanul Jeon | Regarding 2, ZF + AD + DC proves $\omega_1$ is measurable (in the sense that it has a $\omega_1$-complete normal measure.) But I do not think it proves there is an inaccessible: If there is such, then cutting off the universe at the level of the inaccessible would be a model of ZF + AD + DC. | |
| Mar 22, 2023 at 1:03 | history | asked | Anindya | CC BY-SA 4.0 |