UCLA

This dissertation is a contribution to the genre of applications of inner model theory todescriptive set theory. Applying assumptions of determinacy, we investigate the possible lengths of sequences of distinct sets of reals from a fixed pointclass Γ. Substantial work has been done on this question in the case that Γ is a level of the projective hierarchy. In [1], Hjorth shows from ZF + AD + DC that there is no sequence of distinct Σ12 sets of length δ12. Sargsyan extended Hjorth’s technique to prove an analogous result for every even level of the projective hierarchy (see [2]). We show from ZF + AD + DC + V = L(R) that for every inductive-like pointclass Γ in L(R), there is no sequence of distinct Γ sets of length (δΓ)+. This is the optimal result for inductive-like Γ. An essential tool for the proof is Woodin and Steel’s computation of HODL(R) in terms of the direct limit of the system of countable iterates of Mω#. We adapt their method to analyze the direct limit of the system of countable iterates of some Γ-suitable mouse. This allows us code each set in some sequence ⟨Aα : α < λ⟩ ⊂ Γ by a set of conditions in Woodin’s extender algebra at the least Woodin cardinal of this direct limit. The coding sets are contained in the direct limit up to δΓ, bounding |λ| by the successor of δΓ in the direct limit. Our approach also gives a new proof of Sargsyan’s theorem. Chapter 1 surveys prior work in this area. Chapter 2 covers background necessary for the proof of our main result, including some of the descriptive set theory of L(R) and a hasty review of inner model theory. Our main result on inductive-like pointclasses is proven in Chapter 3. Chapter 4 briefly examines how one might apply the techniques of Chapter 3 to obtain analogous results for some projective-like pointclasses in L(R).