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We're working in L(R) under AD.

We know that $\omega_1$ is the least measurable in HOD, $\Theta$ is the least woodin, $\delta^2_1$ is the least strong to the woodin, etc.

My question is about characterizing other L(R) cardinals in HOD, specifically those in the projective hierarchy. What is known? Is there a known characterization of $\omega_2$?

I have heard that Woodin has proved something along the lines of "$\delta^1_{2n}$ is strong to $\delta^1_{2n+1}$" or something like that.....

Any known results along these lines would be appreciated and I would be very grateful if proofs/proof sketches were offered. I do not think this material is published anywhere, but I would happily be corrected.

Thank you,

Cody

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  • $\begingroup$ In $\text{HOD}$, the Suslins of cofinality $\omega$, say $\kappa^1_{2n+3}$ for example, can be characterized as the least $\delta$ such that $\mathcal{M}_{2n}(\text{HOD}\vert\delta) \models \delta \text{ is Woodin}$. This is also a result of Woodin. $\endgroup$
    – Rachid Atmai
    Commented Apr 10, 2015 at 18:35
  • $\begingroup$ Oh and I guess you could also say that for every $n$, each $\delta^1_{2n+1}$ is the least strong to the least Woodin $\delta_{\infty}$ of the direct limit of all iterates of $\mathcal{M}_{2n}$. However I don't know about the even regular Suslins. $\endgroup$
    – Rachid Atmai
    Commented Apr 10, 2015 at 18:41
  • $\begingroup$ The results mentioned by Carlo Von Schnitzel can be found in Steel's paper "$HOD^{L(\mathbb{R})}$ is a core model below $\theta$" and in Sargsyan's paper "On the prewellorderings associated with directed systems of mice". Characterization of $\omega_2$ is a good question, but basically nothing is known. $\endgroup$
    – Yizheng Zhu
    Commented Apr 12, 2015 at 8:48

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IMO, characterization of $\omega_2$ should be the projective version of "$\theta$ is Woodin in HOD": replace $\theta$ with $\omega_\omega$ and HOD with $L(HOD|\omega_\omega)$. It has to do with Jackson's level-2 description analysis, i.e, computing $j_\mu(\omega_n)$ when $\mu$ is a measure on $\omega_\omega$ (cf. new cabal II). Very interesting results are still hidden.

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