Abstract We construct divergent models of $\mathsf {AD}^+$ along with the failure of the Continuum Hypothesis ( $\mathsf {CH}$ ) under various assumptions. Divergent models of $\mathsf {AD}^+$ play an important role in descriptive inner model theory; all known analyses of HOD in $\mathsf {AD}^+$ models (without extra iterability assumptions) are carried out in the region below the existence of divergent models of $\mathsf {AD}^+$ . Our results are the first step toward resolving various open questions concerning the length of definable prewellorderings of the reals and principles implying $\neg \mathsf {CH}$ , like $\mathsf {MM}$ , that divergent models shed light on, see Question 5.1.

Abstract Assume $\mathsf {ZF} + \mathsf {AD}$ and all sets of reals are Suslin. Let $\Gamma $ be a pointclass closed under $\wedge $ , $\vee $ , $\forall ^{\mathbb {R}}$ , continuous substitution, and has the scale property. Let $\kappa = \delta (\Gamma )$ be the supremum of the length of prewellorderings on $\mathbb {R}$ which belong to $\Delta = \Gamma \cap \check \Gamma $ . Let $\mathsf {club}$ denote the collection of club subsets of $\kappa $ . Then the countable length everywhere club uniformization holds for $\kappa $ : For every relation $R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$ with the property that for all $\ell \in {}^{<{\omega _1}}\kappa $ and clubs $C \subseteq D \subseteq \kappa $ , $R(\ell ,D)$ implies $R(\ell ,C)$ , there is a uniformization function $\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$ with the property that for all $\ell \in \mathrm {dom}(R)$ , $R(\ell ,\Lambda (\ell ))$ . In particular, under these assumptions, for all $n \in \omega $ , $\boldsymbol {\delta }^1_{2n + 1}$ satisfies the countable length everywhere club uniformization.

Abstract In this paper, we explore the structure theory ofL(ℝ,μ) under the hypothesisL(ℝ,μ) ⊧ “AD +μis a normal fine measure on ” and give some applications. First we show that “ ZFC + there existω2Woodin cardinals”1has the same consistency strength as “ AD +ω1is ℝ-supercompact”. During this process we show that ifL(ℝ,μ) ⊧ AD then in factL(ℝ,μ) ⊧ AD+. Next we prove important properties ofL(ℝ,μ) including Σ1-reflection and the uniqueness ofμinL(ℝ,μ). Then we give the computation of full HOD inL(ℝ,μ). Finally, we use Σ1-reflection and ℙmaxforcing to construct a certain ideal on (or equivalently on in this situation) that has the same consistency strength as “ZFC+ there existω2Woodin cardinals.”