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Assuming AD, we show that all of the ordinals below $\delta _5^1$ represented by descriptions (c.f. [2], but also defined below) are cardinals. Using this analysis we also get a simple representation for the cardinal structure below $\delta _5^1$. As an application, we compute the cofinalitites of all cardinals below $\delta _5^1$.
Working under AD, we investigate the length of prewellorderings given by the iterates of ℳ2k+1, which is the minimal proper class mouse with 2k + 1 many Woodin cardinals. In particular, we answer some questions from [4] (the discussion of the questions appears in the last section of [2]).
We show that it is relatively consistent with ZFC that there is aprojective wellfounded relation with rank higher than all projective prewellorderings.
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