Mad families and non-meager filters

TL;DR

This paper proves the consistency of the existence of a non-meager filter on ω alongside the non-existence of mad families within ZF+DC+ZFC, addressing a question by Neeman and Norwood.

Contribution

It introduces a weaker form of madness and strengthens previous results by showing no mad families exist in the constructed model.

Findings

Existence of a non-meager filter on ω is consistent with no mad families.

A weaker version of madness is introduced and analyzed.

The result strengthens previous work by eliminating mad families in the model.

Abstract

We prove the consistency of ZF+DC+"there are no mad families"+"there exists a non-meager filter on $\omega$" relative to ZFC, answering a question of Neeman and Norwood. We also introduce a weaker version of madness, and we strengthen the result from [HwSh:1090] by showing that no such families exist in our model.

Full text

Mad families and non-meager filters

Haim Horowitz and Saharon Shelah

Abstract

We prove the consistency of $ZF+DC+"$there are no mad families$"+"$there exists a non-meager filter on $\omega"$ relative to $ZFC$, answering a question of Neeman and Norwood. We also introduce a weaker version of madness, and we strengthen the result from [HwSh:1090] by showing that no such families exist in our model.111Date: January 11, 2017 2000 Mathematics Subject Classification: 03E35, 03E15, 03E25 Keywords: mad families, non-meager filters, amalgamation Publication 1103 of the second author Partially supported by European Research Council grant 338821.

Introduction

This paper is a continuation of [HwSh:1090], which is part of the ongoing effort to investigate the possible non-existence and definability of mad families. In [HwSh:1090] we proved that $ZF+DC+"$there are no mad families$"$ is equiconsistent with $ZFC$ (previous results by Mathias and Toernquist established the consistency of that statement relative to large cardinals, see [Ma1] and [To]). In this paper we extend our results from [HwSh:1090] to address the following question by Neeman and Norwood:

**Question ([NN]): **If there are no mad families, does it follow that every filter is meager?

By a result of Mathias ([Ma2]), if every set of reals has the Ramsey property, then every filter is meager.

We shall construct a model of $ZF+DC$ where there are no mad families, but there is a non-meager filter on $\omega$. Our proof relies heavily on [HwSh:1090], the main change is that now we’re dealing with a class $K_{2}$ consisting of pairs $(\mathbb{P},\mathcal{A})$ such that $\mathbb{P}$ is ccc and forces $MA_{\aleph_{1}}$, and in addition, $\mathbb{P}$ forces that $\mathcal{A}$ is independent (we shall require more, see definition 2). In order to imitate the proof from [HwSh:1090], we need to prove analogous amalgamation results for an appropriate subclass of $K_{2}$. As in [HwSh:1090], our final model is obtained by forcing with $\mathbb{P}$ where $(\mathbb{P},\mathcal{A})$ is a “very large” object in a subclass of $K_{2}$, and the non-meager filter will be constructed from $\mathcal{A}$, which should contain many Cohen reals.

Finally, we consider the notion of nearly mad families (see definition 14), which was also introduced in [NN]. We introduce the notion of a somewhat mad family, which includes both mad and nearly mad families, and we prove that no somewhat mad families exist in our model.

A non-meager filter without mad families

**Hypothesis 1: **We fix $\mu$ and $\lambda$ such that $\aleph_{2}\leq\mu$, $\lambda=\lambda^{<\mu}$, $\mu=cf(\mu)$ and $\alpha<\mu\rightarrow|\alpha|^{\aleph_{1}}<\mu$.

**Definition 2: A. **Let $K_{2}$ be the class of $\mathbb{k}$ such that:

a. Each $\mathbb{k}$ has the form $(\mathbb{P},\mathcal{A})=(\mathbb{P}_{\mathbb{k}},\mathcal{A_{\mathbb{k}}})$.

b. $\mathbb{P}$ is a ccc forcing such that $\Vdash_{\mathbb{P}}MA_{\aleph_{1}}$.

c. $\mathcal{A}$ is a set of canonical $\mathbb{P}$-names of subsets of $\omega$.

d. $\Vdash_{\mathbb{P}}"\mathcal{A}$ is independent, i.e. every finite non-trivial Boolean combination of elements of $\mathcal{A}$ is infinite$"$.

B. For $\mathbb{k}\in K_{2}$ and a $\mathbb{P}_{\mathbb{k}}$-name $\underset{\sim}{b}$, let $\Vdash_{\mathbb{P}_{\mathbb{k}}}"\underset{\sim}{b}\in pos(\mathbb{k})"$ mean $\Vdash_{\mathbb{P}_{\mathbb{k}}}"\underset{\sim}{b}\in[\omega]^{\omega}$and there is no non-trivial Boolean combination of sets from $\mathcal{A}_{\mathbb{k}}$ that is almost disjoint to $\underset{\sim}{b}"$.

C. Let $\leq_{1}$ be the following partial order on $K_{2}$: $\mathbb{k}_{1}\leq_{1}\mathbb{k}_{2}$ if and only if:

a. $\mathbb{P}_{\mathbb{k}_{1}}\lessdot\mathbb{P}_{\mathbb{k}_{2}}$.

b. $\mathcal{A}_{\mathbb{k}_{1}}\subseteq\mathcal{A}_{\mathbb{k}_{2}}$.

D. Let $\leq_{2}$ be the following partial order on $K_{2}$:

$\mathbb{k}_{1}\leq_{2}\mathbb{k}_{2}$ if and only if:

a. As in B(a).

b. As in B(b).

c. If $\underset{\sim}{b}$ is a $\mathbb{P}_{\mathbb{k}_{1}}$-name then $\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}}"\underset{\sim}{b}\in pos(\mathcal{A}_{\mathbb{k}_{1}})"$ implies $\Vdash_{\mathbb{P}_{\mathbb{k}_{2}}}"\underset{\sim}{b}\in pos(\mathcal{A}_{\mathbb{k}_{2}})"$.

E. Let $K_{2}^{+}$ be the class of $\mathbb{k}\in K_{2}$ such that $\Vdash_{\mathbb{P}_{\mathbb{k}}}"\mathcal{A}_{\mathbb{k}}$ is a maximal independent set everywhere$"$, where $\mathcal{A}$ is a maximal independent set everywhere if for every $a_{0},...,a_{n-1}\in\mathcal{A}$ without repetition, $b:=\underset{l<n}{\cap}a_{l}^{\text{if l is even}}\in[\omega]^{\omega}$ and $\{a\cap b:a\in\mathcal{A}\setminus\{a_{0},...,a_{n-1}\}\}$ is a maximal independent set in $[b]^{\omega}$.

F. When we write $"\underset{\sim}{a_{0}},...,\underset{\sim}{a_{n-1}}\in\mathcal{A}"$, we mean that $(\underset{\sim}{a_{i}}:i<n)$ is without repetition, moreover, $i<j<n\rightarrow\Vdash_{\mathbb{P}}"\underset{\sim}{a_{i}}\neq\underset{\sim}{a_{j}}$.

Observation 3: a. $\leq_{1}$ and $\leq_{2}$ are partial orders, and if $\mathbb{k}_{1},\mathbb{k}_{2}\in K_{2}^{+}$ then $\mathbb{k}_{1}\leq_{1}\mathbb{k}_{2}\rightarrow\mathbb{k}_{1}\leq_{2}\mathbb{k}_{2}$.

b. If $\mathbb{k}_{1}\leq_{2}\mathbb{k}_{2}$ and $\underset{\sim}{b}$ is a $\mathbb{P}_{\mathbb{k}_{1}}$-name, then $\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}}"\underset{\sim}{b}\in pos(\mathcal{A}_{\mathbb{k}_{1}})"$ iff $\Vdash_{\mathbb{P}_{\mathbb{k}_{2}}}"\underset{\sim}{b}\in pos(\mathcal{A}_{\mathbb{k}_{2}})"$.

**Proof: **We shall prove the second claim of 3(a), everything else should be clear. Suppose that $\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}}"\underset{\sim}{b}\in pos(\mathcal{A}_{\mathbb{k}_{1}})"$, but for some $\underset{\sim}{a_{0}},...,\underset{\sim}{a_{n-1}}\in\mathcal{A}_{\mathbb{k}_{2}}$ and $p\in\mathbb{P}_{\mathbb{k}_{2}}$, $p\Vdash_{\mathbb{P}_{\mathbb{k}_{2}}}"\underset{\sim}{b}\cap(\underset{l<n}{\cap}\underset{\sim}{a_{l}^{\text{l is even}}})$ is finite$"$. Let $G\subseteq\mathbb{P}_{\mathbb{k}_{1}}$ be generic over $V$ such that $p\in G$ and we shall work over $V[G]$. WLOG there is $n_{1}<n$ such that $\underset{\sim}{a_{l}}\in\mathcal{A}_{\mathbb{k}_{1}}$ iff $l<n_{1}$, and denote $\underset{\sim}{a_{*}}=\underset{l<n_{1}}{\cap}\underset{\sim}{a_{l}^{\text{l is even}}}$. As $\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}}"\underset{\sim}{b}\in pos(\mathcal{A}_{\mathbb{k}_{1}})"$, it follows that $\underset{\sim}{b}\cap\underset{\sim}{a_{*}}$ is infinite. It’s enough to show that for some Boolean combination $\underset{\sim}{a_{**}}$ from $\mathcal{A}_{\mathbb{k}_{1}}$, $\underset{\sim}{a_{**}}\subseteq\underset{\sim}{a_{*}}$ and $\underset{\sim}{a_{**}}\subseteq^{*}\underset{\sim}{b}$, as then $\underset{\sim}{a_{**}}\cap(\underset{n_{1}\leq l<n}{\cap}\underset{\sim}{a_{l}^{\text{l is even}}})\subseteq^{*}\underset{\sim}{b}\cap(\underset{l<n}{\cap}\underset{\sim}{a_{l}^{\text{l is even}}})$, and therefore it’s finite, contradicting the definition of $\mathcal{A}_{\mathbb{k}_{2}}$. As $\mathbb{k}_{1}\in K_{2}^{+}$, it follows that $\{\underset{\sim}{a_{*}}\cap\underset{\sim}{c}:\underset{\sim}{c}\in\mathcal{A}_{\mathbb{k}_{1}}\setminus\{\underset{\sim}{a_{l}}:l<n_{1}\}\}$ is a maximal independent set in $[\underset{\sim}{a_{*}}]^{\omega}$, hence there are $\underset{\sim}{c_{0}},...,\underset{\sim}{c_{m-1}}\in\mathcal{A}_{\mathbb{k}_{1}}\setminus\{\underset{\sim}{a_{l}}:l<n_{1}\}$ such that $(\underset{l<n_{1}}{\cap}\underset{\sim}{a_{l}^{\text{if l is even}}})\cap(\underset{k<m}{\cap}\underset{\sim}{c_{m}^{\text{if m is even}}})\subseteq^{*}\underset{\sim}{b}$, so $\underset{\sim}{a_{**}}=(\underset{l<n_{1}}{\cap}\underset{\sim}{a_{l}^{\text{if l is even}}})\cap(\underset{k<m}{\cap}\underset{\sim}{c_{m}^{\text{if m is even}}})$ is as required. $\square$

Observation 4: $\mathbb{k}_{1}\leq_{2}\mathbb{k}_{2}$ and $\mathbb{k}_{1}\in K_{2}^{+}$ when the following hold for some $\kappa$:

a. $\mathbb{k}_{2}\in K_{2}^{+}$.

b. $\mathbb{k}_{2}\in H(\kappa)$.

c. $M$ is a model such that $\mathbb{k}_{2}\in M\prec_{\mathcal{L}_{\aleph_{2},\aleph_{2}}}(H(\kappa),\in)$.

d. $\mathbb{k}_{1}=\mathbb{k}_{2}^{M}$.

**Proof: **By observation 3, recalling that $"\mathbb{P}\models ccc"$ and $"\mathbb{P}\models MA_{\aleph_{1}}"$ are $\mathcal{L}_{\aleph_{2},\aleph_{2}}$-expressible .

**Claim 5: **For every $\mathbb{k}\in K_{2}$ there is $\mathbb{k}^{\prime}\in K_{2}^{+}$ such that $\mathbb{k}\leq_{1}\mathbb{k}^{\prime}$. Moreover, if $|\mathbb{P}_{\mathbb{k}}|<\mu$ then we can find such $\mathbb{k}^{\prime}$ that satisfies $|\mathbb{P}_{\mathbb{k}^{\prime}}|<\mu$.

**Proof: **If $|\mathbb{P}_{\mathbb{k}}|<\mu$, let $\lambda_{*}=\mu$, otherwise, let $\lambda_{*}$ be a regular cardinal greater than $(2+|\mathbb{P}_{\mathbb{k}}|)^{\aleph_{1}}$, such that $\alpha<\lambda_{*}\rightarrow|\alpha|^{\aleph_{1}}<\lambda_{*}$.

we try to choose a sequence $(\mathbb{k}_{\alpha}:\alpha<\lambda_{*})$ by induction on $\alpha<\lambda_{*}$ such that:

1. $\mathbb{k}_{0}=\mathbb{k}$.

2. $(\mathbb{k}_{\beta}:\beta\leq\alpha)$ is an increasing continuous sequence of members of $K_{2}$ (with respect to $\leq_{1}$).

3. $|\mathbb{P}_{\mathbb{k}_{\alpha}}|<\mu$.

4. For every $\alpha<\lambda_{*}$, if $\mathbb{k}_{\alpha}\notin K_{2}^{+}$, we choose a Boolean combination $\underset{\sim}{a_{\alpha}}$ from $\mathcal{A}_{\mathbb{k}_{\alpha}}$ and $\underset{\sim}{b_{\alpha}}\subseteq\underset{\sim}{a_{\alpha}}$ witnessing the failure of the condition from Definition 2(E). We then define $\mathbb{P}_{\mathbb{k}_{\alpha+1}}$ as an extension (with respect to $\lessdot$) of $\mathbb{P}_{\mathbb{k}_{\alpha}}\star Cohen$ to a ccc forcing that forces $MA_{\aleph_{1}}$, we let $\underset{\sim}{\eta_{\alpha}}$ be the relevant Cohen generic real and we let $\mathcal{A}_{\mathbb{k}_{\alpha+1}}=\mathcal{A}_{\mathbb{k}\alpha}\cup\{\underset{\sim}{b_{\alpha}}\cup\underset{\sim}{\eta_{\alpha}^{-1}}(\{1\})\}$.

5. If $\alpha<\lambda_{*}$ is a limit ordinal, we define $\mathbb{k}_{\alpha}$ as in the proof of claim 7 below.

Why can we carry the induction at stage $\alpha+1$ for $\alpha$ as in (4)? We shall prove that for each $\alpha$, $\Vdash_{\mathbb{P}_{\mathbb{k}_{\alpha+1}}}"\mathcal{A}_{\mathbb{k}_{\alpha+1}}$ is independent$"$. Let $\underset{\sim}{a_{\alpha}^{*}}=\underset{\sim}{b_{\alpha}}\cup\underset{\sim}{\eta_{\alpha}^{-1}}(\{1\})$, note that $\Vdash_{\mathbb{P}_{\mathbb{k}_{\alpha+1}}}"\underset{\sim}{a_{\alpha}^{*}}\notin\mathcal{A}_{\mathbb{k}_{\alpha}}"$, as otherwise there are $p\in\mathbb{P}_{\mathbb{k}_{\alpha+1}}$, $n<\omega$ and $\underset{\sim}{a^{\prime}}\in\mathcal{A}_{\mathbb{k}_{\alpha}}$ such that $p\Vdash_{\mathbb{P}_{\mathbb{k}_{\alpha+1}}}"\underset{\sim}{a^{\prime}}\setminus n=\underset{\sim}{a_{\alpha}^{*}}\setminus n"$, and therefore $p\Vdash_{\mathbb{P}_{\mathbb{k}_{\alpha+1}}}"\underset{\sim}{\eta_{\alpha}^{-1}}(\{1\})\restriction(\omega\setminus\underset{\sim}{b_{\alpha}}\setminus n)=\underset{\sim}{a_{\alpha}^{*}}\setminus\underset{\sim}{b_{\alpha}}\setminus n=\underset{\sim}{a^{\prime}}\setminus\underset{\sim}{b_{\alpha}}\setminus n\in V^{\mathbb{P}_{\mathbb{k}_{\alpha}}}"$, a contradiction (as $\underset{\sim}{\eta_{\alpha}}$ is Cohen and $\omega\setminus\underset{\sim}{b_{\alpha}}\setminus n$ is infinite). Now if $\underset{\sim}{a_{\alpha}}=\underset{l<n}{\cap}\underset{\sim}{a_{\alpha,l}^{\text{if l is even}}}$ and $\underset{\sim}{c}=\underset{\sim}{a_{\alpha}}\cap(\underset{l<m}{\cap}\underset{\sim}{d_{l}^{\text{if l is even}}})$ where $\underset{\sim}{d_{0}},...,\underset{\sim}{d_{m-1}}\in\mathcal{A}_{\mathbb{k}_{\alpha}}\setminus\{\underset{\sim}{a_{\alpha,0}},...,\underset{\sim}{a_{\alpha,n-1}}\}$, then $\underset{\sim}{c}\setminus\underset{\sim}{a_{\alpha}^{*}}$ and $\underset{\sim}{c}\cap\underset{\sim}{a_{\alpha}^{*}}$ are infinite (as $\underset{\sim}{\eta_{\alpha}}$ is Cohen and $\underset{\sim}{c}\setminus\underset{\sim}{b}$ is infinite), so $\mathcal{A}_{\mathbb{k}_{\alpha+1}}$ is forced to be independent.

If for some $\alpha<\lambda_{*}$, $\mathbb{k}_{\alpha}\in K_{2}^{+}$, when we’re done. Otherwise, by Fodor’s lemma, there are $\alpha<\beta<\lambda_{*}$ such that $(\underset{\sim}{a_{\alpha}},\underset{\sim}{b_{\alpha}})=(\underset{\sim}{a_{\beta}},\underset{\sim}{b_{\beta}})$, a contradiction. $\square$

**Definition 6: **We say that $(\mathbb{k}_{\alpha}:\alpha<\beta)$ is increasing continuous if $\alpha_{1}<\alpha_{1}\rightarrow\mathbb{k}_{\alpha_{1}}\leq_{2}\mathbb{k}_{\alpha_{2}}$, and for every limit $\delta<\beta$, $\underset{i<\delta}{\cup}\mathbb{P}_{\mathbb{k}_{i}}\lessdot\mathbb{P}_{\mathbb{k}_{\delta}}$.

**Claim 7: **Every increasing continuous sequence in $(K_{2}^{+},\leq_{2})$ has an upper bound. Moreover, if the length of the sequence has cofinality $>\aleph_{1}$, then the union is an upper bound in $K_{2}^{+}$.

**Proof: **Given an increasing continuous sequence $(\mathbb{k}_{\alpha}:\alpha<\beta)$, we choose $\mathbb{P}_{\mathbb{k}_{\beta}}$ as in [HwSh:1090] and we let $\mathcal{A}_{\mathbb{k}_{\beta}}=\underset{\alpha<\beta}{\cup}\mathcal{A}_{\mathbb{k}_{\alpha}}$. This is enough for $\leq_{1}$, so by claim 5 we’re done. $\square$

**Claim 8: **A. If $\mathbb{k}_{1}\in K_{2}$ then there are $\mathbb{k}_{2}$ and $\underset{\sim}{a}$ such that:

a. $\mathbb{k}_{1}\leq_{1}\mathbb{k}_{2}$.

b. $\mathcal{A}_{\mathbb{k}_{1}}\cup\{\underset{\sim}{a}\}\subseteq\mathcal{A}_{\mathbb{k}_{2}}$.

c. $\underset{\sim}{a}$ is Cohen over $V^{\mathbb{P}_{\mathbb{k}_{1}}}$.

d. $|\mathbb{P}_{\mathbb{k}_{2}}|\leq(2+|\mathbb{P}_{\mathbb{k}_{1}}|)^{\aleph_{1}}$.

B. Moreover, we may require that $\mathbb{k}_{2}\in K_{2}^{+}$.

**Proof: **A. Let $\mathbb{P}=\mathbb{P}_{\mathbb{k}_{1}}\star\mathbb{C}$ where $\mathbb{C}$ is Cohen forcing, now let $\mathbb{P}_{\mathbb{k}_{2}}$ be a ccc forcing such that $\mathbb{P}\lessdot\mathbb{P}_{\mathbb{k}_{2}}$, $\Vdash_{\mathbb{P}_{\mathbb{k}_{2}}}"MA_{\aleph_{1}}"$ and $|\mathbb{P}_{\mathbb{k}_{2}}|\leq(2+|\mathbb{P}|)^{\aleph_{1}}$. Finally, let $\mathcal{A}_{\mathbb{k}_{2}}=\mathcal{A}_{\mathbb{k}_{1}}\cup\{\underset{\sim}{a}\}$ where $\underset{\sim}{a}$ is a name for a Cohen real added by $\mathbb{P}_{\mathbb{k}_{2}}$, it’s easy to see that $(\mathbb{P}_{\mathbb{k}_{2}},\mathcal{A}_{\mathbb{k}_{2}})$ are as required.

B. By claim 5. $\square$

**Definition 9: **We define the amalgamation property in the context of $K_{2}^{+}$ as follows: $K_{2}^{+}$ has the amalgamation property if A implies B where:

A. a. $\mathbb{k}_{l}\in K_{2}^{+}$ $(l=0,1,2)$.

b. $\mathbb{k}_{0}\leq_{2}\mathbb{k}_{l}$ $(l=1,2)$.

c. $\mathbb{P}_{\mathbb{k}_{1}}\cap\mathbb{P}_{\mathbb{k}_{2}}=\mathbb{P}_{\mathbb{k}_{0}}$.

B. There exists $\mathbb{k}_{3}=(\mathbb{P}_{\mathbb{k}_{3}},\mathcal{A}_{\mathbb{k}_{3}})\in K_{2}^{+}$ such that $\mathbb{k}_{l}\leq_{2}\mathbb{k}_{3}$ $(l=1,2)$.

**Claim 10: **a. $(K_{2}^{+},\leq_{2})$ has the amalgamation property.

b. Suppose that $\mathbb{k}_{0},\mathbb{k}_{1}$ and $\mathbb{k}_{2}\in K_{2}^{+}$, $g:\mathbb{P}_{\mathbb{k}_{0}}\rightarrow\mathbb{P}_{\mathbb{k}_{1}}$ is an embedding such that $(g(\mathbb{P}_{\mathbb{k}_{0}}),g(\mathcal{A}_{\mathbb{k}_{0}}))\leq_{2}\mathbb{k}_{1}$ and $\mathbb{k}_{0}\leq_{2}\mathbb{k}_{2}$, then there exist $\mathbb{k},\mathbb{k}_{2}^{\prime}\in K_{2}^{+}$ and $f$ such that:

1. $\mathbb{k}_{1}\leq_{2}\mathbb{k}$ and $|\mathbb{P}_{\mathbb{k}}|\leq(2+|\mathbb{P}_{\mathbb{k}_{1}}|+|\mathbb{P}_{\mathbb{k}_{2}}|)^{\aleph_{1}}$.

2. $(g(\mathbb{P}_{\mathbb{k}_{0}}),g(\mathcal{A}_{\mathbb{k}_{0}}))\leq_{2}\mathbb{k}_{2}^{\prime}\leq_{2}\mathbb{k}$.

3. $f:\mathbb{P}_{\mathbb{k}_{2}}\rightarrow\mathbb{P}_{\mathbb{k}_{2}^{\prime}}$ is an isomorphism mapping $\mathcal{A}_{\mathbb{k}_{2}}$ to $\mathcal{A}_{\mathbb{k}_{2}^{\prime}}$

4. $g\subseteq f$.

**Proof: **a. We shall first prove that $\mathcal{A}_{\mathbb{k}_{1}}\cap\mathcal{A}_{\mathbb{k}_{2}}=\mathcal{A}_{\mathbb{k}_{0}}$. Note that $\mathcal{A}_{\mathbb{k}_{0}}\subseteq\mathcal{A}_{\mathbb{k}_{1}}\cap\mathcal{A}_{\mathbb{k}_{2}}$ is true by the definition of $\leq_{2}$, so suppose that $\underset{\sim}{a}\in\mathcal{A}_{\mathbb{k}_{1}}\setminus\mathcal{A}_{\mathbb{k}_{0}}$, we need to show that $\underset{\sim}{a}\notin\mathcal{A}_{\mathbb{k}_{2}}$. As $\mathbb{k}_{0}\leq_{2}\mathbb{k}_{1}$, $\underset{\sim}{a}$ is not a $\mathbb{P}_{\mathbb{k}_{0}}$-name. Therefore, it’s not a $\mathbb{P}_{\mathbb{k}_{2}}$-name, hence $\underset{\sim}{a}\notin\mathcal{A}_{\mathbb{k}_{2}}$.

Now construct $\mathbb{P}$ as in [HwSh:1090], i.e. we take the amalgamation $\mathbb{P}^{\prime}=\mathbb{P}_{\mathbb{k}_{1}}\underset{\mathbb{P}_{\mathbb{k}_{0}}}{\times}\mathbb{P}_{\mathbb{k}_{2}}$ and then we take $\mathbb{P}\in K$ such that $\mathbb{P}^{\prime}\lessdot\mathbb{P}$ and $|\mathbb{P}|\leq(2+|\mathbb{P}^{\prime}|)^{\aleph_{1}}$. Now let $\mathcal{A}:=\mathcal{A}_{\mathbb{k}_{1}}\cup\mathcal{A}_{\mathbb{k}_{2}}$. We need to show that $\mathcal{A}$ is as required, i.e. we need to prove that $(\mathbb{P},\mathcal{A})$ satisfy requirements (A)(d) and (D)(c) in Definition 2 (in the end, we will use claim 5 for the requirement in Definition (2)(D)(c)). By symmetry, it’s enough to show that if $\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}}"\underset{\sim}{b}\in pos(\mathcal{A}_{\mathbb{k}_{1}})"$ and $\underset{\sim}{a_{0}},...,\underset{\sim}{a_{n-1}}\in\mathcal{A}$, then $\Vdash_{\mathbb{P}}"\underset{\sim}{b}\cap(\underset{l<n}{\cap}\underset{\sim}{a_{l}^{\text{if (l is even)}}})\in[\omega]^{\omega}"$. Let $n_{2}:=n$, wlog there are $n_{0}<n_{1}<n_{2}$ such that $\underset{\sim}{a}_{l}\in\mathcal{A}_{\mathbb{k}_{0}}\iff l<n_{0}$, $\underset{\sim}{a}_{l}\in\mathcal{A}_{\mathbb{k}_{1}}\iff l\in[0,n_{1})$ and $\underset{\sim}{a}_{l}\in\mathcal{A}_{\mathbb{k}_{2}}\iff l\in[0,n_{0})\cup[n_{1},n_{2})$. It’s enough to show that the last statement is forced by $\mathbb{P}^{\prime}$, so let $k<\omega$ and $p=(p_{1},p_{2})\in\mathbb{P}^{\prime}$, we shall find $q\in\mathbb{P}^{\prime}$ and $m<\omega$ such that $p\leq q$, $k\leq m$ and $q\Vdash_{\mathbb{P}^{\prime}}"m\in\underset{\sim}{b}\cap(\underset{l<n}{\cap}\underset{\sim}{a_{l}^{\text{if (l is even)}}})"$. Let $p_{0}\in\mathbb{P}_{\mathbb{k}_{0}}$ witness $"(p_{1},p_{2})\in\mathbb{P}^{\prime}"$, i.e. $p_{0}\Vdash_{\mathbb{P}_{\mathbb{k}_{0}}}"\underset{l=1,2}{\wedge}p_{l}\in\mathbb{P}_{\mathbb{k}_{l}}/\mathbb{P}_{\mathbb{k}_{0}}"$. Let $\underset{\sim}{b^{*}}=\{m:p_{1}\nVdash_{\mathbb{P}_{\mathbb{k}_{1}}/\mathbb{P}_{\mathbb{k}_{0}}}"m\notin\underset{\sim}{b}\cap(\cap\{\underset{\sim}{a}_{l}^{\text{if l is even}}:l\in[n_{0},n_{1})\})"\}$ (so $\underset{\sim}{b^{*}}$ is a $\mathbb{P}_{\mathbb{k}_{0}}$-name) and let $p_{0}\in G_{0}\subseteq\mathbb{P}_{\mathbb{k}_{0}}$ be generic over $V$,** **then $\underset{\sim}{b^{*}}=\underset{\sim}{b^{*}}[G_{0}]\in V[G_{0}]$ and as $p_{1}\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}}"\underset{\sim}{b}\in pos(\mathcal{A}_{\mathbb{k}_{1}})"$, it follows that $p_{0}\Vdash_{\mathbb{P}_{\mathbb{k}_{0}}}"\underset{\sim}{b^{*}}\in[\omega]^{\omega}"$, moreover, $p_{0}\Vdash_{\mathbb{P}_{\mathbb{k}_{0}}}"\underset{\sim}{b}^{*}\in pos(\mathcal{A}_{\mathbb{k}_{0}})"$. Let $b^{**}$ be the $\mathbb{P}_{\mathbb{k}_{0}}$-name defined as $b^{*}$ if $p_{0}$ is in the generic set, and as $\omega$ otherwise. As $\mathbb{k}_{0}\leq_{2}\mathbb{k}_{2}$, it follows that $p_{2}\Vdash_{\mathbb{P}_{\mathbb{k}_{2}}/G_{0}}"\underset{\sim}{b^{**}}\cap(\underset{l\in[0,n_{0})\cup[n_{1},n_{2})}{\cap}\underset{\sim}{a_{l}^{\text{if (l is even)}}})\in[\omega]^{\omega}"$.

Therefore, in $V[G_{0}]$ there are $(p_{2}^{\prime},m)$ such that:

a. $p_{2}\leq p_{2}^{\prime}\in\mathbb{P}_{\mathbb{k}_{2}}/G_{0}$.

b. $m>k$.

c. $p_{2}^{\prime}\Vdash_{\mathbb{P}_{\mathbb{k}_{2}}/G_{0}}"m\in\underset{\sim}{b^{**}}\cap(\underset{l\in[0,n_{0})\cup[n_{1}.n_{2})}{\cap}\underset{\sim}{a_{l}^{\text{if (l is even)}}})"$.

Note that as $\underset{\sim}{b^{**}}\in V[G_{0}]$, $V[G_{0}]\models"m\in\underset{\sim}{b^{**}}[G_{0}]=\underset{\sim}{b^{*}}[G_{0}]"$. Therefore, by the definitions of $\underset{\sim}{b}$ and $\underset{\sim}{b^{*}}$, there is $p_{1}^{\prime}\in\mathbb{P}_{\mathbb{k}_{1}}/G_{0}$ above $p_{1}$ such that $p_{1}^{\prime}\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}/G_{0}}"m\in\underset{\sim}{b}\cap(\cap\{\underset{\sim}{a}_{l}^{\text{if l is even}}:l\in[n_{0},n_{1})\})"$. Therefore, there is $p_{0}\leq p_{0}^{\prime}\in G_{0}$ forcing (in $\mathbb{P}_{\mathbb{k}_{0}}$) all of the aforementioned statements about $(p_{1}^{\prime},p_{2}^{\prime})$ in $V[G_{0}]$. Now it’s easy to check that $q=(p_{1}^{\prime},p_{2}^{\prime})$ is as required. Finally, extend $(\mathbb{P},\mathcal{A})$ (with respect to $\leq_{1}$) to a member of $K_{2}^{+}$. By observation 3, we’re done.

b. Follows from (a) by changing names. $\square$

**Claim 11: **There exists $\mathbb{k}=(\mathbb{P}_{\mathbb{k}},\mathcal{A}_{\mathbb{k}})=(\mathbb{P},\mathcal{A})\in K_{2}^{+}$ such that $|\mathbb{P}_{\mathbb{k}}|=\lambda$ and:

1. For every $X\subseteq\mathbb{P}$ of cardinality $<\mu$, there exists $\mathbb{k}^{\prime}=(\mathbb{Q},\mathcal{A}^{\prime})\in K_{2}^{+}$ such that $X\subseteq\mathbb{Q}$, $\mathbb{k}^{\prime}\leq_{2}\mathbb{k}$ and $|\mathbb{Q}|<\mu$.

2. If $\mathbb{k}_{1},\mathbb{k}_{2}\in K_{2}^{+}$, $|\mathbb{P}_{\mathbb{k}_{1}}|,|\mathbb{P}_{\mathbb{k}_{2}}|<\mu$, $\mathbb{k}_{1}\leq_{2}\mathbb{k}_{2}$ and $f_{1}:\mathbb{P}_{\mathbb{k}_{1}}\rightarrow\mathbb{P}$ is a complete embedding such that $(f_{1}(\mathbb{P}_{\mathbb{k}_{1}}),f_{1}(\mathcal{A}_{\mathbb{k}_{1}}))\leq_{2}(\mathbb{P},\mathcal{A})$, then there is a complete embedding $f_{2}$ such that $f_{1}\subseteq f_{2}$ and $(f_{2}(\mathbb{P}_{\mathbb{k}_{2}}),f_{2}(\mathcal{A}_{\mathbb{k}_{2}}))\leq_{2}(\mathbb{P},\mathcal{A})$.

**Proof: **The first property is satisfied by every $\mathbb{k}\in K_{2}^{+}$ by observation 4. The proof of (2) is as in [HwSh:1090]. $\square$

**Claim 12: **A implies B where:

A. a. $\mathbb{k}_{0},\mathbb{k}_{1},\mathbb{k}_{2}\in K_{2}^{+}$, $\mathbb{k}_{0}\leq_{2}\mathbb{k}_{l}$ $(l=1,2)$ and $\mathbb{P}_{\mathbb{k}_{0}}=\mathbb{P}_{\mathbb{k}_{1}}\cap\mathbb{P}_{\mathbb{k}_{2}}$.

b. $\underset{\sim}{D}$ is a $\mathbb{P}_{\mathbb{k}_{0}}$-name of a nonprincipal ultrafilter on $\omega$.

c. For $l=1,2$, $\underset{\sim}{a}_{l}$ and $\underset{\sim}{b_{l}}$ are canonical $\mathbb{P}_{\mathbb{k}_{l}}$-names of a member of $[\omega]^{\omega}$.

d. For $l=1,2$, $\Vdash_{\mathbb{P}_{\mathbb{k}_{l}}}"\underset{\sim}{a}_{l}\cap\underset{\sim}{b}_{l}$ is infinite and $\underset{\sim}{a}_{l}$ contains no members of $\underset{\sim}{D}$ from $V^{\mathbb{P}_{\mathbb{k}_{0}}}$.

e. $\mathbb{P}_{\mathbb{k}_{1}}\cap\mathbb{P}_{\mathbb{k}_{2}}=\mathbb{P}_{\mathbb{k}_{0}}$.

f. For $l=1,2$, $\Vdash_{\mathbb{P}_{\mathbb{k}_{l}}}"\underset{\sim}{b_{l}}\text{ is a pseudo intersection of }\underset{\sim}{D}"$.

B. There is $\mathbb{k}\in K_{2}^{+}$ such that:

a. $|\mathbb{P}_{\mathbb{k}}|\leq(2+|\mathbb{P}_{\mathbb{k}_{1}}|+|\mathbb{P}_{\mathbb{k}_{2}}|)^{\aleph_{1}}$.

b. $\mathbb{k}_{l}\leq_{2}\mathbb{k}$ $(l=1,2)$.

c. $\Vdash_{\mathbb{P}_{\mathbb{k}}}"\underset{\sim}{a_{2}}\setminus\underset{\sim}{a_{1}}$ and $\underset{\sim}{a_{1}}\setminus\underset{\sim}{a_{2}}$ are infinite$"$.

**Proof: **Let $\mathbb{k}\in K_{2}^{+}$ be the object constructed by the proof of claim 10. We need to prove that $\mathbb{k}$ satisfies clause (B)(c). For $l=0,1,2$, let $\mathbb{P}_{l}=\mathbb{P}_{\mathbb{k}_{l}}$ and let $\mathbb{P}^{\prime}$ be as in the proof of claim 10, so it suffices to prove that $\Vdash_{\mathbb{P}^{\prime}}"\underset{\sim}{a_{2}}\setminus\underset{\sim}{a_{1}}$ and $\underset{\sim}{a_{1}}\setminus\underset{\sim}{a_{2}}$ are infinite$"$. Let $p=(p_{1},p_{2})\in\mathbb{P}^{\prime}$, $k<\omega$ and let $p_{0}\in\mathbb{P}_{0}$ be a witness of $"p=(p_{1},p_{2})\in\mathbb{P}^{\prime}"$. Now let $G_{0}\subseteq\mathbb{P}^{\prime}$ be generic over $V$ such that $p_{0}\in G_{0}$ and let $D=\underset{\sim}{D}[G_{0}]$. By the assumptions, for $l=1,2$, $p_{l}\Vdash_{\mathbb{P}_{1}/G_{0}}"\underset{\sim}{a_{l}}\cap\underset{\sim}{b_{l}}$ is an infinite pseudo intersection of $D"$. In $V[G_{0}]$, let $b_{l}^{*}=\{m:p_{l}\nVdash_{\mathbb{P}_{l}/G_{0}}"m\notin\underset{\sim}{a_{l}}\cap\underset{\sim}{b_{l}}"\}$, then $p_{l}\Vdash_{\mathbb{P}_{l}/G_{0}}"\underset{\sim}{a_{l}}\cap\underset{\sim}{b_{l}}\subseteq\underset{\sim}{b_{l}^{*}}$, hence $b_{l}^{*}$ infinite$"$. As $p_{l}\Vdash_{\mathbb{P}_{l}/G_{0}}"\underset{\sim}{a_{l}}\cap\underset{\sim}{b_{l}}$ is a pseudo intersection of $D"$, necessarily $V[G_{0}]\models b_{l}^{*}\in D$.

Let $a_{l}^{*}=\{m:p_{l}\Vdash_{\mathbb{P}_{l}/G_{0}}m\in\underset{\sim}{a_{l}}\}$, so $a_{l}^{*}\in V[G_{0}]$ and $p_{l}\Vdash_{\mathbb{P}_{l}/G_{0}}"a_{l}^{*}\subseteq\underset{\sim}{a_{l}}"$. Now recall that $\Vdash_{\mathbb{P}_{\mathbb{k}_{l}}}"\underset{\sim}{a}_{l}$ contains no member of $\underset{\sim}{D}$ from $V^{\mathbb{P}_{\mathbb{k}_{0}}}"$, therefore $p_{l}\Vdash_{\mathbb{P}_{l}/G_{0}}"a_{l}^{*}\notin\underset{\sim}{D}"$.

Hence in $V[G_{0}]$ (recalling $D$ is an ultrafilter), $b:=(b_{1}^{*}\cap b_{2}^{*})\setminus(a_{1}^{*}\cup a_{2}^{*})\in D$. Let $m\in b$ be such that $k<m$. By the definition of $b_{l}^{*}$, there is $p_{l}^{\prime}\in\mathbb{P}_{l}/G_{0}$ above $p_{l}$ such that $p_{l}^{\prime}\Vdash_{\mathbb{P}_{l}/G_{0}}"m\in\underset{\sim}{a_{l}}\cap\underset{\sim}{b_{l}}"$. By the definition of $a_{l}^{*}$, there is $p_{l}^{\prime\prime}\in\mathbb{P}_{l}/G_{0}$ above $p_{l}$ such that $p_{l}^{\prime\prime}\Vdash_{\mathbb{P}_{l}/G_{0}}"m\notin\underset{\sim}{a_{l}}"$. Let $p_{0}^{\prime}\in G_{0}$ be a condition above $p_{0}$ forcing the above statements, so $p_{0}^{\prime}$ is witnessing the fact that $(p_{1}^{\prime},p_{2}^{\prime\prime}),(p_{1}^{\prime\prime},p_{2}^{\prime})\in\mathbb{P}^{\prime}$ are above $p=(p_{1},p_{2})$. Now $m>k$, $(p_{1}^{\prime},p_{2}^{\prime\prime})\Vdash"m\in\underset{\sim}{a_{1}}\setminus\underset{\sim}{a_{2}}"$ and $(p_{1}^{\prime\prime},p_{2}^{\prime})\Vdash"m\in\underset{\sim}{a_{2}}\setminus\underset{\sim}{a_{1}}"$, which completes the proof. $\square$

**Definition 13: **Let $\mathbb{P}=\mathbb{P}_{\mathbb{k}}$ be the forcing from claim 11, let $G\subseteq\mathbb{P}$ be generic over $V$ and in $V[G]$, let $V_{1}=HOD(\mathbb{R}^{<\mu}\cup\{\mathcal{A}_{\mathbb{k}}\})$.

**Definition 14 ([NN]): **A family $\mathcal{F}\subseteq[\omega]^{\omega}$ is nearly mad if $|A\cap B|<\aleph_{0}$ or $|A\Delta B|<\aleph_{0}$ for every $A\neq B\in\mathcal{F}$, and $\mathcal{F}$ is maximal with respect to this property.

Theorem 15: $V_{1}\models ZF+DC_{<\mu}+"\text{there are no mad families}"+"\text{there are no nearly mad families}"+"\text{there exists a non-meager filter on }\omega"$.

**Proof: **1. In order to see that there exists a non-meagre filter in $V_{1}$, let $\underset{\sim}{D}$ be the filter generated by $\mathcal{A}_{\mathbb{k}}$ and the cofinite sets. By claim 8 and the choice of $\mathbb{k}$, $\underset{\sim}{D}$ contains many Cohen reals and therefore is non-meager.

2. The proof of the non-existence of mad families is exactly as in [HwSh:1090], where $(K_{2}^{+},\leq_{2})$ here replaces $(K,\lessdot)$ there, and claim 10 is used for the amalgamation arguments. Alternatively, see the proof of (3) below.

3. The non-existence of a nearly mad family in $V_{1}$ will follow from the proofs below. $\square$

Somewhat mad families

**Definition 16: **A family $\mathcal{F}\subseteq[\omega]^{\omega}$ is somewhat mad if:

a. For every $a_{1},a_{2}\in\mathcal{F}$, $|a_{1}\cap a_{2}|<\aleph_{0}$ or $a_{1}\subseteq^{*}a_{2}$ or $a_{2}\subseteq^{*}a_{1}$.

b. If $b\in[\omega]^{\omega}$ then for some $a\in\mathcal{F}$, $|a\cap b|=\aleph_{0}$.

**Observation 17: **Nearly mad families are somewhat mad. $\square$

**Definition 18: **Let $Pr(\mathbb{k}_{1},\mathbb{k}_{2},\underset{\sim}{D},\underset{\sim}{b_{2}})$ mean:

a. $\mathbb{k}_{1}\leq_{2}\mathbb{k}_{2}$, $\underset{\sim}{b_{2}}$ is a $\mathbb{P}_{\mathbb{k}_{2}}$-name and $\underset{\sim}{D}$ is a $\mathbb{P}_{\mathbb{k}_{1}}$-name such that $\Vdash_{\mathbb{P}_{\mathbb{k}_{2}}}"\underset{\sim}{b_{2}}\in[\omega]^{\omega}"$ and $\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}}"\underset{\sim}{D}$ is a nonprincipal ultrafilter on $\omega"$.

b. If $G_{1}\subseteq\mathbb{P}_{\mathbb{k}_{1}}$ is generic over $V$, $p_{1}\in\mathbb{P}_{\mathbb{k}_{2}}/G_{1}$, $b_{0}^{*}=\{n:p_{1}\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}G_{1}}"n\in\underset{\sim}{b_{2}}"\}$ and $b_{1}^{*}=\{n:p_{1}\nVdash_{\mathbb{P}_{\mathbb{k}_{1}}/G_{1}}"n\notin\underset{\sim}{b_{2}}"\}$, then $V[G_{1}]\models b_{1}^{*}\setminus b_{0}^{*}\in D$.

**Claim 19: **(A) implies (B) where:

A. a. $\mathbb{k}_{1}\in K_{2}^{+}$.

b. $\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}}"\underset{\sim}{D}$ is a nonprincipal ultrafilter on $\omega"$.

c. $\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}}"\underset{\sim}{S_{1}}$ is somewhat mad$"$.

d. $\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}}"\underset{\sim}{S_{1}}\cap\underset{\sim}{D}=\emptyset"$.

B. There is $\mathbb{k}_{2}$ such that:

a. $\mathbb{k}_{2}\in K_{2}^{+}$.

b. $\mathbb{k}_{1}\leq_{2}\mathbb{k}_{2}$.

c. $|\mathbb{P}_{\mathbb{k}_{2}}|\leq(2+|\mathbb{P}_{\mathbb{k}_{1}}|)^{\aleph_{1}}$.

d. ($\alpha$) implies ($\beta$) where:

$\alpha$. $(\mathbb{k}_{3},\underset{\sim}{S_{2}})$ satisfy the following properties:

1. $\mathbb{k}_{2}\leq_{2}\mathbb{k}_{3}\in K_{2}^{+}$.

2. $\Vdash_{\mathbb{P}_{\mathbb{k}_{3}}}"\underset{\sim}{S_{2}}$ is somewhat mad and $\underset{\sim}{S_{1}}\subseteq\underset{\sim}{S_{2}}"$.

3. $\Vdash_{\mathbb{P}_{\mathbb{k}_{3}}}"$no member of $\underset{\sim}{S_{2}}\setminus\underset{\sim}{S_{1}}$ contains a member of $\underset{\sim}{D}"$.

$\beta$. For some $\mathbb{P}_{\mathbb{k}_{3}}$-name $\underset{\sim}{a}$, $\Vdash_{\mathbb{P}_{\mathbb{k}_{3}}}"\underset{\sim}{a}\in\underset{\sim}{S_{2}}"$ and $Pr(\mathbb{k}_{1},\mathbb{k}_{3},\underset{\sim}{D},\underset{\sim}{a})$.

**Proof: **Using Mathias forcing restricted to $\underset{\sim}{D}$, it’s easy to see that there is $\mathbb{k}_{2}$ and a $\mathbb{P}_{\mathbb{k}_{2}}$ name $\underset{\sim}{b}$ such that $\mathbb{k}_{1}\leq_{2}\mathbb{k}_{2}\in K_{2}^{+}$, $|\mathbb{P}_{\mathbb{k}_{2}}|\leq(2+|\mathbb{P}_{\mathbb{k}_{1}}|)^{\aleph_{1}}$ and $\Vdash_{\mathbb{P}_{\mathbb{k}_{2}}}"\underset{\sim}{b}$ is a pseudo intersection of $\underset{\sim}{D}"$. Therefore, $\Vdash_{\mathbb{P}_{\mathbb{k}_{2}}}"\underset{\sim}{b}\in[\omega]^{\omega}$ is almost disjoint to every $\underset{\sim}{a}\in\underset{\sim}{S_{1}}"$ (by the fact that $\Vdash_{\mathbb{P}_{\mathbb{k}_{1}}}"\underset{\sim}{S_{1}}\cap\underset{\sim}{D}=\emptyset"$).

We shall now prove that $\mathbb{k}_{2}$ satisfies (B)(d). Suppose that $(\mathbb{k}_{3},\underset{\sim}{S_{2}})$ are as there. By the somewhat madness of $\underset{\sim}{S_{2}}$, there is $\underset{\sim}{a}$ such that $\Vdash_{\mathbb{P}_{\mathbb{k}_{3}}}"\underset{\sim}{a}\in\underset{\sim}{S_{2}}$ and $|\underset{\sim}{a}\cap\underset{\sim}{b}|=\aleph_{0}"$. Therefore, $\Vdash_{\mathbb{P}_{\mathbb{k}_{3}}}"\underset{\sim}{a}\cap\underset{\sim}{b}\in[\omega]^{\omega}$ is a pseudo intersection of $\underset{\sim}{D}"$. Now let $G_{1}\subseteq\mathbb{P}_{\mathbb{k}_{1}}$ be generic over $V$. If $p_{1}\in\mathbb{P}_{\mathbb{k}_{3}}/G_{1}$ then $b^{*}=\{n:p_{1}\nVdash_{\mathbb{P}_{\mathbb{k}_{3}}/G_{1}}"n\notin\underset{\sim}{a}\cap\underset{\sim}{b}"\}\in\underset{\sim}{D}[G_{1}]$ by the fact that $\underset{\sim}{a}\cap\underset{\sim}{b}$ is a pseudo intersection of $\underset{\sim}{D}$. Now let $a^{*}=\{n:p_{1}\Vdash_{\mathbb{P}_{\mathbb{k}_{3}/G_{1}}}"n\in\underset{\sim}{a}"\}$, then $p_{1}\Vdash_{\mathbb{P}_{\mathbb{K}_{3}}/G_{1}}"a^{*}\subseteq\underset{\sim}{a}\text{ is infinite}"$. If $a^{*}\in\underset{\sim}{D}[G_{1}]$, then $p_{1}$ forces that $\underset{\sim}{a}$ (which belongs to $\underset{\sim}{S_{2}}\setminus\underset{\sim}{S_{1}}$) contains a member of $\underset{\sim}{D}[G_{1}]$, contradicting $(\alpha)(3)$. Therefore, $a^{*}\notin\underset{\sim}{D}[G_{1}]$, and $\underset{\sim}{a}$ is as required in the definition of $Pr(\mathbb{k}_{1},\mathbb{k}_{3},\underset{\sim}{D},\underset{\sim}{a})$. $\square$

**Claim 20: **There is no somewhat mad family in $V_{1}$.

**Proof: **Suppose towards contradiction that $\underset{\sim}{S}$ is a $\mathbb{P}$-name of a somewhat mad family. As in [HwSh:1090], let $\underset{\sim}{D}$ be a $\mathbb{P}$-name of a Ramsey ultrafilter on $\omega$ such that $\Vdash_{\mathbb{P}}"\underset{\sim}{S}\cap\underset{\sim}{D}=\emptyset"$. By claim 11(a), there is $\mathbb{k}_{1}\leq_{2}\mathbb{k}$ such that $\mathbb{k}_{1}\in K_{2}^{+}$, $|\mathbb{P}_{\mathbb{k}_{1}}|<\mu$ and $\underset{\sim}{S}$ is definable using a $\mathbb{P}_{\mathbb{k}_{1}}-$name. Let $K_{\mathbb{P}}^{+}$ be the set of $\mathbb{k}^{\prime}\in K_{2}^{+}$ such that $\mathbb{k}^{\prime}\leq_{2}\mathbb{k}$, $|\mathbb{P}_{\mathbb{k}^{\prime}}|<\mu$, $\underset{\sim}{S}\restriction\mathbb{P}_{\mathbb{k}^{\prime}}$ is a canonical $\mathbb{P}_{\mathbb{k}^{\prime}}$-name of a somewhat mad family in $V^{\mathbb{P}_{\mathbb{k}^{\prime}}}$ and $\underset{\sim}{D}\restriction\mathbb{P}_{\mathbb{k}^{\prime}}$ is a $\mathbb{P}_{\mathbb{k}^{\prime}}$-name of a Ramsey ultrafilter on $\omega$. As in [HwSh:1090], $K_{\mathbb{P}}^{+}$ is $\leq_{2}$-dense in $K_{2}^{+}$, so there exists $\mathbb{k}_{2}\in K_{\mathbb{P}}^{+}$ such that $\mathbb{k}_{1}\leq\mathbb{k}_{2}$. Let $\mathbb{k}_{3}\in K_{2}^{+}$ be as in claim 19 for $(\mathbb{k}_{2},\underset{\sim}{S}\restriction\mathbb{P}_{\mathbb{k}_{2}})$, wlog $\mathbb{k}_{3}\leq_{2}\mathbb{k}$ (see claim 11). Choose $\mathbb{k}_{4}\in K_{\mathbb{P}}^{+}$ such that $\mathbb{k}_{3}\leq_{2}\mathbb{k}_{4}$ and let $\underset{\sim}{S_{4}}:=\underset{\sim}{S}\restriction\mathbb{P}_{\mathbb{k}_{4}}$. Let $\underset{\sim}{a}$ be a $\mathbb{P}_{\mathbb{k}_{4}}$-name such that $Pr(\mathbb{k}_{3},\mathbb{k}_{4},\underset{\sim}{D},\underset{\sim}{a})$ holds, as guaranteed by claim 19.

As in [HwSh:1090], there are $\mathbb{k}_{5},\mathbb{k}_{6},\in K_{\mathbb{P}}^{+}$ and an isomorphism $f$ from $\mathbb{k}_{4}$ to $\mathbb{k}_{5}$ over $\mathbb{k}_{2}$ such that $\mathbb{P}_{\mathbb{k}_{5}}$ adds a generic for $\mathbb{M}_{\underset{\sim}{D}\restriction\mathbb{P}_{\mathbb{k}_{4}}}$ (Mathias forcing restricted to the ultrafilter $\underset{\sim}{D}\restriction\mathbb{P}_{\mathbb{k}_{4}}$) and $(\mathbb{k}_{2},\mathbb{k}_{4},\mathbb{k}_{5},\mathbb{k}_{6})$ here are as $(\mathbb{k}_{0},\mathbb{k}_{1},\mathbb{k}_{2},\mathbb{k}_{3})$ in claim 10, and wlog $\mathbb{k}_{6}\leq_{2}\mathbb{k}$. By the choice of $f$, $\Vdash_{\mathbb{P}}"\underset{\sim}{a},f(\underset{\sim}{a})\in\underset{\sim}{S}"$.

By claim 12, with $(\mathbb{k}_{2},\mathbb{k}_{4},\mathbb{k}_{5},\mathbb{k}_{6},\underset{\sim}{a},f(\underset{\sim}{a}))$ standing for $(\mathbb{k}_{0},\mathbb{k}_{1},\mathbb{k}_{2},\mathbb{k}_{,}\underset{\sim}{a}_{1},\underset{\sim}{a}_{2})$ there, it’s forced by $\mathbb{P}_{\mathbb{k}_{5}}$, and hence by $\mathbb{P}$, that $\underset{\sim}{a}\setminus f(\underset{\sim}{a})$ and $f(\underset{\sim}{a})\setminus\underset{\sim}{a}$ are infinite. As in [HwSh:1090], $\Vdash_{\mathbb{P}}"|\underset{\sim}{a}\cap f(\underset{\sim}{a})|=\aleph_{0}"$. As $\Vdash_{\mathbb{P}}"\underset{\sim}{a},f(\underset{\sim}{a})\in\underset{\sim}{S}"$, we get a contradiction. $\square$

References

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

[Ma1] A. R. D. Mathias, Happy families, Ann. Math. Logic **12 **(1977), no. 1, 59-111.

[Ma2] A. R. D.. Mathias, A remark on rare filters.

[NN] Itay Neeman and Zach Norwood, Happy and mad families in $L(\mathbb{R})$, preprint.

[To] Asger Toernquist, Definability and almost disjoint families, arXiv:1503.07577.

(Haim Horowitz) Einstein Institute of Mathematics

Edmond J. Safra campus,

The Hebrew University of Jerusalem.

Givat Ram, Jerusalem, 91904, Israel.

E-mail address: haim.horowitz@mail.huji.ac.il

(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