DIVERGENT MODELS WITH THE FAILURE OF THE CONTINUUM HYPOTHESIS

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.

1 Introduction

In this paper, we identify the reals $\mathbb {R}$ with $\mathbb {N}^{\mathbb {N}}$ , the set of all infinite sequences of natural numbers equipped with the Baire topology.

Definition 1.1. Suppose M and N are transitive models of $\mathsf {AD}^+$ . We say that M and N are divergent models of $\mathsf {AD}^+$ if there are sets of reals $A\in M$ and $B\in N$ such that $A\notin N$ and $B\notin M$ .

If $M,N$ are divergent models of $\mathsf {AD}^+$ , then the Wadge hierarchies of $M, N$ “diverge,” or equivalently ${\wp }(\mathbb {R})\cap M\nsubseteq N$ and ${\wp }(\mathbb {R})\cap N \nsubseteq M$ . Woodin has shown that letting $\Gamma = {\wp }(\mathbb {R})\cap M \cap N$ , then $\Gamma = {\wp }(\mathbb {R})\cap L(\Gamma ,\mathbb {R})$ and furthermore, $L(\Gamma ,\mathbb {R})\models \mathsf {AD}_{\mathbb {R}} + \mathsf {DC}$ . The upper-bound consistency strength of divergent models of $\mathsf {AD}^+$ , as shown by Woodin, is the existence of a Woodin cardinal which is a limit of Woodin cardinals. This bound is conjectured to be exact. ¹ Divergent models of $\mathsf {AD}^+$ play a very important role in descriptive inner model theory; virtually, all known analyses of HOD in strong $\mathsf {AD}^+$ models are carried out below this bound (see cf. [3, 5]).

Working in a universe satisfying $\mathsf {CH}$ , Woodin constructed divergent models of $\mathsf {AD}^+$ [1]. We prove that it is consistent that there are divergent models of $\mathsf {AD}^+$ while $\mathsf {CH}$ fails.

Theorem 1.2. Suppose $\mathsf {CH}$ holds and there are two sets of reals $A,B$ such that $:$

• • $(\mathbb {R},A)^\sharp , (\mathbb {R},B)^\sharp $ exist and are $\aleph _1$ -universally Baire,

• • $L(A,\mathbb {R}), L(B,\mathbb {R})$ are models of $\mathsf {AD}^+$ such that letting $H_A = HOD^{L(A,\mathbb {R})}$ and $H_B = HOD^{L(B,\mathbb {R})}$ , there is some $\alpha < \textrm {min}\{\omega _1^{H_A}, \omega _1^{H_B}\}$ such that the $\alpha $ -th real in the canonical well-order of $H_A$ is different from the $\alpha $ -th real in the canonical well-order of $H_B$ .

Let $\mathbb {P}$ be the standard ccc forcing that adds $\omega _2$ many Cohen reals and $g\subseteq \mathbb {P}$ be V-generic. Then in $V[g]$ , there are $A^*,B^*$ and embeddings $j_A, j_B$ such that $:$

1. 1. $j_A: L(A,\mathbb {R}^V) \rightarrow L(A^*,\mathbb {R}^{V[g]})$ , $j_B: L(B,\mathbb {R}^V) \rightarrow L(B^*,\mathbb {R}^{V[g]})$ fix all ordinals, and

2. 2. $L(A^*,\mathbb {R}^{V[g]}), L(B^*,\mathbb {R}^{V[g]})$ are divergent models of $\mathsf {AD}^+$ .

Corollary 1.3. Con( $\mathsf {ZFC} + $ there is a Woodin limit of Woodin cardinals) implies Con( $\mathsf {CH}$ fails and there are divergent models of $\mathsf {AD}^+)$ .

Proof By results of Woodin’s (see [1]), the hypothesis of Theorem 1.2 is consistent relative to the existence of a Woodin limit of Woodin cardinals. The corollary follows from Theorem 1.2.

The following theorem is folklore. We include the proof here for self-containment. It is used in the proof of Corollary 1.5. A forcing $\mathbb {P}$ is said to be weakly proper if whenever $g\subset \mathbb {P}$ is V-generic, for any ordinal $\alpha $ , ${\wp }_{\omega _1}^{V[g]}(\alpha )\subset {\wp }^V_{\omega _1}(\alpha )$ . $\Gamma _\infty $ denotes the collection of universally Baire sets.

Theorem 1.4. Assume there is a proper class of Woodin cardinals and $A\subseteq \mathbb {R}$ is universally Baire. Suppose $\mathbb {P}$ is weakly proper. Then for any V-generic $g\subseteq \mathbb {P}$ , there is some universally Baire set $B\in V$ such that letting $B^*$ be the canonical interpretation of B in $V[g]$ , A is Wadge reducible to $B^*$ .

Corollary 1.5. Assume there is a proper class of Woodin cardinals. Suppose $A,B$ are as in the hypothesis of Theorem 1.2. Furthermore, assume that $\Gamma _\infty \subset L(A,\mathbb {R})\cap L(B,\mathbb {R})$ . Let $\mathbb {P}$ be the forcing that adds $\omega _2$ Cohen reals and $g\subseteq \mathbb {P}$ be V-generic. Then in $V[g]$ , $\Gamma _\infty \subset L(A^*,\mathbb {R}^{V[g]})\cap L(B^*,\mathbb {R}^{V[g]})$ .

Now we address the question of whether the hypothesis of Corollary 1.5 is consistent. We construct divergent models of $\mathsf {AD}^+$ that contain the collection of universally Baire sets from a strong hypothesis. We are hopeful that with recent advancement in descriptive inner model theory, this hypothesis can be shown to be consistent.

Definition 1.6. Let ${\mathcal {M}}$ be a hybrid premouse. We say that ${\mathcal {M}}$ is appropriate premouse if ${\mathcal {M}} = (|{\mathcal {M}}|,\in , \mathbb {E}, \mathbb {S})$ is an amenable J-structure that satisfies $:$

1. 1. the predicate $\mathbb {S}$ codes $({\mathcal {P}}_0, \Sigma )$ , where ${\mathcal {P}}_0 = ({\mathcal {M}}|{\delta _0})^\sharp $ ² for some Woodin cardinal $\delta _0$ such that ${\mathcal {P}}_0$ is an lsa hod premouse and $\Sigma $ is the short-tree strategy of ${\mathcal {P}}_0;$ ³

2. 2. there is a proper class of Woodin cardinals and a Woodin limit of Woodin cardinals $> \delta _0$ as witnessed by a fine-extender sequence $($ in the sense of [7] $)$ coded by $\mathbb {E};$

3. 3. for any set generic h, $\Sigma $ has a canonical interpretation $\Sigma ^h$ in $V[h];$ more precisely, there is a term-relation $\tau $ such that for all generic h, $\tau ^h = \Sigma ^h;$

4. 4. in all generic extensions $V[g]$ of V for which ${\mathcal {P}}_0$ is countable, $\Sigma ^g\notin (\Gamma _\infty )^{V[g]}$ but letting $\Gamma ({\mathcal {P}}_0,\Sigma ^g)$ be the set of A such that there is a countable ${\mathcal {T}}$ according to $\Sigma ^g$ such that $A \leq _w \Sigma ^g_{{\mathcal {T}}, {\mathcal {M}}({\mathcal {T}})}$ , then $\Gamma ({\mathcal {P}}_0,\Sigma ^g) = (\Gamma _\infty )^{V[g]}$ . This essentially says that all lower-level strategies of $\Sigma ^g$ or its iterates are in $(\Gamma _\infty )^{V[g]}$ .

$({\mathcal {M}},\Psi )$ is an appropriate mouse if ${\mathcal {M}}$ is an appropriate premouse and $\Psi $ is an iteration strategy for ${\mathcal {M}}$ such that if $i: {\mathcal {M}}\rightarrow {\mathcal {N}}$ be an iteration according to $\Psi $ , then for any ${\mathcal {N}}$ -generic g, $i(\tau )^g = (\Psi _N)^{sh}_{{\mathcal {P}}_0}{\mathop{\restriction} } {\mathcal {N}}[g]$ , here $(\Psi _N)_{{\mathcal {P}}_0}^{sh}$ is the restriction of the tail strategy $\Psi _N$ on N to short trees on ${\mathcal {P}}_0$ .

It is not known if the existence of an appropriate mouse is consistent; a weaker version of this is shown to be consistent in [4] and plays a key role in determining the exact consistency strength of Woodin’s Sealing of the Universally Baire sets. Property 4, namely the assumption on $\Sigma $ , is an abstraction of properties of excellent mice defined in [4] and is the key property that allows us to prove Theorem 1.7. The intuition giving rise to 4 comes from the construction of models of $\mathsf {LSA - over - UB}$ in [4], where the $\mathsf {LSA}$ model is generated by a pair $({\mathcal {P}},\Sigma )$ such that $\Sigma $ is a short-tree strategy for an lsa-type hod premouse ${\mathcal {P}}$ and $\Gamma ({\mathcal {P}},\Sigma )=\Gamma _\infty $ . In the proof of Theorem 1.7, we use this property to show that $\Gamma _\infty $ (in a generic extension of the appropriate mouse) is in both divergent models, by showing the interpretation of $\tau $ by the generic is in both models. The main difference between an appropriate mouse and an excellent mouse lies in property 2. We do not yet have a theory of layered-hod mice that reaches the level of “ $\mathsf {ZFC} + $ there is a Woodin cardinal which is a limit of Woodin cardinals” $(\mathsf {WLW})$ , but such a theory exists for least-branch hod mice [8], so it seems very plausible that the existence of appropriate mice is consistent. ⁴

The following property abstracts out some of the features of countable substructures of models obtained by fully backgrounded constructions (see cf. [2, 7]). We say that V satisfies countable self-iterability if for any cardinal $\delta $ and any countable $X\prec V_{\delta +1}$ , the transitive collapse M of X is fully iterable with $\delta $ -universally Baire strategy $\Lambda $ ; furthermore, letting $\tau : M\rightarrow X$ be the uncollapse map, $\Lambda $ is $\tau $ -realizable, i.e., whenever $\pi : M \rightarrow N$ is an iteration map according to $\Lambda $ with $|N| < \omega _1$ , there is some $\sigma : N\rightarrow V_{\delta +1}$ such that $\tau = \sigma \circ \pi $ .

Theorem 1.7. Suppose $V = L[\vec {E}]$ is an extender model such that in V, there is a proper class of Woodin cardinals and countable self-iterability holds. Suppose there is an appropriate mouse $({\mathcal {M}},\Psi )$ such that $\Psi \in \Gamma _\infty $ . Then in some generic extension of ${\mathcal {M}}$ , there are divergent models of $\mathsf {AD}^+ N_1, N_2$ such that $\Gamma _\infty \subset N_1\cap N_2$ .

Remark 1.8. Theorem 1.7 relates to Question 5.1 $(i)$ in light of recent development in the core model induction $;$ in particular, one can show under $\mathsf {MM}$ that $\Gamma _\infty $ contains very complicated mice, e.g., there are Wadge initial segments $\Gamma $ such that $L(\Gamma )\models \mathsf {AD}_{\mathbb {R}} + "\Theta $ is regular” and much more. One can hope that $\mathsf {MM}$ implies the existence of mice that satisfies $\mathsf {WLW}$ with universally Baire iteration strategies. Question 5.1 $(ii)$ is a weakening of Question 5.1 $(i)$ as $\mathsf {MM}$ implies . If Question 5.1 $(ii)$ was true, then $\Gamma _\infty $ is “large” in that $o(\Gamma _\infty )> \omega _2$ . It is open whether $o(\Gamma _\infty )$ could be $>\omega _3$ .

2 Preliminaries

Let $\Theta $ be the supremum of ordinals $\gamma $ such that there is a surjection from $\mathbb {R}$ onto $\gamma $ . A very useful extension of the Axiom of Determinacy, $\mathsf {AD}$ , is a theory called $\mathsf {AD}^+$ isolated by Woodin. $\mathsf {AD}^+$ consists of the following statements.

• • $\mathsf {DC}_{\mathbb {R}}$ .

• • Every set of reals has an $\infty $ -Borel code. (An $\infty $ -Borel code is a pair $(S,\varphi )$ where S is a set of ordinals and $\varphi $ is a formula of set theory. Let $\mathfrak {B}_{(S,\varphi )} = \{r \in \mathbb {R} : L[S,r] \models \varphi (S,r)\}$ . $(S,\varphi )$ is an $\infty $ -Borel code for a set $A \subseteq \mathbb {R}$ if and only if $A = \mathfrak {B}_{(S,\varphi )}$ .)

• • $\mathsf {Ordinal\ Determinacy}$ , which is the statements that for every $\lambda < \Theta $ , $X \subseteq \mathbb {R}$ , and continuous function $\pi : {}^\omega \lambda \rightarrow \mathbb {R}$ , the two player game on $\lambda $ with payoff set $\pi ^{-1}(X)$ is determined.

It is conjectured that under $\mathsf {ZF + DC}_{\mathbb {R}}$ , $\mathsf {AD}$ implies $\mathsf {AD}^+$ . All known models of $\mathsf {AD}$ satisfy $\mathsf {AD}^+$ .

For any model M of $\mathsf {AD}^+$ , the ordinal $\Theta ^M$ is defined to be the supremum of ordinals $\gamma $ such that there is a surjection from $\mathbb {R}$ onto $\gamma $ in M. For any set of reals A in M, let $w(A)$ denote the Wadge rank of A in M. A basic result due to R. Solovay, is that $\Theta ^M$ is supremum of the Wadge ranks of sets of reals A in M.

We summarize basic facts about (weakly) homogeneously Suslin and universally Baire sets we need. For a more detailed discussion, the reader should consult for example [6].

Given an uncountable cardinal $\kappa $ , and a set Z, $meas_\kappa (Z)$ denotes the set of all $\kappa $ -additive measures on $Z^{<\omega }$ . If $\mu \in meas_\kappa (Z)$ , then there is a unique $n<\omega $ such that $Z^n\in \mu $ by $\kappa $ -additivity; we let this $n = dim(\mu )$ . If $\mu ,\nu \in meas_\kappa (Z)$ , we say that $\mu $ projects to $\nu $ if $dim(\nu ) = m \leq dim(\mu ) = n$ and for all $A\subseteq Z^m$ ,

$$ \begin{align*} A\in \nu \Leftrightarrow \{u : u{\mathop{\restriction}} m\in A\} \in \mu. \end{align*} $$

In this case, there is a natural embedding from the ultrapower of V by $\nu $ into the ultrapower of V by $\mu $ :

$$ \begin{align*} \pi_{\nu,\mu} : Ult(V,\nu) \rightarrow Ult(V,\mu) \end{align*} $$

defined by $\pi _{\nu ,\mu }([f]_\nu ) = [f^*]_\mu $ where $f^*(u) = f(u\mathop{\restriction} m)$ for all $u\in Z^n$ . A tower of measures on Z is a sequence $\langle \mu _n : n<k \rangle $ for some $k\leq \omega $ such that for all $m\leq n < k$ , $dim(\mu _n) = n$ and $\mu _n$ projects to $\mu _m$ . A tower $\langle \mu _n : n<\omega \rangle $ is countably complete if the direct limit of $\{Ult(V,\mu _n), \pi _{\mu _m,\mu _n}: m\leq n < \omega \}$ is well-founded. We will also say that the tower $\langle \mu _n : n<\omega \rangle $ is well-founded.

Recall we identify the set of reals $\mathbb {R}$ with the Baire space ${}^\omega \omega $ .

Definition 2.1. Fix an uncountable cardinal $\kappa $ . A function $\bar {\mu }: \omega ^{<\omega } \rightarrow meas_\kappa (Z)$ is a $\kappa $ -complete homogeneity system with support Z if for all $s,t\in \omega ^{<\omega }$ , writing $\mu _t$ for $\bar {\mu }(t):$

1. 1. $dom(\mu _t) = dom(t)$ ,

2. 2. $s\subseteq t \Rightarrow \mu _t$ projects to $\mu _s$ .

Often times, we will not specify the support $Z;$ instead, we just say $\bar {\mu }$ is a $\kappa $ -complete homogeneity system.

A set $A\subseteq \mathbb {R}$ is $\kappa $ -homogeneous iff there is a $\kappa $ -complete homogeneity system $\bar {\mu }$ such that

$$ \begin{align*} A = S_{\mu} =_{def} \{ x : \bar{\mu}_x \textrm{ is countably complete} \}. \end{align*} $$

A is homogeneous if it is $\kappa $ -homogeneous for all $\kappa $ . Let $\textrm {Hom}_\infty $ be the collection of all homogeneous sets.

Definition 2.2. Fix an uncountable cardinal $\kappa $ . A function $\bar {\mu }: \omega ^{<\omega } \rightarrow meas_\kappa (Z)$ is a $\kappa $ -complete weak homogeneity system with support Z if it is injective and for all $t\in \omega ^{<\omega }:$

1. 1. $dom(\mu _t) \leq dom(t)$ ,

2. 2. if $\mu _t$ projects to $\nu $ , then there is some $i<dom(\mu _t)$ such that $\nu = \mu _{t{\mathop{\restriction} } i}$ .

A set $A\subseteq \mathbb {R}$ is $\kappa $ -weakly homogeneous iff there is a $\kappa $ -complete weak homogeneity system $\bar {\mu }$ such that

$$ \begin{align*} A = W_{\bar{\mu}} =_{def} \{ x : \exists (i_k : k<\omega) \in \omega^\omega \langle \mu_{x|i_k} : k<\omega\rangle {is~well\text{-}founded}\}. \end{align*} $$

A is weakly homogeneous if it is $\kappa $ -weakly homogeneous for all $\kappa $ . Let $\textrm {wHom}_\infty $ be the collection of all weakly homogeneous sets.

Definition 2.3. $A\subseteq \mathbb {R}$ is $\kappa $ -universally Baire if there are trees $T,U \subseteq (\omega \times ON)^{<\omega }$ that are $\kappa $ -absolutely complemented, i.e., $A = p[T] = \mathbb {R}\backslash p[U]$ and whenever $\mathbb {P}$ is a forcing such that $|\mathbb {P}|<\kappa $ and $g\subseteq \mathbb {P}$ is V-generic, in $V[g]$ , $p[T] = \mathbb {R}\backslash p[U]$ . In this case, we let $A_g = p[T]$ be the canonical interpretation of A in $V[g]$ .

A is universally Baire if A is $\kappa $ -universally Baire for all $\kappa $ . Let $\Gamma _\infty $ be the collection of all universally Baire sets.

We remark that if A is $\kappa $ -universally Baire as witnessed by pairs $(T_1, U_1)$ and $(T_2, U_2)$ and $\mathbb {P}\in V_\kappa $ and $g\subset \mathbb {P}$ is V-generic, then $A_g = p[T_1] = p[T_2]$ , i.e., $A_g$ does not depend on the choice of absolutely complemented trees that witness A is $\kappa $ -universally Baire. A similar remark applies to $\kappa $ -(weakly) homogeneously Suslin sets.

Suppose there is a proper class of Woodin cardinals. The following are some standard results about universally Baire sets we will use throughout our paper. The proof of these results can be found in [6].

1. 1. Hom $_\infty = \textrm {wHom}_\infty = \Gamma _\infty $ .

2. 2. For any $A\in \Gamma _\infty $ , $L(A,\mathbb {R})\models \mathsf {AD}^+$ ; furthermore, given such an A, there is a $B\in \Gamma _\infty $ such that $B\notin L(A,\mathbb {R})$ and $A\in L(B,\mathbb {R})$ . In fact, $A^\sharp $ is an example of such a B.

3. 3. Suppose $A\in \Gamma _\infty $ . Let B be the code for the first order theory with real parameters of the structure $(HC,\in , A)$ (under some reasonable coding of $HC$ by reals). Then $B\in \Gamma _\infty $ and if g is V-generic for some forcing, then in $V[g]$ , $B_g\in \Gamma _\infty $ is the code for the first order theory with real parameters of $(HC^{V[g]},\in , A_g)$ .

Under the same hypothesis, the results above also imply that:

• • $\Gamma _\infty $ is closed under Wadge reducibility,

• • if $A\in \Gamma _\infty $ , then $\neg A\in \Gamma _\infty $ ,

• • if $A\in \Gamma _\infty $ and g is V-generic for some forcing, then there is an elementary embedding $j: L(A,\mathbb {R})\rightarrow L(A_g, \mathbb {R}_g)$ , where $\mathbb {R}_g = \mathbb {R}^{V[g]}$ .

Finally, the reader should consult [7] for the basics of inner model theory. This is the background needed to follow the proof of Theorem 1.7. Consult [4, 5] for more information on the theory of short-tree strategy mice related to lsa hod mice and appropriate mice; we will not need this material in this paper, however. In the following, we fix a natural coding of $(\omega _1,\omega _1)$ -iteration strategies for countable mice by sets of reals, e.g., we fix a function $\tau : HC\rightarrow \mathbb {R}$ that codes elements of $HC$ by reals as in [9, Chapter 2] and $Code: {\wp }(HC)\rightarrow {\wp }(\mathbb {R})$ is the induced function given by: $Code(A) = \tau [A]$ .

3 Divergent models of $\mathsf {AD}^+$ and the failure of $\mathsf {CH}$

Proof of Theorem 1.2

Fix $A,B, \mathbb {P}, g$ as in the statement of the theorem. Let $\mathbb {R}_g = \mathbb {R}^{V[g]}$ . Let $\alpha $ be the least such that letting $x_A$ be the $\alpha $ -th real in the canonical well-order of $H_A$ and $x_B$ be the $\alpha $ -th real in the canonical well-order of $H_B$ , then $x_A\neq x_B$ .

Let $(U,\varphi )$ and $(W,\psi )$ be $\infty $ -Borel codes for $A, B$ , respectively. Let $s\in ({\wp }_{\omega _1}(\omega _2))^{V[g]}$ . Note that s is added by a countable suborder of $\mathbb {P}$ by the countable chain condition of $\mathbb {P}$ . Let $\mathbb {R}_s = \mathbb {R}^{V[s]}$ and define $A_s$ by: for all $x\in \mathbb {R}_s$ ,

$$ \begin{align*} x\in A_s \Leftrightarrow L[U, x] \models \varphi[x, U]. \end{align*} $$

We define $B_s$ using $(W,\psi )$ in a similar fashion. Let

$$ \begin{align*} M_s = L(A_s, \mathbb{R}_s) \end{align*} $$

and

$$ \begin{align*} N_s = L(B_s, \mathbb{R}_s), \end{align*} $$

Claim 1: Suppose $t\in ({\wp }_{\omega _1}(\omega _2))^{V[g]}$ and $s\subseteq t$ . Then the map $\pi ^A_{s,t}: M_s \rightarrow M_t$ defined by: $\pi ^A_{s,t}{\mathop{\restriction} } \mathbb {R}_s\cup ON = id$ and $\pi ^A_{s,t}(A_s) = A_t$ is an elementary embedding. Similarly, $\pi ^B_{s,t}$ is an elementary embedding.

Proof We prove the statement for A. This follows from [9, Theorems 10.63 and 2.27–2.29] and [1, Theorems 6.3 and 6.4]. The key points are:

• • All sets of reals in $L(A,\mathbb {R})$ are $\aleph _1$ -universally Baire, as $(\mathbb {R},A)^\sharp $ is $\aleph _1$ -universally Baire.

• • The suborder of $\mathbb {P}$ adding s is weakly proper and countable, so $\pi ^A_{\emptyset ,s}{\mathop{\restriction} } ON = id$ and $\pi ^A_{\emptyset ,s}(A) = A_s$ is the canonical interpretation of A in $V[s]$ .

Let $M_\infty $ be the direct limit of $\mathcal {F}_A = \{M_s, \pi ^A_{s,t}: s\subseteq t \in ({\wp }_{\omega _1}(\omega _2))^{V[g]}\}$ and $N_\infty $ be the direct limit of $\mathcal {F}_B = \{N_s, \pi ^B_{s,t}: s\subseteq t \in ({\wp }_{\omega _1}(\omega _2))^{V[g]}\}$ .

Claim 2: $M_\infty , N_\infty $ are well-founded.

Proof The directed systems $\mathcal {F}_A, \mathcal {F}_B$ consist of well-founded models and the directed relation ( $\subseteq $ ) is in fact countably directed, i.e., if $(s_n : n < \omega )$ is such that for all n, $s_n\in ({\wp }_{\omega _1}(\omega _2))^{V[g]}$ , then there is some $s\in ({\wp }_{\omega _1}(\omega _2))^{V[g]}$ such that $s_n \subseteq s$ for all n. Therefore, $M_\infty , N_\infty $ are well-founded as any witness that $M_\infty $ ( $N_\infty $ ) is ill-founded has preimage in some $M_s$ ( $N_s$ ).

Let

$$ \begin{align*} \pi^A: L(A,\mathbb{R}) \rightarrow M_\infty = L(A_\infty, \mathbb{R}_g) \end{align*} $$

and

$$ \begin{align*} \pi^B: L(B,\mathbb{R}) \rightarrow M_\infty = L(B_\infty, \mathbb{R}_g) \end{align*} $$

be the direct ⁵ limit maps. Note that $\pi ^A{\mathop{\restriction} } ON = \pi ^B{\mathop{\restriction} } ON = id$ . Now we claim that $M_\infty , N_\infty $ are divergent models of $\mathsf {AD}^+$ in $V[g]$ . This finishes the proof of the theorem.

We note that $\pi ^A(x_A) = x_A$ is the $\alpha $ -th real in the canonical well order of $HOD^{M_\infty }$ . This follows from the fact that $\pi ^A$ is elementary and fixes all ordinals. Similarly, $\pi ^B(x_B) = x_B$ is the $\alpha $ -th real in the canonical well order of $HOD^{M_\infty }$ . If $M_\infty , N_\infty $ are compatible, then the $\alpha $ -th real in $HOD^{M_\infty }$ must be equal to the $\alpha $ -th real in $HOD^{N_\infty }$ . To see this, suppose without loss of generality ${\wp }(\mathbb {R})^{M_\infty }\subseteq {\wp }(\mathbb {R})^{N_\infty }$ . Suppose $\beta \leq \Theta ^{N_\infty }$ is such that ${\wp }(\mathbb {R})^{M_\infty } = \{A\in N_\infty : w(A) < \beta \}$ . This easily gives $HOD^{M_\infty }$ is $OD$ in $N_\infty $ and that the canonical well-order of $OD$ -reals in $M_\infty $ is compatible with the canonical well-order of $OD$ -reals in $N_\infty $ . So $x_A= x_B$ . Contradiction.⊣

Proof of Theorem 1.4

Fix $A, \mathbb {P}, g$ as in the statement of the theorem. Let $\kappa $ be a measurable cardinal such that:

• • $\mathbb {P}\in V_\kappa $ .

• • A is $\kappa $ -homogeneous.

• • Every $\kappa $ -homogeneously Suslin set in $V[g]$ is universally Baire in $V[g]$ .

Let $\bar {\mu } = (\mu _s : s \in \omega ^{<\omega })$ be a homogeneous system witnessing A is $\kappa $ -homogeneously Suslin, i.e.,

$$ \begin{align*} x\in A \Leftrightarrow (\mu_{x|i} : i < \omega)\ \mathrm{is\ countably\ complete}. \end{align*} $$

Since $\mathbb {P}\in V_\kappa $ , for each $s\in \omega ^{<\omega }$ , there is $\nu \in meas_\kappa (\kappa ^{|s|})$ in V such that $\nu ^* = \mu _s$ , where $\nu ^* = \{A\in V[g] : \exists B\in \nu (B\subseteq A)\}$ is the canonical extension of $\nu $ in $V[g]$ . By the weak properness of $\mathbb {P}$ , there is a countable set of measures $\sigma \subset meas_\kappa (\bigcup _n \kappa ^n)$ in V such that

$$ \begin{align*} \bar{\mu}\subseteq \sigma^* = \{\nu^* : \nu\in\sigma\}. \end{align*} $$

In V, let $\bar {\nu } = (\nu _s : s\in \omega ^{<\omega })$ be an enumeration of $\sigma $ such that:

1. (i) for each $s\in \omega ^{<\omega }$ , $\nu _s$ concentrates on $\kappa ^{|s|}$ ;

2. (ii) if $\nu _t$ projects to $\nu $ , then there is some $i < dom(\nu _t)$ such that $\nu _{t|i} = \nu $ .

Now define the following set B, which is just the $\kappa $ -homogeneously Suslin set given by $\bar {\nu }$ : for $x\in \mathbb {R}$ ,

$$ \begin{align*} x\in B \Leftrightarrow (\nu_{x|k} : k<\omega)\ \mathrm{is\ countably\ complete}. \end{align*} $$

Let $B^*$ be the canonical extension of B induced by $\bar {\nu ^*} = (\nu ^*_s : s\in \omega ^{<\omega })$ in $V[g]$ . Thus, $B^*$ is $\kappa $ -homogeneously Suslin and hence is universally Baire in $V[g]$ . Let $f: \omega ^{<\omega } \rightarrow \omega ^{<\omega }$ be

$$ \begin{align*} f(s) = t\ \text{where}\ t \text{ is such that } \mu_s = \nu_t^*. \end{align*} $$

By the properties of $\bar {\nu }$ and $\bar {\mu }$ , we have:

1. (a) $f(s)$ has the same length as s for every $s\in \omega ^{<\omega }$ .

2. (b) f is order preserving, i.e., if $s_0$ is an initial segment of $s_1$ , then $f(s_0)$ is an initial segment of $f(s_1)$ .

Let $\hat {f}:\mathbb {R}^{V[g]}\rightarrow \mathbb {R}^{V[g]}$ be the continuous map induced by f:

$$ \begin{align*} \hat{f}(x) = \bigcup_{i<\omega} f(x|i). \end{align*} $$

We have, for any $x\in \mathbb {R}^{V[g]}$ ,

$$ \begin{align*} \begin{split} x\in A & \Leftrightarrow (\mu_{x|i} : i<\omega) \textrm{ is countably complete}\\ & \Leftrightarrow (\nu^*_{f(x|i)} : i<\omega) \textrm{ is countably complete}\\ & \Leftrightarrow \hat{f}(x) \in B^*. \end{split} \end{align*} $$

Thus $\hat {f}$ witnesses A is Wadge reducible to $B^*$ .

Proof of Corollary 1.5

First note that $\mathbb {P}$ is weakly proper, so we can apply Theorem 1.4. Now note that

(1) $$ \begin{align} o(\Gamma_\infty)^{V[g]} = \textrm{sup}[j_A{\mathop{\restriction}} o(\Gamma_\infty^V)] = \textrm{sup}[j_B{\mathop{\restriction}} o(\Gamma_\infty^V)]. \end{align} $$

Here, $o(\Gamma _\infty )$ is the length of the Wadge prewellorder on $\Gamma _\infty $ . To see (1), note that for each $X\in \Gamma _\infty $ , $j_A(X), j_B(X) \in \Gamma _\infty ^{V[g]}$ ⁶ and is the canonical interpretation of X, so $j_A(X) = j_B(X)$ . Now apply Theorem 1.4 to see that $j_A{\mathop{\restriction} } \Gamma ^V_\infty = j_B{\mathop{\restriction} }\Gamma ^V_\infty $ is cofinal in $\Gamma _\infty ^{V[g]}$ .

Finally, for each $X\in \Gamma _\infty $ , X is Wadge reducible to A ( $X \leq _w A$ ) in $L(A,\mathbb {R})$ . To see this, note that $A\notin \Gamma _\infty $ . Otherwise, by the facts mentioned at the end of Section 2, there is some $C\in \Gamma _\infty $ such that $A \in L(C,\mathbb {R})$ ; furthermore, $C^\sharp \in \Gamma _\infty $ , so $C^\sharp \notin L(A,\mathbb {R})$ . This contradicts $\Gamma _\infty \subset L(A,\mathbb {R})$ . Since $A\notin \Gamma _\infty , \Gamma _\infty \subset L(A,\mathbb {R})$ , and $L(A,\mathbb {R})\models \mathsf {AD}^+$ , the claim is established.

By elementarity $j_A(X) \leq _w A^*$ . By (1), $\Gamma ^{V[g]}_\infty \subset L(A^*,\mathbb {R}^{V[g]})$ . Similarly, $\Gamma ^{V[g]}_\infty \subset L(B^*,\mathbb {R}^{V[g]})$ .

4 Divergent models of $\mathsf {AD}^+$ over UB

In this section, we give the proof of Theorem 1.7. The proof closely resembles Woodin’s original proof of the existence of divergent models of $\mathsf {AD}^+$ in [1, Section 6]; the reader is advised to consult that proof for details we omit here.

Let ${\mathcal {M}},\Psi $ be as in the statement of the theorem and assume this is a minimal such mouse. Let ${\mathcal {P}}_0 = ({\mathcal {M}}|\delta _0)^\sharp $ be as in clause 1 of Definition 1.6. Let $\lambda = \lambda ^{\mathcal {M}}> \delta _0$ be the Woodin limit of Woodin cardinals of ${\mathcal {M}}$ . Let $c\in V$ be a Cohen real over ${\mathcal {M}}$ and let $A\in \Gamma _\infty $ be such that c is $OD$ in $L(A,\mathbb {R})$ .

The existence of A follows from countable self-iterability and the argument in [1, Section 6.2]. We sketch a proof here. A codes a pair $(P,\Lambda {\mathop{\restriction} } HC)$ where P is the transitive collapse of a countable $X\prec V_{\delta +1}$ such that $c\in X$ and $\delta $ is large enough that $\delta $ -universally Baire sets are universally Baire, and $\Lambda $ is a $\delta $ -universally Baire strategy of P. P is an extender model since $V= L[\vec {E}]$ is an extender model. Therefore, A is universally Baire. So $L(A,\mathbb {R}) \models \mathsf {AD}^+$ . By replacing P by $Hull^P(\{c\})$ we may assume P projects to $\omega $ and $\Lambda $ is the unique iteration strategy for P. Since $c\in P$ , P is an extender model, and $\Lambda {\mathop{\restriction} } HC$ can be extended to a unique $\omega _1+1$ -iteration strategy for P in $L(A,\mathbb {R})$ , the direct limit of all countable nondropping iterates of M via $\Lambda $ is defined and is $OD$ in $L(A,\mathbb {R})$ and hence c is $OD$ in $L(A,\mathbb {R})$ .

We may and do choose A such that $Code(\Psi ) <_w A$ as witnessed by a real $x^*$ . ⁷ To see such an A exists, suppose $Code(\Psi ) = p[T] = \mathbb {R}\backslash p[U]$ , where $T,U$ are trees witnessing $Code(\Psi )$ is $\delta $ -universally Baire for some $\delta $ . By choosing A coding the first order theory of $(HC,\in , (P,\Lambda ))$ with real parameters such that:

• • P is the transitive collapse of some countable $X\prec V_{\gamma +1}$ and

• • $(T,U)\in X$ for $\gamma $ sufficiently large that $\Lambda $ , the strategy for P, is universally Baire,

we can compute $\Psi $ from A as follows. Note that $\Lambda $ exists by countable self-iterability and since $\Lambda \in \Gamma _\infty $ , so is A. Let $x\in Code(\Psi ) = p[T]$ , let $\pi : P \rightarrow N$ be the iteration map that is induced by a genericity iteration according to $\Lambda $ to make x generic for the extender algebra at the first Woodin cardinal of N; we assume the first Woodin cardinal is $< \gamma $ . Let $(T^*,U^*)$ be the image of $(T,U)$ under the transitive collapse map $\tau $ and $(\tilde {T},\tilde {U}) = \pi (T^*,U^*)$ . We claim that $N[x] \models x\in p[\tilde {T}]$ ; otherwise, since $\tilde {T},\tilde {U}$ are absolutely complemented for forcings of size the first Woodin cardinal of N, $N[x]\models x\in p[\tilde {U}]$ . Since $\Lambda $ is a $\tau $ -realizable strategy, there is an embedding $\sigma : N \rightarrow V_{\gamma +1}$ such that $\tau = \sigma \circ \pi $ . This easily gives $x\in p[U]$ . Contradiction. Similarly, if $x\in p[U]$ , then $N[x] \models x\in p[\tilde {U}]$ . The above calculations show that $Code(\Psi )$ is projective in $Code(\Lambda )$ : for any $x\in \mathbb {R}$ , $x\in Code(\Psi )$ if and only if there is a non-dropping, countable tree ${\mathcal {T}}$ with last model N according to $\Lambda $ such that letting $\pi : P\rightarrow N$ be the iteration map, $x\in p[\pi (T^*)]$ . By the choice of A, $Code(\Psi )$ is Wadge reducible to A.

Say c is the $\alpha $ -th real in the canonical well-order of $HOD^{L(A,\mathbb {R})}$ . Let $C=B^\sharp $ , where B codes the first order theory of $(HC, \in , A)$ with real parameters; again, $C\in \Gamma _\infty $ and hence $L(C,\mathbb {R})\models \mathsf {AD}^+$ . Let $\pi : {\mathcal {M}}\rightarrow {\mathcal {N}}$ be the map induced by a countable iteration according to $\Psi $ above ${\mathcal {P}}_0$ such that:

1. 1. letting $\lambda ^* = \pi (\lambda )$ , then $(C{\mathop{\restriction} } \lambda ^*,\mathbb {R}{\mathop{\restriction} } \lambda ^*)$ is in ${\mathcal {N}}[g]$ , where $g\in V$ is ${\mathcal {N}}$ -generic for $\pi (W^{\mathcal {M}}_\lambda ) =_{def} W^{\mathcal {N}}_{\lambda ^*}$ , the $\lambda ^*$ -generator extender algebra of ${\mathcal {N}}$ at $\lambda ^*$ , ⁸

2. 2. $\mathbb {R}\cap L[C{\mathop{\restriction} } \lambda ^*] = \mathbb {R}^{{\mathcal {N}}[g]}$ and $L(C{\mathop{\restriction} } \lambda ^*,\mathbb {R}{\mathop{\restriction} } \lambda ^*)\prec L(C,\mathbb {R})$ ,

3. 3. $c,x^* \in \mathbb {R}^{{\mathcal {N}}[g]}$ .

The proof of these items, making substantial use of the fact that $\lambda $ is Woodin limit of Woodin cardinals, is the same as in [1, Section 6.3]. So in ${\mathcal {N}}[g]$ , there is an $\aleph _1$ -universally Baire set A ⁹ and two reals $c,x$ such that:

1. 4. $L(A,\mathbb {R}) \models \mathsf {AD}^+$ ,

2. 5. c is Cohen over ${\mathcal {N}}$ and c is the $\alpha $ -th real in the canonical well-order of $HOD^{L(A,\mathbb {R})}$ ,

3. 6. $\pi (\tau )^g <_w A$ as witnessed by x. ¹⁰

We note that clauses 4 and 5 follow from clause 2; clause 6 follows from clause 3 and the choice of A.

Say $p\in g$ forces (4)–(6). Note that by appropriateness of ${\mathcal {N}}$ (clauses 3 and 4) and (6), in ${\mathcal {N}}[g]$ , $\Gamma _\infty \subset L(A,\mathbb {R})$ . Let $g_1\times g_2\subset W^{\mathcal {N}}_{\lambda ^*}\times W^{\mathcal {N}}_{\lambda ^*}$ be ${\mathcal {N}}$ -generic and contains $(p,p)$ . In ${\mathcal {N}}[g_1\times g_2]$ , for $i\in \{1,2\}$ , there is a triple $(A_i, c_i, x_i)$ satisfying (4)–(6) for ${\mathcal {N}}[g_i]$ . As in [1, Section 6.3] and the proof of Theorem 1.2, in ${\mathcal {N}}[g_1\times g_2]$ , there are sets $A_1^*, A_2^*$ and embeddings $\pi _i: L(A_i,\mathbb {R}^{{\mathcal {N}}[g_i]}) \rightarrow L(A_i^*,\mathbb {R}^{{\mathcal {N}}[g_1\times g_2]})$ that fix the ordinals.

By (6), we have that $\pi (\tau )^{{\mathcal {N}}[g_1\times g_2]} = \pi (\tau )^{{\mathcal {N}}[g_2\times g_1]} \in L(A_i^*,\mathbb {R}^{{\mathcal {N}}[g_1\times g_2]})$ for $i\in \{1,2\}$ . Therefore, by appropriateness,

(2) $$ \begin{align} \Gamma_\infty^{{\mathcal{N}}[g_1\times g_2]} \subset L(A_1^*,\mathbb{R}^{{\mathcal{N}}[g_1\times g_2]})\cap L(A_2^*,\mathbb{R}^{{\mathcal{N}}[g_1\times g_2]}). \end{align} $$

As in [1, Section 6.3], $\pi _1(c_1) = c_1 \neq \pi _2(c_2) = c_2$ as $c_1, c_2$ are mutually generic over ${\mathcal {N}}$ . So in ${\mathcal {N}}[g_1\times g_2]$

(3) $$ \begin{align} L(A_1^*,\mathbb{R}^{{\mathcal{N}}[g_1\times g_2]}), L(A_2^*,\mathbb{R}^{{\mathcal{N}}[g_1\times g_2]}) \textrm{ are divergent models of } \mathsf{AD}^+. \end{align} $$

By elementarity of $\pi $ applied to (2) and (3), in a generic extension of ${\mathcal {M}}$ , there are divergent models of $\mathsf {AD}^+ M_1, M_2$ such that $\Gamma _\infty \subset M_1\cap M_2$ .

Remark 4.1. We note in the construction above, letting g be a generic over ${\mathcal {M}}$ such that in ${\mathcal {M}}[g]$ there are divergent models $M_1,M_2$ as above, letting $\Delta = M_1\cap M_2\cap {\wp }(\mathbb {R})$ , then $\Gamma _\infty ^{{\mathcal {M}}[g]}\subsetneq \Delta $ . This is because $\tau _g\in M_1\cap M_2$ . By a result of Woodin, $L(\Delta )\cap {\wp }(\mathbb {R}) = \Delta $ and $L(\Delta )\models \mathsf {AD}_{\mathbb {R}}$ ; therefore, there are Suslin co-Suslin sets in $M_1\cap M_2$ that are not universally Baire.

5 Open questions

We collect some open problems concerning divergent models of $\mathsf {AD}^+$ . First, we do not know if divergent models of $\mathsf {AD}^+$ is consistent with or follows from various other strong hypotheses that imply $\mathsf {CH}$ fails.

Question 5.1.

1. 1. Does $\mathsf {MM}$ imply there are divergent models of $\mathsf {AD}^+$ ?

2. 2. Is the theory “there are divergent models of ” consistent?

One way to answer the following question is to show it is possible to construct appropriate mice.

Question 5.2. Is the theory “there is a proper class of Woodin cardinals and there are divergent models of $\mathsf {AD}^+ M$ and N such that $\Gamma _\infty \subset M\cap N$ ” consistent?