Translation procedures in descriptive inner model theory∗†

We develop a basic translation procedure that translates a hod mouse to an equivalent mouse. Unlike the translation procedure used by Steel and Zhu, our procedure works in a coarse setting without assuming AD+. Nevertheless, the procedure resembles the one developed by Steel in [8]. We use the translation procedures to answer a question of Trevor Wilson. Namely, we show that if there is a stationary class of λ such that λ is a limit of Woodin cardinals and the derived model at λ satisfies AD+ + θ0 < Θ then there is a transitive model M such that Ord ⊆M and M “there is a proper class of Woodin cardinals and a strong cardinal”. Using a theorem of Woodin on derived models it is not hard to see that the reverse of the aforementioned theorem is also true, thus proving that the two theories are in fact equiconsistent. In descriptive inner model theory, the role of hod mice is to capture descriptive set theoretic strength of sets of reals inside a fine structural model. Such capturing leads to representation of HOD of models of AD +V = L(P(R)) as a fine structural model (see Theorem 4.24 of [4]). It also leads to partial proofs of the Mouse Set Conjecture (see Theorem 6.19 of [4]). For more on the role of hod mice in descriptive inner model theory see [5]. ∗2000 Mathematics Subject Classifications: 03E15, 03E45, 03E60. †