On the non-existence of $\kappa$-mad families

TL;DR

This paper demonstrates the non-existence of -mad families in a specific set-theoretic model constructed from a supercompact cardinal, advancing understanding of combinatorial set theory under weaker axioms.

Contribution

It constructs a model of ZF+DC_{} where -mad families do not exist, using large cardinal assumptions and forcing techniques.

Findings

No -mad families in the constructed model

Supports the consistency of non-existence of -mad families under ZF+DC_{}

Extends set theory by linking large cardinals to combinatorial properties

Abstract

Starting from a model with a Laver-indestructible supercompact cardinal $\kappa$, we construct a model of $ZF+DC_{\kappa}$ where there are no $\kappa$-mad families.

Full text

On the non-existence of $\kappa$-mad families

Haim Horowitz and Saharon Shelah

Abstract

Starting from a model with a Laver-indestructible supercompact cardinal $\kappa$, we construct a model of $ZF+DC_{\kappa}$ where there are no $\kappa$-mad families.111Date: June 12, 2019 2010 Mathematics Subject Classification: 03E15, 03E25, 03E35, 03E55 Keywords: Generalized descriptive set theory, mad families, supercompact cardinals Publication 1168 of the second author

Introduction

The study of the definability and possible non-existence of mad families has a long tradition, originating with the paper [Ma] of Mathias where it was proven that mad families can’t be analytic and that there are no mad families in the Solovay model constructed from a Mahlo cardinal. It was later shown by Toernquist that an inaccessible cardinal suffices for the consistency of this statement ([To]), and it was then shown by the authors that the non-existence of mad families (in $ZF+DC$) is actually equiconsistent with $ZFC$ ([HwSh:1090]).

The current paper can be seen as a continuation of the line of investigation of [HwSh:1090], as well as of [HwSh:1145], where the definability of $\kappa$-mad families was considered. Recall the following definition:

**Definition 1: **Let $\kappa$ be an infinite regular cardinal. A family $\mathcal{A}\subseteq[\kappa]^{\kappa}$ is $\kappa$-almost disjoint if $|A\cap B|<\kappa$ for every $A\neq B\in\mathcal{A}$. $\mathcal{A}$ will be called $\kappa$-maximal almost disjoint ($\kappa$-mad) if $\mathcal{A}$ is $\kappa$-almost disjoint and can’t be extended to a larger $\kappa$-almost disjoint family.

Assuming the existence of a Laver-indestructible supercompact cardinal $\kappa$, we constructed in [HwSh:1145] a generic extension where $\kappa$ remained supercompact and there are no $\Sigma^{1}_{1}(\kappa)-\kappa-$mad families, thus obtaining a higher analog of Mathias’ result.

Our current main goal is to obtain a higher analog of the main result of [HwSh:1090], i.e. for an uncountable cardinal $\theta>\aleph_{0}$, we would like to construct a model of $ZF+DC_{\theta}$ where there are no $\theta$-mad families. As opposed to [HwSh:1090], we only achieve this goal assuming the existence of a supercompact cardinal. The main result of the paper is the following:

**Theorem 2: **a. Suppose that $\aleph_{0}<cf(\theta)=\theta<cf(\kappa)=\kappa\leq\lambda=\lambda^{<\kappa}$ and $\theta$ is a Laver indestructible supercompact cardinal, then there is a model of $ZF+DC_{<\kappa}+"$there exist no $\theta$-mad families$"$.

b. If we start from a universe $V$, then the final model $V_{1}$ will have the same cardinals and same $H(\theta)$ as $V$.

We shall force with a partial order $\mathbb{P}$ where the conditions themselves are forcing notions (this is somewhat similar to [Sh:218], [HwSh:1093] and [HwSh:1113], as well as to the recent work of Viale in [Vi], where a similar approach is applied to the study of generic absoluteness). Forcing with $\mathbb{P}$ will generically introduce the forcing notion $\mathbb{Q}$ that will give us the desired results. More specifically, we shall fix a Laver-indestructible supercompact cardinal $\theta$. The conditions in $\mathbb{P}$ will be elements from a suitable $H(\lambda^{+})$ that are $(<\theta)$-support iterations along wellfounded partial orders of $(<\theta)$-directed closed forcing notions satisfying a strong version of $\theta^{+}$-cc. Given $\mathbb{q}_{1},\mathbb{q}_{2}\in\mathbb{P}$, we will have $\mathbb{q}_{1}\leq_{\mathbb{P}}\mathbb{q}_{2}$ when the iteration given by $\mathbb{q}_{1}$ is an “initial segment” (in an adequate sense) of the iteration given by $\mathbb{q}_{2}$. Forcing with $\mathbb{P}$ will introduce a generic iteration $\mathbb{q}_{G}$ given by the union of $\mathbb{q}\in\mathbb{P}$ that belong to the generic set. In the further generic extension given by $\mathbb{q}_{G}$, we shall consider $V_{1}=HOD(\mathcal{P}(\theta)^{<\kappa}\cup V)$ (for an adequate fixed $\kappa$). We shall then prove that there are no $\theta$-mad families in $V_{1}$. In order to prove this fact, we shall consider towards contradiction a condition $(\mathbb{q}_{0},\underset{\sim}{p_{0}})$ that forces a counterexample $\mathcal{A}$, where $\mathbb{q}_{0}$ will be “sufficiently closed”. The filter that’s dual to the ideal generated by $\mathcal{A}$ will then be extended to a $\theta$-complete ultrafilter (using the Laver-indestructibility of $\theta$), and we shall obtain a contradiction with the help of an amalgamation argument over $\mathbb{q}_{0}$ using a higher analog of Mathias forcing relative to this ultrafilter.

The rest of the paper will be devoted to the proof of Theorem 2.

Proof of the main result

**Definition 3: **A. Let $K$ be the class of $\mathbb{q}$ that consist of the following objects with the following properties:

a. $U=U_{\mathbb{q}}$ a well-founded partial order whose elements are ordinals. We let $U^{+}=U\cup\{\infty\}$ where $\infty$ is a new element above all elements from $U$, and for $\alpha\in U^{+}$, we let $U_{<\alpha}=\{\beta\in U:\beta<_{U}\alpha\}$.

b. An iteration $(\mathbb{P}_{\mathbb{q},\alpha},\underset{\sim}{\mathbb{Q}_{\mathbb{q},\beta}}:\alpha\in U^{+},\beta\in U)=(\mathbb{P}_{\alpha},\underset{\sim}{\mathbb{Q}_{\beta}}:\alpha\in U^{+},\beta\in U)$. We shall often denote the iteration itself by $\mathbb{q}$.

c. $\mathbb{q}$ is a $(<\theta)$-support iteration, and in addition:

$(\alpha)$ Each $\underset{\sim}{\mathbb{Q}_{\beta}}$ is a $\mathbb{P}_{\beta}$-name of a forcing notion whose set of elements is an object $X_{\beta}$ from $V$.

$(\beta)$ Given $\alpha\in U^{+}$, $p\in\mathbb{P}_{\alpha}$ iff $p$ is a function with domain $dom(p)\in[U_{<\alpha}]^{<\theta}$ such that $p(\beta)$ is a canonical $\mathbb{P}_{\beta}$-name for every $\beta\in dom(p)$.

$(\gamma)$ $\leq_{\mathbb{P}_{\alpha}}$ is defined as usual.

$(\delta)$ If $w\subseteq U$ is downward closed (i.e. $\alpha<_{U}\beta\in w\rightarrow\alpha\in w$) and $\mathbb{P}_{\mathbb{q},w}=\mathbb{P}_{w}=\mathbb{P}_{\infty}\restriction w=\{p\in\mathbb{P}_{\infty}:dom(p)\subseteq w\}$, then $\mathbb{P}_{w}\lessdot\mathbb{P}_{\infty}$.

d. In $V^{\mathbb{P}_{\beta}}$, $\underset{\sim}{\mathbb{Q}_{\beta}}$ satisfies $*_{\theta}^{\epsilon}$ for a fixed limit $\epsilon<\theta$, namely, if $\{p_{\alpha}:\alpha<\theta^{+}\}\subseteq\underset{\sim}{\mathbb{Q}_{\beta}}$, then there is some club $E\subseteq\theta^{+}$ and a pressing down function $f:E\rightarrow\theta^{+}$ such that if $\delta_{1},\delta_{2}\in E$, $cf(\delta_{1})=cf(\delta_{2})$ and $f(\delta_{1})=f(\delta_{2})$, then $p_{\delta_{1}}$ and $p_{\delta_{2}}$ have a common least upper bound.

e. For $\beta\in U$, the following holds in $V^{\mathbb{P}_{\beta}}$: If $I$ is a directed partial order of cardinality $<\theta$ and $(p_{s}:s\in I)\in\mathbb{Q}_{\beta}^{I}$ is $\leq_{\mathbb{Q}_{\beta}}$-increasing, then $\{p_{s}:s\in I\}$ has a $\leq_{\mathbb{Q}_{\beta}}$-least upper bound.

B. Let $\leq_{K}$ be the following partial order on $K$:

$\mathbb{q}_{1}\leq_{K}\mathbb{q}_{2}$ iff the following conditions hold:

a. $U_{\mathbb{q}_{1}}\subseteq U_{\mathbb{q}_{2}}$ as partial orders.

b. If $U_{\mathbb{q}_{2}}\models\alpha<\beta$ and $\beta\in U_{\mathbb{q}_{1}}$, then $\alpha\in U_{\mathbb{q}_{1}}$.

c. If $w\subseteq U_{\mathbb{q}_{1}}$ is downward closed, then $\mathbb{P}_{\mathbb{q}_{1},w}=\mathbb{P}_{\mathbb{q}_{2},w}$.

d. If $\alpha\in U_{\mathbb{q}_{1}}$, then $\underset{\sim}{\mathbb{Q}_{\mathbb{q}_{1},\alpha}}=\underset{\sim}{\mathbb{Q}_{\mathbb{q}_{2},\alpha}}$ (this is well-defined recalling clause (b)).

C. Let $K_{wf}$ be the class of $U$ as in (A)(a), and let $\leq_{wf}$ be the partial order on $K_{wf}$ defined as in clauses (B)(a) and (B)(b).

We shall now observe some easy basic properties of the objects defined above:

**Observation 4: **a. If $(U_{\alpha}:\alpha<\delta)$ is $\leq_{wf}$-increasing, then $\underset{\alpha<\delta}{\cup}U_{\alpha}$ is a $\leq_{wf}$-least upper bound for $(U_{\alpha}:\alpha<\delta)$.

b. $\leq_{K}$ is a partial order on $K$.

c. If $\mathbb{q}_{2}\in K$ and $U_{1}\subseteq U_{\mathbb{q}_{2}}$ is downward closed, then there is a unique $\mathbb{q}_{1}\in K$ such that $\mathbb{q}_{1}\leq_{K}\mathbb{q}_{2}$ and $U_{\mathbb{q}_{1}}=U_{1}$.

d. If $(\mathbb{q}_{\alpha}:\alpha<\delta)$ is $\leq_{K}$-increasing, then there is a unique $\mathbb{q}_{\delta}\in K$ such that $\alpha<\delta\rightarrow\mathbb{q}_{\alpha}\leq_{K}\mathbb{q}_{\delta}$ and $U_{\mathbb{q}_{\delta}}=\underset{\alpha<\delta}{\cup}U_{\mathbb{q}_{\alpha}}$.

e. If $U_{0},U_{1},U_{2}\in K_{wf}$, $U_{0}=U_{1}\cap U_{2}$ and $U_{0}\leq_{wf}U_{l}$ $(l=1,2)$, then there is a unique $U\in K_{wf}$ such that $\underset{l=1,2}{\wedge}U_{l}\leq_{wf}U$, $\alpha\in U$ iff $\alpha\in U_{1}\vee\alpha\in U_{2}$ and $\leq_{U}=\leq_{U_{1}}\cup\leq_{U_{2}}$. We denote this $U$ by $U_{1}+_{U_{0}}U_{2}$.

f. If $\mathbb{q}_{0},\mathbb{q}_{1},\mathbb{q}_{2}\in K$, $\mathbb{q}_{0}\leq_{K}\mathbb{q}_{l}$ $(l=1,2)$ and $U_{\mathbb{q}_{0}}=U_{\mathbb{q}_{1}}\cap U_{\mathbb{q}_{2}}$, then there is a unique $\mathbb{q}\in K$ such that $\underset{l=1,2}{\wedge}\mathbb{q}_{l}\leq_{K}\mathbb{q}$ and $U_{\mathbb{q}}=U_{\mathbb{q}_{1}}+_{U_{\mathbb{q}_{0}}}U_{\mathbb{q}_{2}}$. We shall denote this $\mathbb{q}$ by $\mathbb{q}_{1}+_{\mathbb{q}_{0}}\mathbb{q}_{2}$.

g. If $\alpha\in U_{\mathbb{q}}^{+}$, then $\mathbb{P}_{\mathbb{q},\alpha}$ is a $(<\theta)$-complete forcing satisfying $*_{\theta}^{\epsilon}$ (hence $\theta^{+}$-cc).

h. Suppose that $\mathbb{q}\in K$ and $\underset{\sim}{\mathbb{Q}}$ is a $\mathbb{P}_{\mathbb{q},\infty}$-name of a forcing notion whose universe is from $V$, such that the conditioncs of definitions 3(d) and 3(e) are satisfied, then there is $\mathbb{q}^{\prime}\in K$ such that $\mathbb{q}\leq_{K}\mathbb{q}^{\prime}$, $U_{\mathbb{q}^{\prime}}=U_{\mathbb{q}}\cup\{\gamma\}$, $U_{\mathbb{q}^{\prime}}\models\alpha<\gamma$ for every $\alpha\in U_{\mathbb{q}}$ and $\underset{\sim}{\mathbb{Q}_{\mathbb{q}^{\prime},\gamma}}=\underset{\sim}{\mathbb{Q}}$. $\square$

**Definition 5: **The forcing notion $\mathbb{P}$ will be defined as follows:

a. The conditions of $\mathbb{P}$ are the elements $\mathbb{q}$ of $K\cap H(\lambda^{+})$ such that $U_{\mathbb{q}}\subseteq\lambda^{+}$, and for every $\beta\in U_{\mathbb{q}}$, $\underset{\sim}{\mathbb{Q}_{\beta}}$ is a name for a forcing whose underlying set of conditions is some $X_{\beta}\subseteq\lambda^{+}$.

b. Given $\mathbb{q}_{1},\mathbb{q}_{2}\in\mathbb{P}$, $\mathbb{P}\models"\mathbb{q}_{1}\leq\mathbb{q}_{2}"$ iff $\mathbb{q}_{1}\leq_{K}\mathbb{q}_{2}$.

c. Given a generic set $G\subseteq\mathbb{P}$, we let $\mathbb{q}_{G}=\cup\{\mathbb{q}:\mathbb{q}\in G\}$.

**Claim 6: **a. $\mathbb{P}$ is $(<\kappa)$-strategically complete. Moreover, it’s $(<\lambda^{+})$-complete and $(<\theta)$-directed closed.

b. $\Vdash_{\mathbb{P}}"\mathbb{q}_{\underset{\sim}{G}}\in K"$, hence $\Vdash_{\mathbb{P}}"\mathbb{P}_{\mathbb{q}_{\underset{\sim}{G}},\infty}$ is $(<\theta)$-directed closed and $\theta^{+}$-cc$"$.

c. If $\delta<\lambda^{+}$, $cf(\delta)>\theta$ and $(\mathbb{q}_{\alpha}:\alpha<\delta)$ is $\leq_{\mathbb{P}}$-increasing, then $\mathbb{q}:=\underset{\alpha<\delta}{\cup}\mathbb{q}_{\alpha}$ belongs to $\mathbb{P}$ and $\mathbb{P}_{\mathbb{q}}=\underset{\alpha<\delta}{\cup}\mathbb{P}_{\mathbb{q}_{\alpha}}$. By $\theta^{+}$-c.c., $\underset{\sim}{a}$ is a canonical $\mathbb{P}_{\mathbb{q}}$-name of a member of $[\theta]^{\theta}$ iff $\underset{\sim}{a}$ is a canonical $\mathbb{P}_{\mathbb{q}_{\alpha}}$-name of a member of $[\theta]^{\theta}$ for some $\alpha<\delta$.

**Proof: **The claim follows directly from the definitions. The fact that $\Vdash_{\mathbb{P}}"\mathbb{q}_{\underset{\sim}{G}}\in K"$ follows from the general fact that if $I$ is a directed set, $\{\mathbb{q}_{t}:t\in I\}\subseteq K$ and $s\leq_{I}t\rightarrow\mathbb{q}_{s}\leq_{K}\mathbb{q}_{t}$, then $\cup\{\mathbb{q}_{t}:t\in I\}$ is well-defined and belongs to $K$. This also shows that $\mathbb{P}$ is $(<\theta)$-directed closed. $\square$

We shall now define our desired model:

**Definition 7: **a. In $V^{\mathbb{P}}$, let $\mathbb{Q}=\mathbb{P}_{\mathbb{q}_{\underset{\sim}{G}},\infty}$.

b. Let $V_{2}=V^{\mathbb{P}\star\underset{\sim}{\mathbb{Q}}}$.

c. Let $V_{1}$ be $HOD({\mathcal{P}(\theta)}^{<\kappa}\cup V)$ inside $V_{2}$.

**Claim 8: **a. $V_{1}\models ZF+DC_{<\kappa}$.

b. $(Ord^{<\kappa})^{V_{1}}=(Ord^{<\kappa})^{V_{2}}$, hence $\mathcal{P}(\theta)^{V_{1}}=\mathcal{P}(\theta)^{V_{2}}$.

**Proof: **We shall prove the first part of clause (b), the rest should be clear. Clearly, $(Ord^{<\kappa})^{V_{1}}\subseteq(Ord^{<\kappa})^{V_{2}}$. Now let $\eta\in(Ord^{\gamma})^{V_{2}}$ for some $\gamma<\kappa$, then $\eta=\underset{\sim}{\eta}[G]$ for some name $\underset{\sim}{\eta}$ of a member of $Ord^{\gamma}$, where $G\subseteq\mathbb{P}\star\underset{\sim}{\mathbb{Q}}$ is generic. $G=G_{1}\star G_{2}$ where $G_{1}\subseteq\mathbb{P}$ is generic and $G_{2}\subseteq\underset{\sim}{\mathbb{Q}}[G_{1}]$ is generic. Working in $V[G_{1}]$, $\underset{\sim}{\eta}/G_{1}$ is a $\underset{\sim}{\mathbb{Q}}[G_{1}]$-name. As $\underset{\sim}{\mathbb{Q}}[G_{1}]$ is $\theta^{+}$-cc, for every $\beta<\gamma$ there is a maximal antichain $\{p_{\beta,i}:i<\theta\}\subseteq\underset{\sim}{\mathbb{Q}}[G_{1}]$ of conditions that force a value to $\underset{\sim}{\eta}/G_{1}(\beta)$. Let $\{\zeta_{\beta,i}:i<\theta\}$ be the set corresponding values forced by the above conditions. Let $\Gamma=\{\underset{\sim}{p_{\beta,i}},\underset{\sim}{\zeta_{\beta,i}}:\beta<\gamma,i<\theta\}$ be the corresponding $\mathbb{P}$-names for the above objects (so we can regard them as $\mathbb{P}$-names for ordinals). As there are $<\kappa$ such names and $\mathbb{P}$ is $(<\kappa)$-strategically complete, there is a dense set of $\mathbb{q}\in\mathbb{P}$ that force values to all elements of $\Gamma$. Therefore, there is some $\mathbb{q}\in\mathbb{P}\cap G_{1}$ that forces values to all elements of $\Gamma$ (and the values forced are necessarily $\{p_{\beta,i},\zeta_{\beta,i}:\beta<\gamma,i<\theta\}$). It follows that $\{p_{\beta,i},\zeta_{\beta,i}:\beta<\gamma,i<\theta\}\in V$. In $V_{2}$, there is a function $f:\gamma\rightarrow\theta$ such that for every $\beta<\gamma$, $\eta(\beta)=\zeta_{\beta,f(\beta)}$. As $f\in\mathcal{P}(\theta)^{<\kappa}$ and $\{p_{\beta,i},\zeta_{\beta,i}:\beta<\gamma,i<\theta\}\in V$, it follows that $\eta\in V_{1}$. $\square$

**Main Claim 9: **There are no $\theta$-mad families in $V_{1}$.

The rest of the paper will be devoted to the proof of Claim 9.

Suppose towards contradiction that there is a $\theta$-mad family in $V_{1}$, so there is some $(\mathbb{q}_{0},\underset{\sim}{p_{0}})\in\mathbb{P}\star\underset{\sim}{\mathbb{Q}}$ forcing this statement about $\underset{\sim}{\mathcal{A}}$ where $\underset{\sim}{\mathcal{A}}$ is a canonical $\mathbb{P}\star\underset{\sim}{\mathbb{Q}}$-name of a $\theta$-mad family definable using $\underset{\sim}{\eta}$, and $\underset{\sim}{\eta}$ is a canonical $\mathbb{P}\star\underset{\sim}{\mathbb{Q}}$-name of a parameter (so $\underset{\sim}{\eta}=((\underset{\sim}{a_{\epsilon}}:\epsilon<\underset{\sim}{\epsilon(*)}),\underset{\sim}{x})$, where $\Vdash"\underset{\sim}{\epsilon(*)}<\kappa"$, each $\underset{\sim}{a_{\epsilon}}$ is a $\mathbb{P}\star\underset{\sim}{\mathbb{Q}}$-name of a subset of $\theta$ and $\Vdash"\underset{\sim}{x}\in V"$). Let $G_{0}\subseteq\mathbb{P}$ be generic over $V$ such that $\mathbb{q}_{0}\in G_{0}$. In $V[G_{0}]$, $\underset{\sim}{\eta}$ is a $\mathbb{P}_{\mathbb{q}_{G_{0}},\infty}$-name, and by increasing $\mathbb{q}_{0}$, we may assume wlog that $p_{0}:=\underset{\sim}{p_{0}}[G_{0}]\in\mathbb{P}_{\mathbb{q}_{0}}$, $x=\underset{\sim}{x}[G_{0}]\in V$, $\epsilon(*)=\underset{\sim}{\epsilon(*)}[G_{0}]\in\kappa$ and that each $\underset{\sim}{a_{\epsilon}}$ ($\epsilon<\epsilon(*)$) is a canonical $\mathbb{P}_{\mathbb{q}_{0}}$-name of a subset of $\theta$. Given $\mathbb{q}\in\mathbb{P}$ above $\mathbb{q}_{0}$, let $\mathcal{A}_{\mathbb{q}}$ be the set of canonical $\mathbb{P}_{\mathbb{q}}$-names $\underset{\sim}{a}$ such that $(\mathbb{q},\underset{\sim}{p_{0}})\Vdash_{\mathbb{P}\times\underset{\sim}{\mathbb{Q}}}"\underset{\sim}{a}\in\underset{\sim}{\mathcal{A}}"$, so $\mathbb{q}_{0}\leq\mathbb{q}_{1}\leq\mathbb{q}_{2}\rightarrow\mathcal{A}_{\mathbb{q}_{1}}\subseteq\mathcal{A}_{\mathbb{q}_{2}}$. Note that if $\mathbb{q}_{0}\leq\mathbb{q}_{1}$, $\mathbb{P}_{\mathbb{q}_{1},\infty}\models"p_{0}\leq p_{1}"$ and $(\mathbb{q}_{1},p_{1})\Vdash"\underset{\sim}{b}\in[\theta]^{\theta}"$, then for some $(\mathbb{q}_{2},\underset{\sim}{a})$ we have $\mathbb{q}_{1}\leq_{\mathbb{P}}\mathbb{q}_{2}$, $\underset{\sim}{a}\in\mathcal{A}_{\mathbb{q}_{2}}$ and $(\mathbb{q}_{2},p_{0})\Vdash"\underset{\sim}{b}\cap\underset{\sim}{a}\in[\theta]^{\theta}"$. By extending any given $\mathbb{q}_{1}\in\mathbb{P}$ above $\mathbb{q}_{0}$ in this way sufficiently many times to add witnesses for madness, and recalling Claim 6(c), we establish that the set $\{\mathbb{q}_{1}:\mathbb{q}_{0}\leq_{\mathbb{P}}\mathbb{q}_{1}$ and $\Vdash_{\mathbb{P}_{\mathbb{q}_{1}}}"\mathcal{A}_{\mathbb{q}_{1}}$ is $\theta$-mad$"\}$ is dense in $\mathbb{P}$ above $\mathbb{q}_{0}$.

Now, in $V_{2}$, let $I=\{A\subseteq\theta:A$ is contained in a union of $<\theta$ members of $\mathcal{A}\}$, then $I$ is a $\theta$-complete ideal and $\theta\notin I$. Let $F$ be the dual filter of $I$, then $F$ is $\theta$-complete, and as $\theta$ is supercompact in $V_{2}$ (recalling that $\theta$ is Laver indestructible and that $\mathbb{P}\star\underset{\sim}{\mathbb{Q}}$ is $(<\theta)$-directed closed), there is a $\mathbb{P}\star\underset{\sim}{\mathbb{Q}}$-name $\underset{\sim}{D}$ such that $(\mathbb{q}_{0},p_{0})\Vdash_{\mathbb{P}\star\underset{\sim}{\mathbb{Q}}}"\underset{\sim}{D}$ is a $\theta$-complete ultrafilter on $\theta$ that extends $F$, and hence is disjoint to $\underset{\sim}{\mathcal{A}}"$. By Claim 6 and a previous observation, we may assume wlog that $\mathbb{q}_{0}\Vdash_{\mathbb{P}}"\mathcal{A}_{\mathbb{q}_{0}}$ is $\theta$-mad and $\underset{\sim}{D_{\mathbb{q}_{0}}}:=\underset{\sim}{D}\cap\mathcal{P}(\theta)^{V^{\mathbb{P}_{\mathbb{q}_{0},\infty}}}$ is a $\mathbb{P}_{\mathbb{q}_{0},\infty}$-name of an ultrafilter on $\theta"$.

Given an ultrafilter $U$ on $\theta$, the forcing $\mathbb{Q}_{U}$ is defined as follows: the conditions of $\mathbb{Q}_{U}$ have the form $(u,A)$ where $u\in[\theta]^{<\theta}$ and $A\in U$. the order is defined naturally, i.e. $(u_{1},A_{1})\leq(u_{2},A_{2})$ iff $u_{1}\subseteq u_{2}$, $u_{2}\setminus u_{1}\subseteq A_{1}$ and $A_{2}\subseteq A_{1}$.

We may assume wlog that $\mathbb{P}_{\mathbb{q}_{0},\infty}$ forces $2^{\theta}=\lambda$, hence there is a canonical $\mathbb{P}_{\mathbb{q}_{0},\infty}$-name $\underset{\sim}{f}$ of a bijection from $\mathbb{Q}_{\underset{\sim}{D}_{\mathbb{q}_{0}}}$ onto $\lambda$. Let $\underset{\sim}{\mathbb{Q}^{\prime}}$ be a name for the forcing such that $\Vdash_{\mathbb{P}_{\mathbb{q}_{0}}}"\underset{\sim}{f}$ is an isomorphism from $\mathbb{Q}_{\underset{\sim}{D}_{\mathbb{q}_{0}}}$ onto $\underset{\sim}{\mathbb{Q}^{\prime}}"$. Let $\underset{\sim}{B}=\underset{\sim}{B_{\underset{\sim}{D}_{\mathbb{q}_{0}}}}$ be the $\mathbb{Q}_{\underset{\sim}{D}_{\mathbb{q}_{0}}}$-name $\cup\{u:(u,A)\in G_{\mathbb{Q}_{\underset{\sim}{D}_{\mathbb{q}_{0}}}}\}$, so $\Vdash_{\mathbb{P}_{\mathbb{q}_{0},\infty}\star\mathbb{Q}_{\underset{\sim}{D}_{\mathbb{q}_{0}}}}"\underset{\sim}{B}\in[\theta]^{\theta}$ is $\theta$-almost disjoint to $\mathcal{A}_{\mathbb{q}_{0}}"$. Let $\underset{\sim}{B}^{\prime}$ be the canonical $\mathbb{P}_{\mathbb{q}_{0},\infty}\star\mathbb{Q}_{\underset{\sim}{D}_{\mathbb{q}_{0}}}$-name for the image of $\underset{\sim}{B}$ under $\underset{\sim}{f}$.

Now observe that there is $\mathbb{q}^{\prime}\in\mathbb{P}$ such that $\mathbb{q}_{0}\leq_{\mathbb{P}}\mathbb{q}^{\prime}$, $U_{\mathbb{q}^{\prime}}=U_{\mathbb{q}_{0}}\cup\{\gamma\}$, $\alpha<_{U_{\mathbb{q}^{\prime}}}\gamma$ for every $\alpha\in U_{\mathbb{q}_{0}}$ and $\underset{\sim}{\mathbb{Q}_{\mathbb{q}^{\prime},\gamma}}=\underset{\sim}{\mathbb{Q}^{\prime}}$. As before, there is $\mathbb{q}^{\prime\prime}\in\mathbb{P}$ above $\mathbb{q}^{\prime}$ such that $p_{0}\Vdash_{\mathbb{P}_{\mathbb{q}^{\prime\prime},\infty}}"\mathcal{A}_{\mathbb{q}^{\prime\prime}}$ is $\theta$-mad$"$. Therefore, there is some canonical $\mathbb{P}_{\mathbb{q}^{\prime\prime}}$-name $\underset{\sim}{A}\in\mathcal{A}_{\mathbb{q}^{\prime\prime}}$ such that $p_{0}\Vdash_{\mathbb{P}_{\mathbb{q}^{\prime\prime},\infty}}"\underset{\sim}{A}\cap\underset{\sim}{B}^{\prime}\in[\theta]^{\theta}$, so $\underset{\sim}{A}$ has intersection of size $\theta$ with every member of $\underset{\sim}{D}_{\mathbb{q}_{0}}$ and $\underset{\sim}{A}\notin\mathcal{A}_{\mathbb{q}_{0}}"$.

Now let $(\mathbb{q}_{1},\underset{\sim}{B_{1}},\underset{\sim}{A_{1}})=(\mathbb{q}^{\prime\prime},\underset{\sim}{B^{\prime}},\underset{\sim}{A})$ and let $(\mathbb{q}_{2},\underset{\sim}{B_{2}},\underset{\sim}{A_{2}})$ be an isomorphic copy of $(\mathbb{q}_{1},\underset{\sim}{B_{1}},\underset{\sim}{A}_{1})$ over $\mathbb{q}_{0}$ such that $U_{\mathbb{q}_{1}}\cap U_{\mathbb{q}_{2}}=U_{\mathbb{q}_{0}}$ and $\mathbb{q}_{2}\in\mathbb{P}$.

**Claim 10: **Let $\mathbb{q}_{0}$, $(\mathbb{q}_{1},\underset{\sim}{B_{1}},\underset{\sim}{A_{1}})$ and $(\mathbb{q}_{2},\underset{\sim}{B_{2}},\underset{\sim}{A_{2}})$ be as above (so $\mathbb{q}_{0}\leq_{K}\mathbb{q}_{l}$ $(l=1,2)$, $U_{\mathbb{q}_{1}}\cap U_{\mathbb{q}_{2}}=U_{\mathbb{q}_{0}}$ and $\underset{l=1,2}{\wedge}\Vdash_{\mathbb{P}_{\mathbb{q}_{l},\infty}}"\underset{\sim}{A_{l}}\in\underset{\sim}{\mathcal{A}}\setminus\mathcal{A}_{\mathbb{q}_{0}}"$) and let $G\subseteq\mathbb{P}_{\mathbb{q}_{0},\infty}$ be generic over $V$, then $\Vdash_{\mathbb{P}_{\mathbb{q}_{1},\infty}/G\times\mathbb{P}_{\mathbb{q}_{2},\infty}/G}"\underset{\sim}{A_{2}}\setminus\underset{\sim}{A_{1}},\underset{\sim}{A_{1}}\setminus\underset{\sim}{A_{2}}\in[\theta]^{\theta}"$.

**Proof: **We shall prove the claim for $A_{2}\setminus A_{1}$, the other case is similar. Suppose towards contradiction that $(p_{1},p_{2})$ forces that $\underset{\sim}{A_{2}}\setminus\underset{\sim}{A_{1}}\subseteq\gamma<\theta$. For $l\in\{1,2\}$, let $B_{l}=\{\epsilon<\theta:p_{l}\nVdash_{\mathbb{P}_{\mathbb{q}_{l}},\infty/G}"\epsilon\notin\underset{\sim}{A_{l}}"\}\in V[G]$. By the assumption of the claim, $B_{l}\in[\theta]^{\theta}$. By the $\theta$-madness of $\underset{\sim}{\mathcal{A}_{0}}[G]$ in $V[G]$, there is some $Y\in\underset{\sim}{\mathcal{A}_{0}}[G]$ such that $|Y\cap B_{2}|=\theta$. As $p_{1}\Vdash_{\mathbb{P}_{\mathbb{q}_{1},\infty}/G}"|\underset{\sim}{A_{1}}\cap Y|<\theta"$, there are $q_{1}$ and $\beta_{1}<\theta$ such that $p_{1}\leq q_{1}\in\mathbb{P}_{\mathbb{q}_{1},\infty}/G$ and $q_{1}\Vdash_{\mathbb{P}_{\mathbb{q}_{1},\infty}/G}"\underset{\sim}{A_{1}}\cap Y\subseteq\beta_{1}"$. Let $\beta_{2}\in Y\cap B_{2}$ such that $max\{\gamma,\beta_{1}\}<\beta_{2}$ (recalling that $|Y\cap B_{2}|=\theta$). By the definition of $B_{2}$, there is $q_{2}\in\mathbb{P}_{\mathbb{q}_{2},\infty}/G$ above $p_{2}$ that forces $"\beta_{2}\in\underset{\sim}{A_{2}}"$. Therefore, $(p_{1},p_{2})\leq(q_{1},q_{2})\in\mathbb{P}_{\mathbb{q}_{1},\infty}/G\times\mathbb{P}_{\mathbb{q}_{2},\infty}/G$ and $(q_{1},q_{2})\Vdash_{\mathbb{P}_{\mathbb{q}_{1},\infty}/G\times\mathbb{P}_{\mathbb{q}_{2},\infty}/G}"\beta_{2}\in\underset{\sim}{A_{2}}\setminus\underset{\sim}{A_{1}}"$, a contradiction. It follows that $\Vdash_{\mathbb{P}_{\mathbb{q}_{1},\infty}/G\times\mathbb{P}_{\mathbb{q}_{2},\infty}/G}"\underset{\sim}{A_{2}}\setminus\underset{\sim}{A_{1}}\in[\theta]^{\theta}"$. $\square$

**Claim 11: **Under the assumptions of Claim 10 (recalling that $\Vdash_{\mathbb{P}_{\mathbb{q}_{l},\infty}}"\underset{\sim}{A_{l}}\cap B\neq\emptyset$ for every $B\in\underset{\sim}{D_{\mathbb{q}_{0}}}"$ $(l=1,2)$), we have $\Vdash_{\mathbb{P}_{\mathbb{q}_{1},\infty}/G\times\mathbb{P}_{\mathbb{q}_{2},\infty}/G}"\underset{\sim}{A_{1}}\cap\underset{\sim}{A_{2}}\in[\theta]^{\theta}"$.

**Proof: **Assume towards contradiction that $(p_{1},p_{2})\in\mathbb{P}_{\mathbb{q}_{1},\infty}/G\times\mathbb{P}_{\mathbb{q}_{2},\infty}/G$ forces that $\underset{\sim}{A_{1}}\cap\underset{\sim}{A_{2}}\subseteq\gamma$ for some $\gamma<\theta$. It’s forced by $(p_{1},p_{2})$ that $\underset{\sim}{A_{l}}\subseteq B_{l}$ $(l=1,2)$ where $B_{l}$ is as in the proof of the previous claim, hence it’s forced by $(p_{1},p_{2})$ that each $B_{l}$ intersects each member of $\underset{\sim}{D_{\mathbb{q}_{0}}}$. As $B_{1},B_{2}\in V[G]$, it follows that $B_{1},B_{2}\in\underset{\sim}{D_{\mathbb{q}_{0}}}[G]$. Therefore, there is some $\beta\in(B_{1}\cap B_{2})\setminus\gamma$, hence there is $q_{l}\in\mathbb{P}_{\mathbb{q}_{l},\infty}/G$ above $p_{l}$ that forces $"\beta\in\underset{\sim}{A_{l}}"$ $(l=1,2)$. It follows that $(p_{1},p_{2})\leq(q_{1},q_{2})\in\mathbb{P}_{\mathbb{q}_{1},\infty}/G\times\mathbb{P}_{\mathbb{q}_{2},\infty}/G$ and $(q_{1},q_{2})\Vdash_{\mathbb{P}_{\mathbb{q}_{1},\infty}/G\times\mathbb{P}_{\mathbb{q}_{2},\infty}/G}"\beta\in\underset{\sim}{A_{1}}\cap\underset{\sim}{A_{2}}"$, contradicting the choice of $\gamma$ and $(p_{1},p_{2})$. It follows that $\Vdash_{\mathbb{P}_{\mathbb{q}_{1},\infty}/G\times\mathbb{P}_{\mathbb{q}_{2},\infty}/G}"\underset{\sim}{A_{1}}\cap\underset{\sim}{A_{2}}\in[\theta]^{\theta}"$. $\square$

Now given $\mathbb{q}_{0}$, $(\mathbb{q}_{1},\underset{\sim}{B_{1}},\underset{\sim}{A_{1}})$ and $(\mathbb{q}_{2},\underset{\sim}{B_{2}},\underset{\sim}{A_{2}})$ as above, let $\mathbb{q}_{3}=\mathbb{q}_{1}+_{\mathbb{q}_{0}}\mathbb{q}_{2}$. Then $\mathbb{q}_{3}\in\mathbb{P}$, $\mathbb{q}_{1},\mathbb{q}_{2}\leq_{K}\mathbb{q}_{3}$, and by claims 10 and 11, we get a contradiction. This completes the proof of Main Claim 9 and hence of Theorem 2. $\square$

We conclude with the following natural question:

**Question: **What’s the consistency strength of $ZF+DC_{\theta}+"$there are no $\theta$-mad families$"$ for some $\theta>\aleph_{0}$?

References

[HwSh:1090] Haim Horowitz and Saharon Shelah, Can you take Toernquist’s inaccessible away? arXiv:1605.02419

[HwSh:1093] Haim Horowitz and Saharon Shelah, Transcendence bases, well-orderings of the reals and the axiom of choice, arXiv:1901.01508

[HwSh:1113] Haim Horowitz and Saharon Shelah, Madness and regularity properties, arXiv:1704.08327

[HwSh:1145] Haim Horowitz and Saharon Shelah, $\kappa$-Madness and Definability, arXiv:1805.07048

[Ma] A. R. D. Mathias, Happy families, Ann. Math. Logic **12 **(1977), no. 1, 59-111

[Sh:218] Saharon Shelah, On measure and category, Israel J. Math. 52 (1985) 110-114

[To] Asger Toernquist, Definability and almost disjoint families, Advances in Mathematics 330, 61-73, 2018

[Vi] Matteo Viale, Category forcings, $MM^{+++}$, and generic absoluteness for the theory of strong forcing axioms, J. Amer. Math. Soc. 29 (2016), no. 3, 675-728

(Haim Horowitz) Department of Mathematics

University of Toronto

Bahen Centre, 40 St. George St., Room 6290

Toronto, Ontario, Canada M5S 2E4

E-mail address: haim@math.toronto.edu

(Saharon Shelah) Einstein Institute of Mathematics

Edmond J. Safra Campus,

The Hebrew University of Jerusalem.

Givat Ram, Jerusalem 91904, Israel.

Department of Mathematics

Hill Center - Busch Campus,

Rutgers, The State University of New Jersey.

110 Frelinghuysen Road, Piscataway, NJ 08854-8019 USA

E-mail address: shelah@math.huji.ac.il