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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 A set of reals isuniversally Baireif all of its continuous preimages in topological spaces have the Baire property.$$\mathsf {Sealing}$$is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. The$$\mathsf {Largest\ Suslin\ Axiom}$$($$\mathsf {LSA}$$) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let$$\mathsf {LSA-over-uB}$$be the statement that in all (set) generic extensions there is a model of$$\mathsf {LSA}$$whose Suslin, co-Suslin sets are the universally Baire sets. We show that over some mild large cardinal theory,$$\mathsf {Sealing}$$is equiconsistent with$$\mathsf {LSA-over-uB}$$. In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see Definition 2.7). As a consequence, we obtain that$$\mathsf {Sealing}$$is weaker than the theory ‘$$\mathsf {ZFC} +$$there is a Woodin cardinal which is a limit of Woodin cardinals’. A variation of$$\mathsf {Sealing}$$, called$$\mathsf {Tower\ Sealing}$$, is also shown to be equiconsistent with$$\mathsf {Sealing}$$over the same large cardinal theory. The result is proven via Woodin’s$$\mathsf {Core\ Model\ Induction}$$technique and is essentially the ultimate equiconsistency that can be proven via the current interpretation of$$\mathsf {CMI}$$as explained in the paper. 

Abstract This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals. The following summarizes the main results proved under suitable partition hypotheses.•If$$\kappa $$is a cardinal,$$\epsilon < \kappa $$,$${\mathrm {cof}}(\epsilon ) = \omega $$,$$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$$and$$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$$, then$$\Phi $$satisfies the almost everywhere short length continuity property: There is a club$$C \subseteq \kappa $$and a$$\delta < \epsilon $$so that for all$$f,g \in [C]^\epsilon _*$$, if$$f \upharpoonright \delta = g \upharpoonright \delta $$and$$\sup (f) = \sup (g)$$, then$$\Phi (f) = \Phi (g)$$.•If$$\kappa $$is a cardinal,$$\epsilon $$is countable,$$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$$holds and$$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$$, then$$\Phi $$satisfies the strong almost everywhere short length continuity property: There is a club$$C \subseteq \kappa $$and finitely many ordinals$$\delta _0, ..., \delta _k \leq \epsilon $$so that for all$$f,g \in [C]^\epsilon _*$$, if for all$$0 \leq i \leq k$$,$$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$$, then$$\Phi (f) = \Phi (g)$$.•If$$\kappa $$satisfies$$\kappa \rightarrow _* (\kappa )^\kappa _2$$,$$\epsilon \leq \kappa $$and$$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$$, then$$\Phi $$satisfies the almost everywhere monotonicity property: There is a club$$C \subseteq \kappa $$so that for all$$f,g \in [C]^\epsilon _*$$, if for all$$\alpha < \epsilon $$,$$f(\alpha ) \leq g(\alpha )$$, then$$\Phi (f) \leq \Phi (g)$$.•Suppose dependent choice ($$\mathsf {DC}$$),$${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$$and the almost everywhere short length club uniformization principle for$${\omega _1}$$hold. Then every function$$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$$satisfies a finite continuity property with respect to closure points: Let$$\mathfrak {C}_f$$be the club of$$\alpha < {\omega _1}$$so that$$\sup (f \upharpoonright \alpha ) = \alpha $$. There is a club$$C \subseteq {\omega _1}$$and finitely many functions$$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$$so that for all$$f \in [C]^{\omega _1}_*$$, for all$$g \in [C]^{\omega _1}_*$$, if$$\mathfrak {C}_g = \mathfrak {C}_f$$and for all$$i < n$$,$$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$$, then$$\Phi (g) = \Phi (f)$$.•Suppose$$\kappa $$satisfies$$\kappa \rightarrow _* (\kappa )^\epsilon _2$$for all$$\epsilon < \kappa $$. For all$$\chi < \kappa $$,$$[\kappa ]^{<\kappa }$$does not inject into$${}^\chi \mathrm {ON}$$, the class of$$\chi $$-length sequences of ordinals, and therefore,$$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$$. As a consequence, under the axiom of determinacy$$(\mathsf {AD})$$, these two cardinality results hold when$$\kappa $$is one of the following weak or strong partition cardinals of determinacy:$${\omega _1}$$,$$\omega _2$$,$$\boldsymbol {\delta }_n^1$$(for all$$1 \leq n < \omega $$) and$$\boldsymbol {\delta }^2_1$$(assuming in addition$$\mathsf {DC}_{\mathbb {R}}$$). 

Assume [Formula: see text]. If [Formula: see text] is an ordinal and X is a set of ordinals, then [Formula: see text] is the collection of order-preserving functions [Formula: see text] which have uniform cofinality [Formula: see text] and discontinuous everywhere. The weak partition properties on [Formula: see text] and [Formula: see text] yield partition measures on [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. The following almost everywhere continuity properties for functions on partition spaces with respect to these partition measures will be shown. For every [Formula: see text] and function [Formula: see text], there is a club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. For every [Formula: see text] and function [Formula: see text], there is an [Formula: see text]-club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. The previous two continuity results will be used to distinguish the cardinalities of some important subsets of [Formula: see text]. [Formula: see text]. [Formula: see text]. [Formula: see text]. It will also be shown that [Formula: see text] has the Jónsson property: For every [Formula: see text], there is an [Formula: see text] with [Formula: see text] so that [Formula: see text]. 

We show that Woodin’s AD+ Conjecture follows from various hypotheses extending the Continuum Hypothesis (CH). These results complement Woodin’s original result that the AD+ Conjecture follows from MM(c). 

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. 

A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The 𝖫𝖺𝗋𝗀𝖾𝗌𝗍 𝖲𝗎𝗌𝗅𝗂𝗇 𝖠𝗑𝗂𝗈𝗆 ( 𝖫𝖲𝖠 ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let 𝖫𝖲𝖠 - 𝗈𝗏𝖾𝗋 - 𝗎𝖡 be the statement that in all (set) generic extensions there is a model of 𝖫𝖲𝖠 whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 is equiconsistent with 𝖫𝖲𝖠 - 𝗈𝗏𝖾𝗋 - 𝗎𝖡 . In fact, we isolate an exact theory (in the hierarchy of strategy mice) that is equiconsistent with both (see Definition 3.1). As a consequence, we obtain that 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 is weaker than the theory “ 𝖹𝖥𝖢 + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 by Woodin. A variation of 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 , called 𝖳𝗈𝗐𝖾𝗋 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 , is also shown to be equiconsistent with 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that 𝖫𝖲𝖠 - 𝗈𝗏𝖾𝗋 - 𝗎𝖡 is not equivalent to 𝖲𝖾𝖺𝗅𝗂𝗇𝗀 .