Fact 2.4 [Reference Weiß19, Proposition 3.4].

Let $\lambda '> \lambda $ . $\operatorname {\mathrm {ITP}}(\mu , \lambda ')$ implies $\operatorname {\mathrm {ITP}}(\mu , \lambda )$ .

Fact 2.5 [Reference Fontanella5, Lemma 3.4].

Let $\lambda '> \lambda $ . If every thin $\mathcal {P}_\mu (\lambda ')$ -list has a cofinal branch, then so does every thin $\mathcal {P}_\mu (\lambda )$ list.

Proof Work in $W[G]$ , and consider $j(d)$ . Since $j(\kappa )> \lambda $ , we have that $j"\lambda \in \mathcal {P}_{j(\kappa )}(j(\lambda ))$ . Let $b = \{\alpha < \lambda \mid j(\alpha ) \in j(d)_{j"\lambda }\}$ .

We first claim that b is a cofinal branch. Let $z \in \mathcal {P}_\kappa (\lambda )^W$ . We want to show that $b \cap z \in L_z$ . Note that $j(d)_{j"\lambda }\cap j(z) \in j(L_z)$ . Since d is a thin list, $|L_z|< \kappa $ , so $j(L_z) = j"(L_z) = \{j(d)_{j(y)} \cap j(z) \mid z\subseteq y, y \in \mathcal {P}_\kappa (\lambda )^W\}$ . Since $j(d)_{j"\lambda }\cap j(z) \in j(L_z)$ , there must be some y such that $j(d)_{j"\lambda }\cap j(z) = j(d)_{j(y)}\cap j(z)$ . It follows that $b\cap z = d_y\cap z$ .

Now, suppose that $b \in W$ . Let U be the normal measure on $\mathcal {P}_\kappa (\lambda )^W$ corresponding to j. Note that in the ultrapower of W by U, $j"\lambda $ is represented by $[x\to x]$ , and so $j(d)_{j"\lambda }$ is represented by $[x \mapsto d]_{[x\mapsto x]} = [x \mapsto d_x]$ . On the other hand, since $b \in W$ , we can take $j(b)$ ; indeed, $j(d)_{j"\lambda } = j(b) \cap j"\lambda $ . Since $j(b)$ is represented by $[x \mapsto b]$ , $j(d)_{j"\lambda } = [x \mapsto b] \cap [x \mapsto x]$ . We conclude that $d_x = b \cap x$ for U-many x. Since U is a normal measure on W, all measure one sets must be stationary in W, so b must be an ineffable branch.

Proof Let $\dot {b}$ be a name for a cofinal branch, and let $x \in (\mathcal {P}_\kappa (\lambda ))^V$ . Since $\dot {b}$ is forced to be cofinal, $\dot {b}\cap x$ is forced to be in $L_x$ for all x. Since $L_x$ is in V, $\dot {b}\cap x$ is likewise forced to be in V. Since the list is thin, $|L_x| < \kappa $ , and so there are fewer than $\kappa $ possibilities for $\dot {b}\cap x$ .

Proof Let G be generic for $\mathbb {P}$ over V. Since $\mathbb {P}$ is $\kappa $ -cc, $\mathcal {P}_\kappa (\mu )^V$ is cofinal in $\mathcal {P}_\kappa (\mu )^{V[G]}$ . In $V[G]$ , if $x \in \mathcal {P}_\kappa (\mu )$ is a subset of some $y \in \mathcal {P}_\kappa (\mu )^{V}$ , then $\dot {b}\cap x$ is determined solely by $\dot {b}\cap y$ and x. Since $\dot {b}$ is $\kappa $ -approximated by $\mathbb {P}\ast \dot {\mathbb {Q}}$ over V, $\dot {b}\cap y$ must be in V. It follows that $\mathbb {Q}$ forces that $\dot {b}\cap x \in V[G]$ , as desired.

Now suppose that $\dot {b}$ is thinly $\kappa $ -approximated by $\mathbb {P}\ast \dot {\mathbb {Q}}$ , not just $\kappa $ -approximated. For every possible value of $\dot {b}\cap x$ , we can always choose an extension that is a possible value for $\dot {b}\cap y$ . This defines an injection from the possible values (in $V[G]$ ) of $\dot {b}\cap x$ to the possible values (in $V)$ of $\dot {b}\cap y$ . Since $\dot {b}$ is thinly $\kappa $ -approximated in V by $\mathbb {P}\ast \dot {\mathbb {Q}}$ , there $<\kappa $ -many possible values for $\dot {b}\cap y$ , and thus there will be $<\kappa $ -many possible values in $V[G]$ for $\dot {b}\cap x$ . So $\dot {b}$ is thinly $\kappa $ -approximated by $\mathbb {Q}$ over $V[G]$ .

Proof Let $z \in (\mathcal {P}_\kappa (\mu ))^{V[\mathbb {P}\ast \dot {\mathbb {Q}}]}$ . Since $\mathbb {Q}$ does not add sequences of length $< \kappa $ , $z \in (\mathcal {P}_\kappa (\mu ))^{V[\mathbb {P}]}$ . By Lemma 2.9, $\Vdash _{\mathbb {Q}\ast \dot {\mathbb {S}}} z \cap \dot {b} \in V[\mathbb {P}]$ . We conclude that, working in $V[\mathbb {P} \ast \dot {\mathbb {Q}}]$ , $\Vdash _{\mathbb {S}} z \cap \dot {b} \in V[\mathbb {P}] \subseteq V[\mathbb {P}\ast \dot {\mathbb {Q}}]$ .

Proof Let $\lambda>\kappa $ be some ordinal, and let $\dot {d}$ be a $\mathbb {P}\times \mathbb {Q}$ -name for a subset of $\lambda $ that is thinly $\kappa ^+$ -approximated by $\mathbb {Q}$ over the generic extension of V by $\mathbb {P}$ . Suppose that $1 \Vdash _{\mathbb {P}\times \mathbb {Q}} \dot {d}\notin V[\dot {G}_{\mathbb {P}}]$ , where $\dot {G}_{\mathbb {P}}$ is the canonical name for a generic of $\mathbb {P}$ . Let $\delta \leq \tau $ be the least such that $2^\delta> \kappa $ . For each $\sigma \in 2^{<\delta }$ we will build a triple $(A_\sigma , q_\sigma , x_\sigma )$ where $A_\sigma $ is a maximal antichain in $\mathbb {P}$ , $q_\sigma \in \mathbb {Q}$ , and $x_\sigma \in (\mathcal {P}_{\kappa ^+}(\lambda ))^V$ , such that $q_{\sigma '} \leq q_\sigma $ whenever $\sigma '$ extends $\sigma $ , and for all $\sigma \in 2^{<\delta }$ and all $p \in A_\sigma $ , $(p,q_{\sigma ^\smallfrown 0})$ and $(p,q_{\sigma ^\smallfrown 1})$ force contradictory values for $\dot {d}\cap x_\sigma $ . Note that it suffices to consider $x_\sigma \in (\mathcal {P}_{\kappa ^+}(\lambda ))^V$ , since $(\mathcal {P}_{\kappa ^+}(\lambda ))^V$ is cofinal in $(\mathcal {P}_{\kappa ^+}(\lambda ))^{V[\mathbb {P}]}$ by the $\kappa $ -cc of $\mathbb {P}$ .

We will build this inductively, using the following claim.

Let $q \in \mathbb {Q}$ . We wish to build $(A,q_0, q_1, x)$ where A is a maximal antichain in $\mathbb {P}$ , $q_0, q_1 \leq q$ , and for all $p \in A$ , p forces that $q_0$ and $q_1$ force contradictory information about $\dot {d}\cap x$ .

For each $q \in \mathbb {Q}$ , we construct $(A, q_0, q_1, x)$ recursively. Set $q_0^0 = q_0^1 = q$ and ${A_0 = \emptyset }$ . At successor stages $\alpha +1$ we choose a new p incompatible with all conditions in our antichain-in-construction $A_\alpha $ if such exists, then strengthen it to $p' \leq p$ so that there exist some $q^0_{\alpha +1} \leq q^0_\alpha $ , $q^1_{\alpha +1} \leq q^1_\alpha $ , and $x_{\alpha +1}\supset x_\alpha $ as desired. We then define $A_{\alpha +1}$ to be $A_\alpha \cup \{p'\}$ . At limit stages $\gamma $ , we take unions (for $A_\gamma $ and $x_\gamma $ ) and lower bounds for $q^0_\gamma $ and $q_\gamma ^1$ . Note that since $\mathbb {P}$ is $\kappa $ -cc, this process will end at some $\gamma < \kappa $ , and so since $\mathbb {Q}$ is $\kappa $ -closed we will always be able to take these lower bounds. This process gives $(A, q_0, q_1, x)$ . Applying this construction repeatedly, we can build the binary tree structure as desired.

Now let $x = \bigcup _{\sigma \in 2^{<\delta }} x_\sigma .$ For each $f \in 2^\delta $ , by closure we can find $q^{\prime }_f$ such that $q^{\prime }_f \leq q_{f\upharpoonright \alpha }$ for all $\alpha < \delta $ . Let G be generic for $\mathbb {P}$ . Since G is generic, $G \cap A_\sigma $ is nonempty for all $\sigma $ . Working in $V[G]$ , we define $q_f \leq q^{\prime }_f$ to be a refinement deciding $\dot {d}\cap x$ .

We claim that if $f \neq g$ are elements of $2^\delta $ , then $q_f$ and $q_g$ force contradictory things about $\dot {d}\cap x$ . Let $\alpha $ be the largest such that $f\upharpoonright \alpha = g\upharpoonright \alpha $ . Since $G \cap A_{f\upharpoonright \alpha }$ is nonempty, $q_{f\upharpoonright \alpha ^\smallfrown 0}$ and $q_{f\upharpoonright \alpha ^\smallfrown 1}$ force contradictory information about $x_{f\upharpoonright \alpha }\subseteq x$ . Since f extends one of these and g extends the other, we see that $q_f$ must disagree with $q_g$ on $\dot {d}\cap x$ . Thus every $q_f$ for $f\in 2^\delta $ forces a different value for $\dot {d}\cap x$ , and so there must be at least $2^\delta> \kappa $ potential values for $\dot {d}\cap x$ . But since $\dot {d}$ is thinly $\kappa ^+$ -approximated, there are fewer than $\kappa ^+$ potential values for $\dot {d}$ , and thus for $\dot {d}\cap x$ . This gives a contradiction.

Claim 2.18. Suppose $\nu $ is a singular strong limit cardinal with cofinality $\tau $ , and let $\lambda \geq \nu ^+$ . Let $\mathbb {S}$ be a notion of forcing. Let $\dot {d}$ be a $\mathbb {S}$ -name for a subset of $\lambda $ that is thinly $\nu ^+$ -approximated by $\mathbb {S}$ over V, such that $\dot {d}$ is forced not to be in V. Then for any $s \in \mathbb {S}$ , there is a sequence $\langle s_\alpha \mid \alpha < \nu \rangle $ of conditions in $\mathbb {S}$ and a set $x \in \mathcal {P}_{\nu ^+}(\lambda )$ , such that $s_\alpha \leq s$ for each $\alpha < \nu $ and each $s_\alpha $ decides a different value for $\dot {d}\cap x$ .

Proof Fix $s \in \mathbb {S}$ . For $x \in \mathcal {P}_{\nu ^+}(\lambda )$ , let $T(x)$ be the set of all $y \subseteq x$ such that some $s' \leq s$ forces $\dot {d}\cap x = y$ . Since $\dot {d}$ is thinly $\nu ^+$ -approximated, $|T(x)| \leq \nu $ for all x. We wish to construct $x \in \mathcal {P}_{\nu ^+}(\lambda )$ such that $|T(x)| = \nu $ .

Fix $x_0 \in \mathcal {P}_{\nu ^+}(\lambda )$ . We will build a $\subseteq $ -increasing sequence $\langle x_\alpha \mid \alpha < \nu \rangle $ such that for all $\alpha < \nu $ and all $y \in T(x_\alpha )$ , there are distinct $y_0$ and $y_1$ in $T(x_{\alpha +1})$ such that $y_0 \cap x_\alpha = y_1 \cap x_\alpha = y$ . We do so recursively.

Fix $\alpha < \nu $ , and let $y \in T(x_\alpha )$ as witnessed by a condition $s' \leq s$ . Since $\dot {d}$ is forced not to be in V, we can find a set $x_{\alpha +1}^y \supseteq x_\alpha $ , distinct $y^{\prime }_0, y^{\prime }_1 \in \mathcal {P}_{\nu ^+}(\lambda )$ , and conditions $s_0, s_1 \leq s'$ such that $s_0 \Vdash \dot {d}\cap x_{\alpha +1}^y = y^{\prime }_0$ and $s_1 \Vdash \dot {d}\cap x_{\alpha +1}^y = y^{\prime }_1$ . (If such objects did not exist, then a single condition could decide $\dot {d}$ , contradicting the assumption that $\dot {d} \notin V$ .) Since $s_0$ and $s_1$ extend $s'$ , we must have $y^{\prime }_0 \cap x_\alpha = y^{\prime }_1 \cap x_\alpha = y$ . Let $x_{\alpha +1} = \bigcup _{y \in T(x_\alpha )} x_{\alpha +1}^y$ . For each $y \in T(x_\alpha )$ , let $s_0$ , $s_1$ , $y^{\prime }_0$ , and $y^{\prime }_1$ be as above. Extend $s_0$ to decide the value of $\dot {d}\cap x_{\alpha +1}$ to be some $y_0 \supseteq y^{\prime }_0$ ; similarly, extend $s_1$ to force $\dot {d}\cap x_{\alpha +1} = y_1$ for some $y_1 \supseteq y^{\prime }_1$ . Then $y_0$ and $y_1$ are distinct elements of $T(x_{\alpha +1})$ , and $y_0\cap x_\alpha = y_1\cap x_\alpha = y$ , so $x_{\alpha +1}$ has the desired properties. This covers successor stages; at limit stages we take unions.

Now let $x = \bigcup _{\alpha < \nu } x_\alpha $ . Let $y \in T(x)$ . Then for all $\alpha < \nu $ , $y\cap x_\alpha \in T(x_\alpha )$ . By construction, we have distinct $y_0$ and $y_1$ in $T(x_\alpha +1)$ (as witnessed by conditions $s_0$ and $s_1$ ) such that $y_0 \cap x_\alpha = y_1 \cap x_\alpha = y\cap x_\alpha $ . Moreover, any extensions of these conditions that decide $\dot {d}\cap x$ must decide incompatible values. Since there are $\nu $ -many of these splittings, there must be at least $\nu $ possible values for $\dot {d}\cap x$ . We conclude that $|T(x)| = \nu $ .

Claim 2.19. Suppose $\nu $ is a singular strong limit cardinal with cofinality $\tau $ , and let $\lambda \geq \nu ^+$ . Let $\mathbb {P}$ and $\mathbb {Q}$ be notions of forcing such that $|\mathbb {P}|\leq \mu $ and $\mathbb {Q}$ is $\mu ^+$ -closed for some $\mu $ with $\tau < \mu < \nu $ . Let $\dot {d}$ be a $\mathbb {P}\times \mathbb {Q}$ -name for a subset of $\lambda $ that is thinly $\nu ^+$ -approximated by $\mathbb {Q}$ over the generic extension of V by $\mathbb {P}$ , such that $1_{\mathbb {P}\times \mathbb {Q}} \Vdash \dot {d}\notin V[\dot {G}_{\mathbb {P}}]$ .

Suppose $p_0 \in \mathbb {P}$ , and let $\langle q_\alpha \mid \alpha < \nu \rangle $ be a sequence of conditions in $\mathbb {Q}$ . Then there is a sequence $\langle q^{\prime }_\alpha \mid \alpha < \nu \rangle $ with $q^{\prime }_\alpha \leq q_\alpha $ for each $\alpha < \nu $ , along with a set $x \in \mathcal {P}_{\nu ^+}(\lambda )$ , such that for all $\alpha < \beta $ there is some $p_{\alpha \beta } \leq p_0$ so that $(p_{\alpha \beta },q^{\prime }_\alpha )$ and $(p_{\alpha \beta }, q^{\prime }_\beta $ ) force incompatible information about $\dot {d}\cap x$ .

Proof Let G be generic for $\mathbb {P}$ containing $p_0$ , and work for the moment in $V[G]$ . For all $\alpha < \nu $ , by Claim 2.18 there is $x_\alpha \in \mathcal {P}_{\nu ^+}(\lambda )$ such that there are $\nu $ -many possible values for $\dot {d}\cap x_\alpha $ that can be forced by extensions of $q_\alpha $ . Returning to V, we note that these possible values are also possible values of $\dot {d}\cap x_\alpha $ that can be forced by extensions of $(p_0, q_\alpha )$ . Since $\mathbb {P}$ does not collapse $\nu $ or $\nu ^+$ , we conclude that (in V) there are $\nu $ -many possibilities for $\dot {d}\cap x_\alpha $ that can be forced by extensions of $(p_0, q_\alpha )$ .

Let $x = \bigcup _{\alpha < \nu } x_\alpha $ ; note that $x \in \mathcal {P}_{\nu ^+}(\lambda )$ . Since $(p_0,q_\alpha )$ has $\nu $ -many extensions deciding pairwise-distinct values for $\dot {d}\cap x_\alpha $ , by extending each further we can obtain $\nu $ -many extensions of each $(p_0, q_\alpha )$ deciding pairwise-distinct values for $\dot {d}\cap x$ . (To do this, simply take the $\nu $ -many extensions that disagree on $\dot {d}\cap x_\alpha $ , and extend them further to decide $\dot {d}\cap x$ . Since $x_\alpha \subseteq x$ , these further extensions must also disagree on $\dot {d}\cap x$ .)

For all $\alpha < \nu $ , we will inductively define a condition $q^{\prime }_\alpha \leq q_\alpha $ and functions $e_\alpha :\mathbb {P}\to \mathbb {P}$ and $f_\alpha :\mathbb {P}\to \mathcal {P}(x)$ such that the following hold:

• • $e_\alpha (p) \leq p.$

• • $(e_\alpha (p), q^{\prime }_\alpha )\Vdash _{\mathbb {P}\times \mathbb {Q}} \dot {d}\cap x = f_\alpha (p).$

• • For all $\alpha < \beta < \nu $ , $f_\alpha (p) \neq f_\beta (p_0)$ for all $p\in \mathbb {P}$ .

The sequence $\langle q^{\prime }_\alpha \mid \alpha < \nu \rangle $ will be the desired object; for each pair $\alpha < \beta $ we will choose $p_{\alpha \beta }$ to be $e_\alpha (e_\beta (p_0))$ . We build these objects with a double induction, inducting over both $\alpha $ and the elements of $\mathbb {P}$ .

Enumerate $\mathbb {P}$ by $\langle p_\xi \mid \xi < \mu \rangle $ , starting with the already determined condition $p_0$ . Suppose we have defined $q^{\prime }_\delta , e_\delta ,$ and $f_\delta $ for all $\delta < \alpha $ . Inducting on $\xi $ , we wish to define $q_\alpha ^\xi \leq q_\alpha $ , $e_\alpha (p_\xi ) \leq p_\xi $ , and $f_\alpha (p_\xi )$ such that:

• • $\langle q_\alpha ^\xi \mid \xi < \mu \rangle $ forms a decreasing sequence below $q_\alpha $ ,

• • $(e_\alpha (p_\xi ), q_\alpha ^\xi )\Vdash _{\mathbb {P}\times \mathbb {Q}}\dot {d}\cap x = f_\alpha (p_\xi )$ , and

• • $f_\delta (p) \neq f_\alpha (p_0)$ for all $\delta < \alpha $ and all $p \in \mathbb {P}$ .

Since $(p_0, q)$ has $\nu $ -many extensions deciding $\dot {d}\cap x$ differently, and we have used at most $\max (\alpha , |\mathbb {P}|) < \nu $ of these extensions in earlier stages, we can pick $e_\alpha (p_0), q_\alpha ^0,$ and $f_\alpha (p_0)$ as desired. At successor stages $\xi +1$ , we choose an extension $(e_\alpha (p_{\xi +1}), q_\alpha ^{\xi +1})$ of $(p_{\xi +1}, q_\alpha ^\xi )$ deciding the value of $\dot {d}\cap x$ , and set $f_\alpha (p_{\xi +1})$ to be that value. At limit stages $\gamma $ , using the $\mu ^+$ -closure of $\mathbb {Q}$ we select $e_\alpha (p_\gamma ) \leq p_\gamma $ and a lower bound $q_\alpha ^{\gamma }$ of $\langle q_\alpha ^\xi \mid \xi < \gamma \rangle $ , such that $(e_\alpha (p_\gamma ), q_\alpha ^\gamma )$ decides the value of $\dot {d}\cap x$ . Set $f_\alpha (p_{\gamma })$ to be this value. When this induction on the elements of $\mathbb {P}$ is finished, we define $q^{\prime }_\alpha $ to be a lower bound of $\langle q_\alpha ^\xi \mid \xi < \mu \rangle $ . This process constructs functions $e_\alpha $ and $f_\alpha $ , as well as the sequence $\langle q^{\prime }_\alpha \mid \alpha < \nu \rangle $ , with the desired properties.

Finally we verify that the sequence $\langle q^{\prime }_\alpha \mid \alpha < \nu \rangle $ meets the requirements in the statement of the lemma. Fix $\alpha < \beta < \nu $ . Let $p_{\alpha \beta } = e_\alpha (e_\beta (p_0))$ . By construction, $e_\beta (p_0)\Vdash \dot {d}\cap x = f_\beta (p_0)$ ; since $p_{\alpha \beta } \leq e_\beta (p_0)$ , $(p_{\alpha \beta }, q^{\prime }_\beta )$ also forces this. On the other hand, $(p_{\alpha \beta }, q^{\prime }_\alpha )\Vdash \dot {d}\cap x = f_\alpha (e_\beta (p_0))$ . By construction, these must be different values.

Proof Let $\lambda $ be a some ordinal, and let $\dot {d}$ be a $\mathbb {P}\times \mathbb {Q}$ -name for a subset of $\lambda $ that is thinly $\nu ^+$ -approximated by $\mathbb {Q}$ over the generic extension by $\mathbb {P}$ . Suppose further that the empty condition in $\mathbb {P}\times \mathbb {Q}$ forces $\dot {d}\notin V[\dot {G}_{\mathbb {P}}]$ . For each $\sigma \in \nu ^{<\tau }$ we will build a pair $(q_\sigma , x_\sigma )$ with $q_\sigma \in \mathbb {Q}$ and $x_\sigma \in (\mathcal {P}_{\nu ^+}(\lambda ))^V$ , along with a collection of dense sets $\langle D_\sigma ^{\alpha \beta }\mid \alpha < \beta < \nu \rangle $ , such that $q_{\sigma '} \leq q_\sigma $ whenever $\sigma '$ extends $\sigma $ , and for all $\sigma \in \nu ^{<\tau }$ , all $\alpha < \beta $ , and all $p \in D^{\alpha \beta }_{\sigma }$ , $(p, q_{\sigma ^\smallfrown \alpha })$ and $(p, q_{\sigma ^\smallfrown \beta })$ force contradictory information about $\dot {d}\cap x_{\sigma }$ .

Fix $q \in \mathbb {Q}$ . We will construct $\langle q_\alpha \mid \alpha < \nu \rangle , x \in \mathcal {P}_{\nu ^+}(\lambda )$ , and a dense sets $D_{\alpha \beta }$ , such that each $q_\alpha \leq q$ , and for all $\alpha < \beta $ and all $p \in D_{\alpha \beta }$ , p forces that $q_\alpha $ and $q_\beta $ force contradictory information about $\dot {d}\cap x$ .

We do so by recursion, iterating over all elements of $\mathbb {P}$ . Enumerate the elements of $\mathbb {P}$ by $\langle p_\delta \mid \delta < \mu \rangle $ . Set $D_{\alpha \beta }^0 = \emptyset $ , and for all $\alpha < \nu $ , let $q_\alpha ^0 = q$ . First we consider the successor stage $\delta +1$ . Applying Claim 2.19 to $p_{\delta }$ and the sequence $\langle q^\delta _\alpha \mid \alpha < \nu \rangle $ , we obtain $\langle q^{\delta +1}_\alpha \mid \alpha < \nu \rangle $ with $q^{\delta +1}_\alpha \leq q^\delta _\alpha $ for each $\alpha < \nu $ , $x_{\delta +1} \in \mathcal {P}_{\nu ^+}(\lambda )$ , and strengthenings $p_{\alpha \beta }^{\delta } \leq p_{\delta }$ for all $\alpha < \beta < \nu $ , so that $(p_{\alpha \beta }^{\delta }, q^{\delta +1}_\alpha )$ and $(p_{\alpha \beta }^{\delta }, q^{\delta +1}_\beta )$ force contradictory information about $\dot {d}\cap x_{\delta +1}$ . We set $D_{\alpha \beta }^{\delta +1} = D_{\alpha \beta }^\delta \cup \{p_{\alpha \beta }^{\delta }\}$ .

At each limit stage $\gamma $ , we take unions for $x_\gamma $ and each $D_{\alpha \beta }^\gamma $ , and lower bounds to produce each $q^\gamma _\alpha $ . Since $\mathbb {Q}$ is $\mu ^+$ -closed, we will always be able to take these lower bounds. Let $q_\alpha $ be a lower bound of the sequence $\langle q^\delta _\alpha \mid \delta < \mu \rangle $ . Finally, setting $x = \bigcup _{\delta <\mu }x_\delta $ , we obtain the desired objects. Repeating this construction, we can obtain $(q_\sigma , x_\sigma )$ and $\langle D_\sigma ^{\alpha \beta }\mid \alpha < \beta <\nu \rangle $ as desired for any $\sigma \in \nu ^{<\tau }$ and any $\alpha < \beta < \nu $ .

Let $x = \bigcup _{\sigma \in \nu ^{<\tau }} x_\sigma $ . For all $f \in \nu ^\tau $ , by the closure of $\mathbb {Q}$ we can find $q^{\prime }_f$ such that $q^{\prime }_f\leq q_{f\upharpoonright \alpha }$ for all $\alpha <\tau $ . Let G be generic for $\mathbb {P}$ , and note that it will intersect every $D^{\alpha \beta }_{\sigma }$ . Working in $V[G]$ , let $q_f \leq q_f'$ be a strengthening that decides $\dot {d}\cap x$ .

Let $f \neq g$ be elements of $\nu ^\tau $ , and let $\alpha $ be the largest such that $f\upharpoonright \alpha = g\upharpoonright \alpha $ . Let $\zeta = f(\alpha )$ , and $\xi = g(\alpha )$ . Since G intersects $D^{\zeta \xi }_{f\upharpoonright \alpha }$ , we see that $q_{(f\upharpoonright \alpha ) ^\smallfrown \zeta }$ and $q_{(f\upharpoonright \alpha ) ^\smallfrown \xi }$ force contradictory information about $\dot {d}\cap x_{f\upharpoonright \alpha } \subseteq \dot {d}\cap x$ . Since f extends $(f\upharpoonright \alpha ) ^\smallfrown \zeta $ and g extends $(f\upharpoonright n) ^\smallfrown \xi $ , it follows that $q_f$ and $q_g$ disagree on $\dot {d} \cap x$ .

We conclude that each $f \in \nu ^\tau $ forces a distinct value for $\dot {d}\cap x$ , so there must be at least $\nu ^\tau> \nu $ potential values for $\dot {d}\cap x$ . But since $\dot {d}$ is thinly $\nu ^+$ -approximated, there are at most $\nu $ potential values for $\dot {d}$ and thus for $\dot {d}\cap x$ . This is a contradiction.