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



Characterizing L(R) Cardinals in HOD
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.


