Fact 1. For $n>0$ , $\lim ^n\mathbf {A}=0$ if and only if every n-coherent family is n-trivial.

Proof. If $n=1$ , let $\Psi $ be a trivialization of $\Phi $ and set

$$\begin{align*}\Phi^*_x=\left\{\begin{array}{@{}cc} \Phi_x&x\in X \\ \Psi\upharpoonright I_x&\text{otherwise}. \end{array}\right.\end{align*}$$

Then $\Phi ^*$ is $1$ -trivial since $\Psi $ is a trivialization.

If $n>1$ , let $\Psi $ be a trivialization of $\Phi $ . Extend $\Psi $ arbitrarily to an alternating family indexed by $\omega ^\omega $ , and for each $\vec x\in (\omega ^\omega )^n\setminus X^n$ let

$$\begin{align*}\Phi_{\vec x}^*=d\Psi_{\vec x}.\end{align*}$$

Then $\Phi ^*$ is n-trivial since $\Psi $ is a trivialization.

Proof. We cite [Reference Todorčevíc14, Theorem 3.3.13], whose original use was to give a walks construction of a Hausdorff gap.

Now, set $A_\alpha =I_{f_\alpha \wedge g}$ . The sequence $\langle A_\alpha \mid \alpha <\omega _1\rangle $ is strictly $\subseteq ^*$ -increasing: for weakly increasing, if $\alpha <\beta $ and $f_\alpha \leq ^mf_\beta $ then $A_\alpha \setminus (m\times \omega )\subseteq A_\beta $ and for the strict part, whenever $f_\alpha (k)<f_\beta (k)<g(k)$ , $(k,f_\alpha (k)+1)\in A_\beta \setminus A_\alpha $ . Note that $A_\alpha \subseteq \omega \times \omega $ rather than $\omega $ , though the difference is only cosmetic. Let $\langle B_\alpha \mid \alpha <\omega _1\rangle $ be as in Theorem 3.5 and set

$$\begin{align*}\Phi_{f_\alpha}(i,j)=\left\{\begin{array}{@{}cc} 1&(i,j)\in B_\alpha \\ 0&\text{otherwise}. \end{array}\right.\end{align*}$$

Then $\Phi $ is coherent by the first conclusion of Theorem 3.5 and nontrivial below g by the second.

Proof. Fix $\langle M_\alpha \mid \alpha <\omega _2\rangle $ increasing, continuous elementary substructures of $H_\theta $ such that:

• • for each $\alpha <\omega _2$ , $|M_\alpha |=\aleph _1$ ;

• • $\omega ^\omega =\bigcup _{\alpha <\omega _2}M_\alpha \cap \omega ^\omega $ ;

• • for each $\alpha <\omega _2$ , $\langle M_i\mid i\leq \alpha \rangle \in M_{\alpha +1}$ (that is, the $M_\alpha $ are internally approaching).

We define $\Phi _{xy}$ for $x,y\in M_\alpha \cap \omega ^\omega $ by recursion on $\alpha $ . An easy coding argument yields that $\diamondsuit (S^2_1)$ implies there are $\langle \Psi ^\alpha _x\mid \alpha <\omega _2,x\in M_\alpha \cap \omega ^\omega \rangle $ with ${\Psi ^\alpha _x\colon I_x\to \mathbb {Z}}$ such that for any family $\langle \Psi _x\mid x\in \omega ^\omega \rangle $ with each $\Psi _x\colon I_x\to \mathbb {Z}$ , there are stationarily many $\alpha \in S^2_1$ satisfying that for all $x\in M_\alpha \cap \omega ^\omega $ , $\Psi _x=\Psi ^\alpha _x$ . To see this, fix compatible bijections $f_\alpha \colon M_\alpha \to \alpha $ for each $\alpha <\omega _2$ satisfying $\omega _1\cdot \alpha =\alpha $ . Whenever $f_\alpha ^{-1}A_\alpha $ has the correct type to be a $\Psi ^\alpha $ as above, we set $\Psi ^\alpha =f_\alpha ^{-1}A_\alpha $ and otherwise define $\Psi ^\alpha $ arbitrarily. Then for any $\Psi $ , there are stationarily many $\alpha \in S^2_1$ satisfying $\omega _1\cdot \alpha =\alpha $ and $A_\alpha =f_\alpha [A_\alpha \cap M_\alpha ]$ .

For $x,y\in M_0$ , we define $\Phi _{x,y}$ to be the constant $0$ function. Suppose now that $\alpha \not \in S^2_1$ or that $\Psi ^\alpha $ does not trivialize $\Phi \upharpoonright M_\alpha $ . By Theorem 2.2, $\Phi \upharpoonright M_\alpha $ is trivial and we extend $\Phi $ to a $2$ -coherent family indexed by $M_{\alpha +1}\cap \omega ^\omega $ using Proposition 2.3.

Now suppose $\alpha \in S^2_1$ and $\Psi ^\alpha $ trivializes $\Phi \upharpoonright M_\alpha $ . We will extend $\Phi $ to indices in $M_{\alpha +1}$ in such a way that $\Psi ^\alpha $ does not extend to a trivialization of $\Phi \upharpoonright M_{\alpha +1}$ using a $1$ -coherent family indexed by $M_\alpha $ produced using Lemma 3.4. To this end, first obverse that there is a $<^*$ -increasing chain of order type $\omega _1$ which is cofinal in $M_\alpha \cap \omega ^\omega $ since $\operatorname {cof}(\alpha )=\omega _1$ and if $\beta <\alpha $ then any countable subset of $M_\beta \cap \omega ^\omega $ has a $<^*$ upper bound in $M_{\beta +1}$ . Since $\mathfrak {d}=\aleph _2$ , we may fix some $x\in M_{\alpha +1}\cap \omega ^\omega $ which is not $<^*$ -dominated by any element of $M_\alpha $ . By Lemma 3.4, there is a $1$ -coherent family $\Theta =\langle \theta _{y}\mid y\in M_{\alpha }\cap \omega ^\omega \rangle $ which is nontrivial below x. For $y,z$ not both in $M_{\alpha }$ , let $\Phi _{yz}=\Psi ^\alpha _z+\theta _z-\Psi ^\alpha _y-\theta _y$ , where $\Psi ^\alpha ,\theta $ are defined as the constant $0$ functions on indices not in $M_\alpha $ —note that extending $\Theta $ in this way does not maintain coherence. We now claim that $\Psi ^\alpha $ does not extend to a trivialization of $\Phi $ on $M_{\alpha +1}$ . Indeed, if $\Psi ^*$ were any such trivialization then for any $y\in M_\alpha \cap \omega ^\omega $ ,

$$\begin{align*}\begin{aligned} -\Psi^*_x&=\ -\Psi^*_x+\Psi^\alpha_y-\Psi^\alpha_y\\ &=\, ^*\Phi_{xy}-\Psi^\alpha_y\\ &=\Psi^\alpha_y+\theta_y-\Psi^\alpha_y\\ &=\theta_y, \end{aligned} \end{align*}$$

where the first equality is adding $0$ , the second is using that $\Psi ^*$ is a trivialization of $\Phi $ extending $\Psi ^\alpha $ , the third is the definition of $\Phi _{xy}$ , and the last is canceling identical terms with opposite signs. That is, $-\Psi ^*_x$ is a trivialization of $\Theta $ below x, contradicting the choice of $\Theta $ to be nontrivial below x. Now, if $\Phi $ were trivial and $\Psi $ were a trivialization, we could find some $\alpha \in S^2_1$ such that $\Psi ^\alpha _{y}=\Psi _{y}$ for all $y\in M_\alpha \cap \omega ^\omega $ . But then $\Psi $ would be a trivialization of $\Phi $ on $M_{\alpha +1}$ extending $\Psi ^\alpha $ , which we just showed was impossible.

Proof. The following preservation fact is standard and not due to the author; however, we have been unable to find a proof in the literature of this exact statement and so provide one here.

Proof. We may assume that the underlying set of $\mathbb {P}$ is $\kappa $ . By $\diamondsuit (S)$ , there are $B_\alpha \subseteq \alpha \times \alpha $ for each $\alpha <\kappa $ such that whenever $X\subseteq \kappa \times \kappa $ , there are stationarily many $\alpha \in S$ such that $X\cap (\alpha \times \alpha )=B_\alpha $ . Now, in $V[G]$ , we let $A_\alpha $ be

$$\begin{align*}\{\beta<\alpha\mid\exists \gamma\in G\cap\alpha\;(\gamma,\beta)\in B_\alpha\}.\end{align*}$$

Given $X\subseteq \kappa $ , let $\dot \tau $ be a name such that $\dot \tau (G)=X$ . Let $T\subseteq S$ be

$$\begin{align*}\{\alpha\in S\mid B_\alpha=\{(\beta,\gamma)\mid\beta,\gamma<\alpha\wedge \beta\Vdash \gamma\in\dot\tau\}\}.\end{align*}$$

Note that T is stationary in V and therefore in $V[G]$ . Let

$$\begin{align*}C=\{\gamma<\kappa\mid \forall\alpha<\gamma\;\exists \beta\in G\cap\gamma\;(\beta\,||\,\alpha\in\dot\tau)\}.\end{align*}$$

Then C is a club in $V[G]$ and if $\gamma \in T\cap C$ then $A_\gamma =X\cap \gamma $ .

Now, suppose $\mathbb {P}$ is as in Corollary 4.2. If $\mathbb {P}$ collapses $\aleph _1$ then by Fact 2, $V^{\mathbb {P}}$ satisfies $\diamondsuit _{\omega _1}$ and, in particular, $CH$ so that $\lim ^1\mathbf {A}\neq 0$ . Otherwise, $\diamondsuit (S^2_1)$ holds in $V^{\mathbb {P}}$ by Fact 2 and the proof is complete by Corollary 4.1.

Proof. For $\alpha \in S$ , let $S_\alpha =\mathcal {P}(\alpha )\cap V[G_\alpha ]$ ; we claim the $S_\alpha $ form a $\diamondsuit ^+$ sequence. Note that by inaccessibility of $\kappa $ , $|S_\alpha |\leq \aleph _1$ in $V[G]$ . Now, suppose that $x=\dot x(G)$ is a subset of $\kappa $ . For each $\alpha <\kappa $ , let $A_\alpha $ be a maximal antichain of conditions deciding whether $\alpha \in \dot x$ . Since Mitchell forcing has the $\kappa $ -cc, $|A_\alpha |<\kappa $ . Let

$$\begin{align*}f(\alpha)=\sup\bigcup_{(p,g)\in A_\alpha}\operatorname{dom}(p)\cup\operatorname{dom}(g);\end{align*}$$

observe that $f(\alpha )<\kappa $ . Let $C_f$ be the club of all ordinals closed under f. Then for $\alpha \in C_f$ , $x\cap \alpha \in \mathcal {P}(\alpha )\cap V[G_\alpha ]$ . Moreover, since $C_f\in V$ , $C\cap \alpha \in \mathcal {P}(\alpha )\cap V[G_\alpha ]$ .

Proof. If $n=1$ then a trivialization is by definition an extension of $\Phi $ to $X\cup \{g\}$ so we assume $n>1$ .

Suppose that $\Phi $ extends to some $\Phi ^*$ indexed by $X\cup \{g\}$ . Then $\Psi _{\vec y}=(-1)^{n+1}\Phi ^*_{\vec y^\frown g}$ is a trivialization of $\Phi $ below g as for any $\vec x\in X^n$ ,

$$\begin{align*}\sum_{i=0}^{n-1}(-1)^i\Psi_{\vec x^i}=\sum_{i=0}^{n-1}(-1)^i\Phi^*_{(\vec x^i)^\frown g}=\left(\sum_{i=0}^{n}(-1)^i\Phi^*_{(\vec x^\frown g)^i}\right)-(-1)^n\Phi_{\vec x}=^*(-1)^{n+1}\Phi_{\vec x},\end{align*}$$

where the first equality is the definitions of $\Psi $ , the second is adding $0$ , and the third is coherence of $\Phi ^*$ .

Now suppose that $\Phi $ is trivial below g as witnessed by $\langle \Psi _{\vec y}\mid \vec y\in X^{n-1}\rangle $ . Let $\Psi ^*$ to be an arbitrary alternating extension of $\Psi $ to $X\cup \{g\}$ and set

$$\begin{align*}\Phi^*_{\vec y}=\left\{\begin{array}{@{}cc} \Phi_{\vec y}&\vec y\in X^n \\ d\Psi^*_{\vec y}&\text{otherwise}. \end{array}\right.\end{align*}$$

Then $\Phi ^*$ is readily verified to be an n-coherent family indexed by $X\cup \{g\}$ extending $\Phi $ .

Proof. By recursion on the length of s, we define for each $s\in 2^{<\omega _{n+1}}$ an $(n+1)$ -coherent family $\Phi ^s$ indexed by $A_{|s|}$ , ensuring additionally that:

• • if $t\sqsubseteq s$ then $\Phi ^s\upharpoonright A_{|t|}=\Phi ^t$ ;

• • if $t\upharpoonright S=s\upharpoonright S$ and $|t|=|s|$ then $\Phi ^t=\Phi ^s$ .

If $|s|\not \in S$ , then since $\Phi ^s$ is trivial by Theorem 2.2, we may extend $\Phi ^s$ to an $(n+1)$ -coherent family indexed by $A_{|s|+1}$ using Proposition 2.3. We choose the same extension for both $\Phi ^{s^\frown 0}$ and $\Phi ^{s^\frown 1}$ . Now suppose $|s|\in S$ . By Theorem 2.2, we may fix a trivialization $\Psi $ of $\Phi ^s$ . By hypothesis, we may fix an n-coherent family $\Theta $ indexed by $A_{|s|}$ and $y\in A_{|s|+1}$ such that $\Theta $ is nontrivial below $y\wedge g$ . For $\vec x\in (A_{\alpha +1})^{n+1}\setminus (A_\alpha )^n$ , we let

$$\begin{align*}\Phi_{\vec x}^{s^\frown0}=d\Psi_{\vec x}\end{align*}$$

and

$$\begin{align*}\Phi_{\vec x}^{s^\frown 1}=d[\Psi+\Theta]_{\vec x},\end{align*}$$

where we define $\Psi $ and $\Theta $ as the constant $0$ functions at indices not all in $A_{|s|}$ (note that this extension of $\Theta $ is not coherent).

Proof. Suppose not and let $\Psi ^0$ and $\Psi ^1$ be extensions of $\Psi '$ which trivialize $\Phi ^{s^\frown 0}$ and $\Phi ^{s^\frown 1}$ below $g,$ respectively. The proof is slightly different depending on whether $n=1$ or $n>1$ . If $n=1$ , let $\Upsilon \colon I_{y\wedge g}\to \mathbb {Z}$ be $\Psi ^0_y-\Psi ^1_y$ . Then for each $z\in A_{|s|}$ ,

$$\begin{align*}\begin{aligned} \Upsilon&=\Psi^0_y-\Psi^1_y\\ &=(\Psi^0_y-\Psi^{\prime}_z)-(\Psi^1_y-\Psi^{\prime}_z)\\ &=^*\Phi_{zy}^{s^\frown 0}-\Phi_{zy}^{s^\frown 1}\\ &=-\Psi_z+(\Psi_z+\Theta_z)\\ &=\Theta_z, \end{aligned}\end{align*}$$

where the first line is the definition of $\Upsilon $ , the second is adding $0$ , the third is that $\Psi ^0$ and $\Psi ^1$ are trivializations of $\Phi ^0$ and $\Phi ^1$ below g extending $\Psi '$ , respectively, and the last is the definition of $\Phi $ . So $\Upsilon $ trivializes $\Theta $ below $y\wedge g$ , contrary to our choice of $\Theta $ .

Now, suppose $n>1$ . For each $\vec z\in (A_{|s|})^{n-1}$ , let $\Upsilon _{\vec z}\colon I_{\bigwedge \vec z\wedge y\wedge g}\to \mathbb {Z}$ be $\Psi ^0_{\vec z^\frown y}-\Psi ^1_{\vec z^\frown y}$ . Then for each $\vec x\in (A_{|s|})^n$ ,

$$\begin{align*}\begin{aligned} d\Upsilon_{\vec x}&= \sum_{i=0}^{n-1}(-1)^i(\Psi^0_{(\vec x^i)^\frown y}-\Psi^1_{(\vec x^i)^\frown y})\\ &=(d\Psi^0_{\vec x^\frown y}-(-1)^n\Psi^0_{\vec x})-(d\Psi^1_{\vec x^\frown y}-(-1)^n\Psi^1_{\vec x})\\ &=^*\Phi^{s^\frown 0}_{\vec x^\frown y}-\Phi^{s^\frown 1}_{\vec x^\frown y}\\ &=(-1)^n\Psi_{\vec x}-(-1)^n(\Psi_{\vec x}+\Theta_{\vec x})\\ &=(-1)^{n+1}\Theta_{\vec x}, \end{aligned}\end{align*}$$

where the first line is the definition of d and $\Upsilon $ , the second is the definition of d applied to $\Psi ^0$ and $\Psi ^1$ , the third is that $\Psi ^0$ and $\Psi ^1$ extend $\Psi '$ and trivialize $\Phi ^{s^\frown 0}$ and $\Phi ^{s^\frown 1}$ , respectively, the fourth is the definition of $\Phi $ , and the last is canceling like terms with opposite signs. So $(-1)^{n+1}\Upsilon $ trivializes $\Theta $ below $y\wedge g$ , contrary to our choice of $\Theta $ .

We claim that for some $h\in 2^{\omega _1}$ , $\Phi ^h=\bigcup _{\alpha <\omega _{n+1}} \Phi ^{h\upharpoonright \alpha }$ is nontrivial below g.

For any reasonable choice of encoding attempts at trivialization, if $b\in 2^{<\omega _{n+1}}$ codes a trivialization of $\Phi ^s$ for some s with $|b|=|s|\in S$ , then note that there is a unique such $\Phi ^s$ . Set $F(b)\in \{0,1\}$ to be such that the trivialization coded by b does not extend to a trivialization of $\Phi ^{s^\frown F(b)}$ ; we define $F(b)$ arbitrarily otherwise. Now by $w\diamondsuit (S)$ , we may fix some $h\in 2^{\omega _{n+1}}$ such that for all $b\colon \omega _{n+1}\to 2$ , there are stationarily many $\alpha \in S$ such that $h(\alpha )=F(b\upharpoonright \alpha )$ ; it is for this h that we claim $\Phi ^h$ is nontrivial. Indeed, if b codes a trivialization for $\Phi ^h$ , we may fix some $\alpha \in S$ such that $h(\alpha )=F(b\upharpoonright \alpha )$ . But then $b\upharpoonright \alpha $ extends to a trivialization of $\Phi ^{(h\upharpoonright \alpha )^\frown F(b\upharpoonright \alpha )}$ , contrary to our choice of $F(b\upharpoonright \alpha )$ .

Proof. Suppose not and fix $A\subseteq M\cap \omega ^\omega $ with $|A|<|M|$ and $\{y\wedge g\mid y\in A\}$ cofinal in $(\{y\wedge g\mid y\in M\cap \omega ^\omega \},<^*)$ . Then fix $B\in M$ such that ${A\subseteq B\subseteq M\cap \omega ^\omega }$ . Then B is not $<^*$ -cofinal in $\omega ^\omega $ since g is not dominated by any element of B so by elementarity we may fix $z\in M$ which is not dominated by any $y\in B$ . By changing z if necessary, we assume that z is increasing. Since A was chosen to be cofinal in $(\{y\wedge g\mid y\in M\cap \omega ^\omega \},<^*)$ , we now fix some $y\in A$ such that $g\wedge z\leq ^* y\wedge z\leq y$ . Let $E\in [\omega ]^\omega \cap M$ be $\{n\mid y(n)<z(n)\}$ ; note E is infinite since $y\in A\subseteq B$ and z is not $<^*$ -dominated by any element of B. Let $n_0<n_1<\ldots $ enumerate E. Let $a\in M\cap \omega ^\omega $ be defined by $a(n)=z(n_{i+1})$ where $n_i< n\leq n_{i+1}$ . Since g is not $<^*$ -dominated by any element of M, there are infinitely many i such that for some n with $n_i<n\leq n_{i+1}$ , $g(n)>a(n)$ . For each such i and n,

$$\begin{align*}g(n_{i+1})\geq g(n)>a(n)= z(n_{i+1})>y(n_{i+1}).\end{align*}$$

This contradicts that $g\wedge z\leq ^* y$ as $g\wedge z(n_{i+1})=z(n_{i+1})>y(n_{i+1})$ for each such i.

Proof. By induction on n. When $n=1$ , the hypotheses on M yield a cofinal $\omega _1$ sequence in $(M\cap \omega ^\omega ,<^*)$ so the lemma follows from Lemma 3.4.

Suppose now $n>1$ . Build an increasing continuous chain $\langle M_\alpha \mid \alpha <\omega _n\rangle $ of elementary substructures of M such that:

• • $|M_\alpha |=\aleph _{n-1}$ ;

• • whenever $\alpha $ is a successor ordinal and $A\subseteq M_\alpha \cap \omega ^\omega $ with $|A|<\aleph _{n-1}$ there is a $B\in M_\alpha $ with $|B|=|A|$ and $A\subseteq B\subseteq M_\alpha $ ;

• • $\bigcup _\alpha M_\alpha =M$ ;

• • for each $\alpha $ there is a $y_\alpha \in M_{\alpha +1}\cap \omega ^\omega $ such that $g\wedge y_\alpha $ is not $<^*$ -dominated by any element of $M_\alpha $ .

Note the last item is possible by Lemma 5.7 if $g\in \omega ^\omega $ or from the hypothesis that $\operatorname {cof}(M\cap \omega ^\omega ,<^*)=\aleph _n$ if g is the constant function $\omega $ . We verify the hypotheses of Lemma 5.5 for $A_\alpha =M_\alpha \cap \omega ^\omega $ and $S=S^n_{n-1}$ .

The first and second items are immediate. For the third, by replacing $y_\alpha $ if necessary, we may assume that $y_\alpha $ is weakly increasing. The inductive hypothesis yields an $(n-1)$ -coherent family indexed by $M_\alpha \cap \omega ^\omega $ which is nontrivial below $g\wedge y_\alpha $ .

Proof. Fix $M\prec H_\theta $ as in Lemma 5.8 such that some cofinal subset of $(\omega ^\omega ,\leq )$ is contained in M. Then there is a nontrivial n-coherent family indexed by ${M\cap \omega ^\omega }$ . The proof is finished since derived limits may be computed along any cofinal suborder.

Proof. We define $\langle \Phi _{\vec x}\mid \vec x\in (A_\alpha )^{n+1}\rangle $ by recursion on $\alpha $ such that if $\vec x\in A_{\alpha }^{n+1}$ then $\Phi _{\vec x}$ takes values in $\bigoplus _{i<\omega _n\alpha }G_i$ . If $\alpha \not \in S$ then since $\Phi \upharpoonright A_\alpha $ is trivial by Theorem 2.2, we extend $\Phi $ to $A_{\alpha +1}$ using Proposition 2.3. Suppose now that $\alpha \in S$ and let $\Psi ^\alpha $ be a trivialization of $\langle \Phi _{\vec x}\mid \vec x\in (A_\alpha )^{n+1}\rangle $ such that each $\Psi ^\alpha _{\vec y}$ takes values in $\bigoplus _{i<\omega _n\alpha }G_i$ . By hypothesis, let $\Theta ^\alpha $ be a $\bigoplus _{\omega _n\alpha \leq i<\omega _{n}(\alpha +1)}G_i$ -valued n-coherent family indexed by $A_\alpha $ which is nontrivial below $y\wedge g$ for some $y\in A_{\alpha +1}$ . For each $\vec x\in (A_{\alpha +1})^{n+1}\setminus (A_\alpha )^{n+1}$ , let $\Phi _{\vec x}=d(\Psi ^\alpha +\Theta ^\alpha )_{\vec x}$ , where we extend $\Psi ^\alpha $ and $\Theta ^\alpha $ to $(A_{\alpha +1})^n\setminus (A_\alpha )^n$ as the constant $0$ functions (note this extension of $\Theta $ is not coherent).

Suppose for contradiction that $\Phi $ were trivial below g and let $\Psi $ be a trivialization of $\Phi $ below g. By stationarity of S, we may fix $\alpha \in S$ such that $\omega _n\alpha =\alpha $ and for all $\vec x\in (A_{\alpha })^{n}$ , $\Psi _{\vec x}$ takes values in $\bigoplus _{i<\alpha }G_i$ . Then whenever $\vec x\in (A_\alpha )^{n}$ ,

$$\begin{align*}(-1)^{n}\Psi_{\vec x}+\sum_{i=0}^{n-1}(-1)^i\Psi_{(\vec x^i)^\frown y}=d\Psi_{\vec x^\frown y}=^*\Phi_{\vec x^\frown y}=(-1)^{n}(\Psi^\alpha_{\vec x}+\Theta^\alpha_{\vec x}).\end{align*}$$

Since $\Psi _{\vec x}$ and $\Psi ^\alpha _{\vec x}$ take values in $\bigoplus _{i<\alpha }G_i$ , the restriction of the values of $(-1)^{n}\Psi _{\vec z^\frown y}$ to $\bigoplus _{\alpha \leq i<\alpha +\omega _n}G_i$ trivializes $\Theta ^\alpha $ below $y\wedge g$ , contradicting our choice of $\Theta ^\alpha $ .

Proof. Note that the “moreover” statement implies the original statement so it suffices to show that $\Phi $ does not extend to $A\cup \{c\}$ . Suppose not and fix p forcing that $\Phi _c\colon I_c\to \mathbb {Z}$ satisfies $\Phi _c=^*\Phi _x$ for each $x\in A$ . For each $x\in A$ , fix $q_x\leq p$ and $n_x<\omega $ such that $q_x\Vdash \Phi _x=^{n_x}\Psi _c$ . Since A is countably directed under $<^*$ and Cohen forcing is countable, there is a condition q and an $n<\omega $ such that $B=\{x\in A\mid q_x=q, n_x=n\}$ is cofinal in $(A,<^*)$ . We now note that if $x,y\in B$ then $\Phi _x=^{\max (n,|q|)}\Phi _y$ . Indeed, for any $m>\max (n,|q|)$ and $i\leq x(m),y(m)$ , we may extend q to some r such that $r(m)>x(m),y(m)$ . Then

$$\begin{align*}r\Vdash\Phi_x(m,i)=\Phi_c(m,i)=\Phi_y(m,i).\end{align*}$$

Then

$$\begin{align*}\Psi(m,i)=\left\{\begin{array}{@{}cc} \Phi_x(m,i)&m>\max(n,|q|), \text{for some }x\in B, x(m)\geq i \\ 0&\text{there is no such }x \end{array}\right.\end{align*}$$

trivializes $\Phi $ .

Proof. Suppose for simplicity that $\dot \Phi $ is a name for a $1$ -coherent family. We show that there is a condition q forcing that $\dot \Phi $ is trivial. First, by inaccessibility of $\kappa $ , there is a club $C\subseteq \kappa $ such that whenever $\alpha \in C$ has uncountable cofinality, $\dot \Phi $ reflects to an $\mathbb {M}_\alpha $ name $\dot \Phi _\alpha $ for a $1$ -coherent family. By Lemma 6.3, for each such $\alpha $ , $\dot \Phi _\alpha $ must name a trivial $1$ -coherent family since it admits an extension to the next Cohen real and, since all reals added by Mitchell forcing are added on the Cohens coordinate, extends to a Cohen generic real in an extension by Cohen forcing.

For each $\alpha \in C$ of uncountable cofinality, pick a condition $q_\alpha \in \mathbb {M}_\alpha $ and a $\mathbb {P}_\alpha $ name $\dot x_\alpha $ such that $q_\alpha \Vdash \dot x_\alpha $ trivializes $\dot \Phi _\alpha $ . Since $\alpha $ has uncountable cofinality, there is a $\beta _\alpha <\alpha $ such that $q_\alpha \in \mathbb {M}_\beta $ and $\dot x_\alpha $ is a $\mathbb {P}_\beta $ name. By Fodor’s lemma, the function $\alpha \mapsto \beta _\alpha $ is constant with value $\beta ^*$ on a stationary $S\subseteq C\cap \operatorname {cof}(\geq \omega _1)$ . By inaccessibility of $\kappa $ , $|M_{\beta ^*}|<\kappa $ and there are fewer than $\kappa $ many $\mathbb {P}_{\beta ^*}$ names for real numbers. In particular, the function $\alpha \mapsto (q_\alpha ,\dot x_\alpha )$ is constant on a stationary $T\subseteq S$ with value $(q,\dot x)$ . Then q forces that $\dot x$ trivializes $\dot \Phi $ , as desired.

Proof. We show each item in turn.

1. (1) A trivialization is coded by a real.

2. (2) Every real added by a strongly proper forcing is (equivalent to) a Cohen real.

3. (3) Suppose $\Vdash \dot \psi $ is a trivialization for $\Phi $ . Let $\langle A_\alpha \mid \alpha <\kappa \rangle $ be a partition of $\mathbb {P}$ by linked sets with $\kappa <\mathfrak {b}$ . For each $x\in \omega ^\omega $ , pick $p_x,k_x$ such that $p_x\Vdash \dot \psi =^{k_x}\Phi _x$ and let $\alpha _x$ be such that $p_x\in A_{\alpha _x}$ . We color $\omega ^\omega $ by

   $$\begin{align*}x\mapsto(k_x,\alpha_x).\end{align*}$$
   By the $\mathfrak {b}$ -directedness of $\leq ^*$ , there is a $\leq ^*$ -cofinal X and a k such that

   • • $k_x=k$ for each $x\in X$ and

   • • whenever $x,y\in X$ , $\alpha _x=\alpha _y$ ; in particular, $p_x$ is compatible with $p_y$ .

   Then the function

   $$\begin{align*}\psi(i,j)=\left\{\begin{array}{@{}cc} \Phi_x(i,j)&i\geq k\text{ and there is some }x\in X\text{ with }j\leq x(i) \\ 0&\text{otherwise} \end{array}\right.\end{align*}$$
   trivializes $\Phi $ ; note that $\psi $ is well defined since if $x,y\in X$ then $\Phi _x=^k\Phi _y$ .

4. (4) Suppose $\mathbb {P}^2$ is $\omega ^\omega $ bounding and $\mathbb {P}$ adds a trivialization for $\Phi $ . Let $G\times H$ be generic for $\mathbb {P}^2$ and let $\psi _1$ and $\psi _2$ be trivializations for $\Phi $ added by G and H respectively. Then for some k, $\psi _1=^k\psi _2$ as otherwise there is some x such that for infinitely many i and $j\leq x(i)$ , $\psi _1(i,j)\neq \psi _2(i,j)$ ; such an x cannot be bounded by any real in V. In particular, $\psi _1\upharpoonright (k,\infty )\times \omega =\psi _2\upharpoonright (k,\infty )\times \omega \in V[G]\cap V[H]=V$ and trivializes $\Phi $ .

5. (5) Recall that Miller forcing consists of trees $T\subseteq \omega ^<\omega $ such that every $\sigma \in T$ has an extension which splits infinitely often, ordered by containment. If p is a Miller condition, we write $\operatorname {split}(p)$ to mean the set of splitting nodes in p. If s is a node of p we write $p^{[s]}$ to mean all the nodes of p which are either an initial segment of s or which end extend s; note that $p^{[s]}$ is a Miller condition extending p. Suppose for simplicity that $\Vdash \dot \Psi $ is a trivialization of $\Phi $ . The key claim is the following.

   Claim 7.2. There is a condition p, functions $\langle f_s\mid s\in \operatorname {split}(p)\rangle $ , and numbers $\langle n^s_i\mid s\in \operatorname {split}(p),s^\frown i\in p\rangle $ such that:

   • • for each $s,i,$ and j, if $i<j$ then $n^s_i<n^s_j$ ;

   • • for each s and i,

     $$\begin{align*}p^{[s^\frown i]}\Vdash\dot\Psi=^{n_i^s}\Phi_{f_s};\end{align*}$$

   • • for each s and each $i<j$ ,

     $$\begin{align*}p^{[s^\frown j]}\Vdash\dot\Psi\neq^{n_i^s}\Phi_{f_s}.\end{align*}$$
   We first note that the claim is sufficient. Indeed, let f be a $<^*$ upper bound for $\{f_s\mid s\in \operatorname {split}(p)\}$ and fix $q\leq p$ and n such that $q\Vdash \dot \Psi =^n\Phi _f$ . For $s\in \operatorname {split}(p)$ , let $k_s=\max (n,\min \{m\mid f_s\leq ^mf\}, \min \{m\mid \Phi _{f_s}=^m\Phi _f\})$ . Then whenever $s\in \operatorname {split}(q)$ ,
   $$\begin{align*}q\Vdash \dot \Psi=^{k_s}\Phi_{f_s}\end{align*}$$
   since q forces that for each $i\geq k$ and $j\leq f_s(k)$ ,
   $$\begin{align*}\Psi(i,j)=\Phi_f(i,j)=\Phi_{f_s}(i,j).\end{align*}$$
   But this means that q can split only finitely often at s as q is incompatible with each $p^{[s^\frown i]}$ satisfying $n^s_i> k_s$ , a contradiction.

   The proof of Claim 7.2 is, of course, a fusion argument. We argue that for a single condition q and $s\in \operatorname {split}(q)$ , there is a condition extending q in which s is still a splitting node but there are $f_s$ and $\langle n_i^s\rangle $ as in Claim 7.2. We use the following general fact which holds for every forcing whatsoever.

   Subclaim 7.3. For every condition r, there is an $f\in {}^\omega \omega $ such that for all $n<\omega $ ,

   $$\begin{align*}r\not\Vdash\dot\Psi=^n\Phi_f.\end{align*}$$

   Proof. Suppose not and let r be a counterexample. For each $f\in {}^\omega \omega $ fix $n_f$ such that $r\Vdash \dot \Psi =^{n_f}\Phi _f$ . Since $(\omega ^\omega ,<^*)$ is countably directed, there is a $<^*$ -cofinal set $A\subseteq \omega ^\omega $ and an $n<\omega $ such that $n_f=n$ for each $f\in A$ . Then let

   $$\begin{align*}\Psi(i,j)=\left\{\begin{array}{@{}cc} \Phi_f(i,j)&i>n \text{ and for some }f\in A\text{, } f(i)\geq j\\ 0&\text{otherwise} \end{array}\right.\end{align*}$$

   $\Psi $ trivializes $\Phi $ since for each $f\in \omega ^\omega $ , we may find $g\in A$ and $m<\omega $ such that $f<^mg$ so that whenever $i\geq m$ , $j\geq m$ , and $\Phi _f(i,j)=\Phi _g(i,j)$ , $\Psi (i,j)=\Phi _f(i,j)$ .

   Now, for each i with $s^\frown i\in q$ , fix an $f_i$ such that for all $n<\omega $ , $q^{[s^\frown i]}\not \Vdash \dot \Psi =^n\Phi _f$ . Let $f_s$ be a $<^*$ upper bound for $\{f_i\mid s^\frown i\in q\}$ . Then whenever $s^\frown i\in q$ and $n<\omega $ , $q^{[s^\frown i]}\not \Vdash \dot \Psi =^n\Phi _{f_s}$ . Now, recursively define $q_i\leq q^{[s^\frown i]}$ and $n_i$ as in the claim, first forcing that $\dot \Psi \neq ^{\max _{j<i}n_j}\Phi _f$ and then using that $\dot \Psi $ is forced to be a trivialization of $\Phi $ to decide some $n_i$ with $\dot \Psi =^{n_i}\Phi _f$ . Then $\bigcup _{i\in \operatorname {succ}_q(s)}q_i$ is a condition extending q which still has s as a splitting node and $f_s$ and $\langle n_i\rangle $ satisfy Claim 7.2.

Question 8.1. Does $\mathfrak {d}=\aleph _2$ imply $\lim ^2\mathbf {A}\neq 0$ ?

Question 8.2. Does $\lim ^n\mathbf {A}=0$ for all $n>0$ imply $\mathfrak {d}\geq \aleph _{\omega +1}$ ?

Question 8.3. Can Cohen forcing add a trivialization to a nontrivial $2$ -coherent family?

Question 8.4. Are there any forcings which add new sets of reals but do not trivialize any nontrivial $2$ -coherent families?

Question 8.5. Does $\lim ^1\mathbf {A}=0$ in the Miller model?