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    $\begingroup$ 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. $\endgroup$
    – Hanul Jeon
    Commented Mar 22, 2023 at 7:08
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    $\begingroup$ 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.) $\endgroup$
    – Hanul Jeon
    Commented Mar 22, 2023 at 7:11
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    $\begingroup$ 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 $\endgroup$
    – Holo
    Commented Mar 22, 2023 at 9:49
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    $\begingroup$ Furthermore, in $ZF+AD$ every regular $\kappa<\aleph_{\omega_1}$ is Jonsson $\endgroup$
    – Holo
    Commented Mar 22, 2023 at 9:58
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    $\begingroup$ 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. $\endgroup$
    – Andrés E. Caicedo
    Commented Mar 22, 2023 at 14:05