2 The forcing posets

In this section, we describe the two main posets in our construction: a Mitchell style forcing and a diagonal Prikry forcing.

We will use the following version of the Mitchell forcing from [Reference Mitchell3]. Given regular cardinals $\kappa ,\, \mu ,\, \lambda $ where $\kappa < \mu < \lambda $ , $\kappa ^{<\kappa } = \kappa $ and $\lambda $ is strongly inaccessible, we define a forcing poset $\mathbb M = {\mathbb M}(\kappa ,\, \mu ,\, \lambda )$ . Conditions are pairs $(p,\, q)$ such that:

1. 1. $p \in \operatorname {\mathrm {Add}}(\kappa ,\, \lambda )$ .

2. 2. q is a partial function on $\lambda $ such that $\vert q \vert < \mu $ .

3. 3. For all $\alpha \in \operatorname {\mathrm {dom}}(q)$ , $q(\alpha )$ is an $ \operatorname {\mathrm {Add}}(\kappa ,\, \alpha )$ -name for a condition in $ \operatorname {\mathrm {Add}}(\mu ,\, 1)$ .

The ordering is given by $(p_1,\, q_1) \le (p_0,\, q_0)$ iff $p_1 \le p_0$ , $ \operatorname {\mathrm {dom}}(q_0) \subseteq \operatorname {\mathrm {dom}}(q_1)$ , and $p_1 \upharpoonright \alpha \operatorname {\mathrm {\Vdash }} q_1(\alpha ) \le q_0(\alpha )$ for all $\alpha \in \operatorname {\mathrm {dom}}(q_0)$ .

We refer the reader to [Reference Cummings and Foreman1] for a detailed account of the properties of $\mathbb M$ . We will use the following facts:

• • $\mathbb M$ is $\kappa $ -directed closed and $\lambda $ -cc.

• • $\mathbb M$ is the projection of $ \operatorname {\mathrm {Add}}(\kappa ,\, \lambda ) \times {\mathbb R}$ , where $\mathbb R$ is $\mu $ -closed.Footnote ¹

• • All $<\mu $ -sequences of ordinals in the generic extension by $\mathbb M$ lie in the subextension by $ \operatorname {\mathrm {Add}}(\kappa ,\, \lambda )$ .

• • $\mathbb M$ preserves all cardinals except those in the interval $(\mu ,\, \lambda )$ , which are collapsed to $\mu $ .

• • $\mathbb M$ forces $2^\kappa = \lambda = \mu ^+$ .

If $\lambda $ is measurableFootnote ² then $\lambda $ still has the tree property in the extension by $\mathbb M$ . The key points are that if $j: V \rightarrow M$ is an embedding with critical point $\lambda $ into a model M such that ${}^\lambda M \subseteq M$ , then:

• • In M, $\mathbb M$ is an initial segment of $j(\mathbb M)$ .

• • $j \restriction {\mathbb M}$ is a complete embedding of ${\mathbb M}$ into $j({\mathbb M})$ .

• • If G is $\mathbb M$ -generic then in $M[G]$ the quotient forcing $j({\mathbb M})/G$ is the projection of a product ${\mathbb A} \times {\mathbb Q}$ , where $\mathbb A = \operatorname {\mathrm {Add}}(\kappa ,\, j(\lambda ) - \lambda )$ .

• • In $V[G]$ , $\mathbb A$ is $\kappa ^+$ -Knaster and $\mathbb Q$ is $\mu $ -closed.

Next, we describe the diagonal supercompact Prikry forcing $\mathbb {P}$ . This forcing was first defined by Gitik and Sharon [Reference Gitik and Sharon2] to prove the consistency of failure of SCH at $\kappa $ with failure of $\square ^*_\kappa $ , and then modified by Neeman [Reference Neeman4] to prove the consistency of failure of SCH at $\kappa $ with the tree property holding at $\kappa ^+$ .

Let $\langle \kappa _n \mid n < \omega \rangle $ be an increasing sequence of regular cardinals. Let $\kappa =\kappa _0$ and assume that $\kappa $ is a supercompact cardinal. Let $\mu = (\text {sup}_n\kappa _n)^+$ . Let U be a normal measure on $\mathcal {P}_\kappa (\mu )$ , and for every n let $U_n$ be the projection of U to $\mathcal {P}_\kappa (\kappa _n)$ . The measure $U_n$ concentrates on x such that $x \cap \kappa $ is an inaccessible cardinal: we only consider $x \in P_\kappa (\kappa _n)$ with this property, and write $\kappa _x$ for $x \cap \kappa $ .

Conditions in $\mathbb {P}$ are of the form $p =\langle x_0,\, \ldots ,\, x_{n-1},\, A_n,\, A_{n+1} \ldots \rangle $ , where each $x_i \in \mathcal {P}_\kappa (\kappa _i)$ , $x_i\prec x_{i+1}$ (i.e. $x_i\subseteq x_{i+1}$ and $|x_i|<\kappa _{x_{i+1}}$ ), and each $A_k \in U_k$ . We say that $ \operatorname {\mathrm {lh}}(p) = n$ and the stem of p is $ \operatorname {\mathrm {s}}(p)= \langle x_0,\, \ldots ,\, x_{n-1} \rangle $ .

The ordering is given by $q\leq p$ iff:

• • $ \operatorname {\mathrm {lh}}(q) \ge \operatorname {\mathrm {lh}}(p)$ .

• • For all i with $i < \operatorname {\mathrm {lh}}(p)$ , $x^q_i = x^p_i$ .

• • For all i with $ \operatorname {\mathrm {lh}}(p) \leq i< \operatorname {\mathrm {lh}}(q)$ , $x^q_i \in A^p_i$ .

• • For all i with $ \operatorname {\mathrm {lh}}(q) \le i$ , $A^q_i\subseteq A^p_i$ .

We say that q is a direct extension of p, and write $q \leq ^* p$ , if $q \leq p$ and they have the same length,

The following are standard facts about $\mathbb {P}$ :

1. 1. $\mathbb {P}$ has the Prikry property: for any statement in the forcing language $\phi $ and any $p\in \mathbb {P}$ , there is $q \leq ^* p$ deciding $\phi $ . In particular $\mathbb P$ adds no bounded subsets of $\kappa $ .

2. 2. Conditions with the same stem are compatible, and so $\mathbb {P}$ has the $\mu $ -chain condition.

3. 3. $\mathbb {P}$ forces that $ \operatorname {\mathrm {cf}}(\kappa _n)=\omega $ for all $n\geq 0$ .

4. 4. $\mathbb {P}$ preserves $\kappa $ and $\mu $ , and forces that $\mu =\kappa ^+$ .

Given a formula $\phi $ and a stem h, we will say that $h\Vdash ^*\phi $ if there is a condition p with stem h which forces $\phi $ . Note that by the Prikry property, for every $\phi ,\, h$ either we have $h\Vdash ^*\phi $ or $h\Vdash ^*\neg \phi $ .

3 The main construction

Let $\langle \kappa _n \mid n < \omega \rangle $ be an increasing sequence of indestructible supercompact cardinals with limit $\kappa _\omega $ . Let $\mu =\kappa _\omega ^+$ and $\kappa = \kappa _0$ . Suppose also that $\lambda $ is a measurable cardinal above $\mu $ . Let $\mathbb {M}$ be the Mitchell forcing ${\mathbb M}(\kappa ,\, \mu ,\, \lambda )$ from the last section, and G be $\mathbb {M}$ -generic. Since $\mathbb M$ is $\kappa $ -directed closed, $\kappa $ is still supercompact in $V[G]$ .

In $V[G]$ let $\mathbb {P}$ be the diagonal Prikry forcing described in the preceding section, built using normal measures $U_n$ on $\mathcal {P}_\kappa (\kappa _n)$ for $n<\omega $ . Let H be $\mathbb {P}$ -generic over $V[G]$ . In $V[G][H]$ , $\kappa $ is preserved, $ \operatorname {\mathrm {cf}}(\kappa )=\omega $ , $\kappa ^+=\mu $ , $\kappa ^{++}=\lambda =2^\kappa $ .

Lemma 3.2. In $V[G]$ there exist an unbounded set $J\subseteq \lambda $ , a stem h, and a function $f:J\rightarrow \mu $ , such that for all $\alpha <\beta $ both in J,

\begin{align*} h\Vdash^*\langle\alpha,\, f(\alpha)\rangle<_{\dot{T}}\langle\beta,\, f(\beta)\rangle.\end{align*}

Since there are only $\kappa _\omega $ -many stems and the cofinality of $\lambda $ is $\mu $ in $M[G^*]$ , there exist an unbounded $J\subseteq \lambda $ and a stem $\bar {h}$ , such that for all $\alpha \in J$ we have $\bar {h}=h_\alpha $ . For all $\alpha <\lambda $ , if $\alpha \in J$ then $\bar {h}\Vdash ^* \langle \alpha ,\,\xi _\alpha \rangle <_{j(\dot {T})} u$ . Increasing J if needed we may assume that the converse also holds.

Define $f:J\rightarrow \mu $ by $f(\alpha )=\xi _\alpha $ . Then if $\beta \in J$ , we have that for all $\alpha <\beta $ , $\alpha \in J$ if and only if there is some $\xi <\mu $ such that $\bar {h}\Vdash ^*\langle \alpha ,\,\xi \rangle <_{\dot {T}}\langle \beta ,\, f(\beta )\rangle $ , and in this case $\xi _\alpha $ is the unique $\xi $ with this property. This implies that for all $\gamma < \lambda $ both $J \cap \gamma $ and $f \restriction \gamma $ are in $V[G]$ .

Next we want to find such a J and f in $V[G]$ . Since $\mathbb {A}^2$ has the $\kappa ^+$ -approximation property in $V[G][K]$ , it is easy to see that the versions of J and f which we just constructed lie in $V[G][K]$ . In $V[G][K\times A]$ , for every stem h extending $\bar {h}$ , let $J_h = \{\alpha \in J\mid \exists \xi (h\Vdash ^* \langle \alpha ,\, \xi \rangle <_{j(\dot {T})}u)\}$ and define $f_h:J_h\rightarrow \mu $ by setting $f_h(\alpha )$ to be the unique $\xi $ witnessing that $\alpha \in J_h$ . As above, if $J_h$ is unbounded then $J_h$ and $f_h$ are in $V[G][K]$ . Let $\bar {\alpha }<\lambda $ be forced by $\mathbb A$ to be a bound for all $J_h$ which are bounded in $\lambda $ .

For each h, let $\dot {J}_h,\,\dot {f}_h\in V[G][K]$ be $\mathbb {A}$ -names for $J_h,\,f_h$ . In $V[G][K]$ let $\mathcal {C}_h =\{ C\subseteq \lambda \mid C \setminus \bar {\alpha }\neq \emptyset ,\, \exists b\in \mathbb {A}(b\Vdash C=\dot {J}_h)\}$ . Then by the above remark and since $\mathbb {A}$ is $\kappa ^+$ -Knaster, if $J_h$ is unbounded in $\lambda $ , we have that $1\leq |\mathcal {C}_h|\leq \kappa $ . Enumerate $\mathcal {C}_h$ as $\langle C_{h,\,\eta } \mid \eta <\kappa \rangle $ (possibly with repetitions) when it is not empty. For every $\eta <\kappa $ pick some $b_\eta \in \mathbb {A}$ forcing that $C_{h,\,\eta }=\dot {J}_h$ , and let $f_{h,\,\eta } : C_{h,\,\eta } \rightarrow \mu $ be defined by setting $f_{h,\,\eta }(\alpha )$ equal to the unique $\xi $ witnessing $\alpha \in C_{h,\,\eta }$ as forced by $b_\eta $ . That is to say, $b_\eta \Vdash _{\mathbb {A}}^{V[G][K]} (h \Vdash ^* \langle \alpha ,\, \xi \rangle <_{j(\dot {T})} u)$ .

Working in $V[G]$ , for each h such that $\dot {\mathcal {C}}_h$ is not the empty set and for each $\eta <\kappa $ , fix ${\mathbb Q}$ -names $\dot {C}_{h,\,\eta }$ and $\dot {f}_{h,\, \eta }$ . We want to show that for some h and $\eta $ the pair $(\dot {C}_{h,\,\eta },\, \dot {f}_{h,\,\eta })$ can be forced to be in $V[G]$ . Towards a contradiction, suppose that for all h and $\eta $ we have $1_{\mathbb {Q}}\Vdash (\dot {C}_{h,\,\eta },\, \dot {f}_{h,\,\eta })\notin V[G]$ .

We say that $q_0,\, q_1$ in $\mathbb {Q}$ force contradictory information about $\dot {f}_{h,\,\eta }(\alpha )$ if $q_0,\, q_1$ both decide “ $\alpha \in \dot {C}_{h,\,\eta }$ ” with at least one of them forcing a positive decision, and $(q_0,\, q_1)\Vdash _{\mathbb {Q}\times \mathbb {Q}}\ \dot {f}_{h,\,\eta }[\dot {G}_L](\alpha )\neq \dot {f}_{h,\,\eta }[\dot {G}_R](\alpha )$ .

Since we have assumed that each $\dot {f}_{h,\,\eta }$ is forced to be new, we have the following in $V[G]$ :

Working in $V[G]$ we build a binary tree of conditions in $\mathbb Q$ , forcing contradictory information about $f_{h,\, \eta }$ for every pair $(h,\,\eta )$ at every splitting. More precisely we build $\langle q_\sigma ,\, \alpha _{\sigma ,\, h,\,\eta } \mid \sigma \in 2^{<\kappa },\, h\text { a stem },\, h\sqsupseteq \bar {h},\, \eta <\kappa \rangle $ , such that for all $(\sigma ,\, h,\, \eta )$ the conditions $q_{\sigma ^\frown 0}$ and $q_{\sigma ^\frown 1}$ force contradictory information about $f_{h,\,\eta }(\alpha _{\sigma ,\, h,\,\eta })$ . We build this tree by induction on $|\sigma |$ , and at every stage we apply Claim 3.3 repeatedly for each h and $\eta $ . We use that $\mathbb {Q}$ is $\mu $ -closed and the number of pairs $(h,\,\eta )$ is $\kappa _\omega $ .

Let $\alpha ^*=\text {sup}_{\sigma ,\, h,\,\eta }\alpha _{\sigma ,\, h,\,\eta }$ . For each $i\in 2^\kappa $ , let $q_i\leq q_{i\upharpoonright \eta }$ for all $\eta <\kappa $ . Now let $q^{\prime }_i\leq q_i$ be such that there are $h_i$ , $\eta _i<\kappa $ , $\xi _i<\mu $ and $b\in \mathbb {A}$ such that:

• • ,

• • $(q^{\prime }_i,\,b)\Vdash _{\mathbb {Q}\times \mathbb {A}}\dot {C}_{h_i,\,\eta _i}=\dot {J}_{h_i},\, \dot {f}_{h_i}=\dot {f}_{h_i,\,\eta _i}$ Footnote ³ (and so $q^{\prime }_i\Vdash \alpha ^*\in \dot {C}_{h_i,\,\eta _i}$ ).

Note that for all $\alpha <\alpha ^*$ , $q^{\prime }_i\Vdash \dot {f}_{h_i,\,\eta _i}(\alpha )=\delta $ iff $h_i\Vdash ^*\langle \alpha ,\, \delta \rangle <_{\dot {T}}\langle \alpha ^*,\, \xi _i\rangle $ .

Since $2^\kappa =\lambda>\mu $ , there exist distinct $i,\, j \in 2^\kappa $ and $(h,\, \eta ,\, \xi )$ such that $h_i=h_j = h$ , $\eta _i = \eta _j = \eta $ and $\xi _i = \xi _j = \xi $ . Let $\sigma $ be the node where i and j split. By construction the conditions $q_i$ and $q_j$ cannot force contradictory information about $\dot {f}_{h,\,\eta }(\alpha _{h,\,\sigma ,\,\eta })$ . This is a contradiction, so in $V[G]$ we may find a stem h, set J and function f as required.⊣

Let $J,\, f,\,h$ be given by Lemma 3.2, and let $n=|h|$ . As above, let $\xi _\alpha = f(\alpha )$ .

We can now apply this lemma inductively as in [Reference Neeman4] to get conditions $\langle p_\alpha \mid \alpha \in J\setminus \rho \rangle $ so that for all $\alpha <\beta $ , $p_\alpha \land p_\beta \Vdash \langle \alpha ,\,\xi _\alpha \rangle <_{\dot {T}}\langle \beta ,\, \xi _\beta \rangle $ . By the $\mu $ chain condition, there are unboundedly many $\alpha $ ’s such that $p_\alpha $ is forced into a generic for $\mathbb {P}$ . This gives a branch.⊣

The arguments of [Reference Sinapova and Unger6, Section 3] can be used to show that $\mu $ has the tree property in $V[G][H]$ . The setting here is rather simpler and we just outline the argument. Let $\dot T \in V[G]$ be a $\mathbb P$ -name for a $\mu $ -tree. Recalling that $\mathbb M$ can be written as a projection of $ \operatorname {\mathrm {Add}}(\kappa ,\, \lambda ) \times {\mathbb R}$ for a suitable term forcing $\mathbb R$ , we force to obtain $V[G] \subseteq V[A \times B]$ where $A \times B$ is $ \operatorname {\mathrm {Add}}(\kappa ,\, \lambda ) \times {\mathbb R}$ -generic. The Prikry poset $\mathbb P$ has the same definition in $V[A \times B]$ as in $V[G]$ , and in this model the cardinals $\kappa _n$ are generically supercompact via reasonably nice forcing. We can use this to argue as in [Reference Neeman4] that T has a branch in $V[A \times B][H]$ , and then use the arguments of [Reference Sinapova and Unger6, Lemma 3.6] and [Reference Sinapova and Unger6, Lemma 3.8] to get a branch of T in $V[G][H]$ .

We have therefore given a new proof of the main theorem of [Reference Sinapova5].

Next, we use the above arguments to give another proof of the main theorem of [Reference Sinapova and Unger6].

Proof We use the same forcing as in [Reference Sinapova and Unger6]. Namely, we prepare the ground model V, so that:

1. 1. $\kappa _n=\kappa ^{+n}$ , and $\kappa _n$ is generically supercompact.

2. 2. After we force with $\mathbb {M}\times \operatorname {\mathrm {Add}}(\kappa ,\, \lambda ^+\setminus \lambda )$ , for each n we have normal measures $U_n$ on $\mathcal {P}_\kappa (\kappa _n)$ , and “guiding generics” $K_n$ , for $Col(\kappa ^{+\omega +3}, <j_{U_n}(\kappa ))$ over the ultrapower by $U_n$ .

3. 3. The guiding generics above are for a $\lambda ^+=\mu ^{++}$ -closed forcing since, they are pulled back from a generic for a $Col(\kappa ^{+\omega +3}, <j^*(\kappa ))$ where $j^*$ is a $\lambda $ -supercompact embedding with critical point $\kappa $ , and there the ultrapower is closed under $\lambda $ -sequences. For more details, see Section 2 of [Reference Sinapova and Unger6].

Let G be $\mathbb {M}\times \operatorname {\mathrm {Add}}(\kappa ,\, \lambda ^+\setminus \lambda )$ -generic. In $V[G]$ define the diagonal Prikry forcing $\mathbb {P}$ with interleaved collapses to make $\kappa =\aleph _{\omega ^2}$ as follows. Conditions are of the form $p=\langle d,\, x_0,\, c_o,\, \ldots ,\, x_{n-1},\, c_{n-1},\, A_n,\, C_n,\,\ldots \rangle $ , where,

1. 1. $\langle x_0,\, \ldots ,\, x_{n-1},\, A_n\ldots \rangle $ is in the diagonal Prikry forcing defined from the measures $\langle U_i\mid i<\omega \rangle $ , as defined in Section 2,

2. 2. $d\in Col(\omega ,\,\kappa _{x_0}^{+\omega })$ ,

3. 3. for $i<n-1$ , $c_i\in Col(\kappa _{x_i}^{+\omega +3},\, <\kappa _{x_{i+1}})$ , $c_{n-1}\in Col(\kappa _{x_{n-1}}^{+\omega +3}, <\kappa )$ ,

4. 4. for $i\geq n$ , $ \operatorname {\mathrm {dom}}(C_i)=A_i$ , each $C_i(x)\in Col(\kappa _x^{+\omega +3}, <\kappa )$ , $[C_i]_{U_i}\in K_i$ .

Let H be $\mathbb {P}$ -generic over $V[G]$ . In $V[G][H]$ , $\kappa =\aleph _{\omega ^2}$ , $\mu =\aleph _{\omega ^2+1},\, \lambda =\aleph _{\omega ^2+2},\, 2^\kappa =\aleph _{\omega ^2+3}$ . By the arguments of [Reference Sinapova and Unger6, Section 3], which we already sketched above in the setting of Theorem 3.1, we have the tree property at $\mu =\aleph _{\omega ^2+1}$ .

To prove the tree property at $\aleph _{\omega ^2+2}$ , we use the same approach as in Theorem 3.1. Note that we still have that the number of stems is $\kappa _\omega $ . Also, any two conditions with the same stem are compatible, as witnessed by a common lower bound with that same stem. So $\mathbb {P}$ has the $\mu $ -chain condition, and we can define as before the notion of $h\Vdash ^*\phi $ .

Let $j:V\rightarrow M$ be an elementary embedding with critical point $\lambda $ . As in Theorem 3.1, we lift j to $j:V[G]\rightarrow M[G^*]$ in $V[G][K\times A]$ , where K is generic for a $\mu $ -closed forcing $\mathbb {Q}$ and A is generic for a $\kappa ^+$ -Knaster forcing. The critical point is still above any fixed condition $p\in \mathbb {P}$ , that is to say $j(p)=p$ . So we have the analogue of Lemma 3.2: in $V[G]$ , there is an unbounded $J\subseteq \lambda $ , a stem h, and a function $f:J\rightarrow \mu $ , such that for all $\alpha <\beta $ both in J,

$$\begin{align*}h\Vdash^*\langle\alpha,\, f(\alpha)\rangle<_{\dot{T}}\langle\beta,\, f(\beta)\rangle.\end{align*}$$

Next we prove the following analogue of Lemma 3.4.

Lemma 3.10. There are $\rho <\lambda $ and a sequence $\langle A_\alpha ,\, C_\alpha \mid \alpha \in J\setminus \rho \rangle $ in $V[G]$ , such that:

• • $A_\alpha \in U_n$ , $ \operatorname {\mathrm {dom}}(C_\alpha )=A_\alpha $ , $C_\alpha (x)\in Col(\kappa _x^{+\omega +3}, <\kappa )$ for all $x \in A_\alpha $ , and $[C_\alpha ]$ is in the guiding generic $K_n$ .

• • For all $x \in A_\alpha \cap A_\beta $ , if $C_\alpha (x)$ and $C_\beta (x)$ are compatible, then

  $$ \begin{align*} h^\frown \langle x,\, C_\alpha(x)\cup C_\beta(x) \rangle \Vdash^* \langle \alpha,\, \xi_\alpha\rangle <_{\dot{T}} \langle \beta,\, \xi_\beta \rangle.\end{align*} $$

Proof For every $\alpha <\beta $ both in J, let $r_{\alpha ,\,\beta } \in \mathbb {P}$ be a condition with stem h, such that $r_{\alpha ,\,\beta }\Vdash \langle \alpha ,\, \xi _\alpha \rangle <_{\dot {T}}\langle \beta ,\, \xi _\beta \rangle $ . Denote $r_{\alpha ,\,\beta }=h^\frown (A_{\alpha ,\,\beta },\, C_{\alpha ,\,\beta })^\frown r_{\alpha ,\,\beta }\upharpoonright [n+1,\, \omega )$ . Since each $[C_{\alpha ,\, \beta }]$ belongs to a $\mu ^{++}$ -closed guiding generic $K_n$ , there is a lower bound $[C]$ in $K_n$ for all of them. So we may assume that $ \operatorname {\mathrm {dom}}(C)=\mathcal {P}_\kappa (\kappa _n)$ and that $C_{\alpha ,\,\beta }=C \upharpoonright A_{\alpha ,\,\beta }$ for all $\alpha $ and $\beta $ , by shrinking measure one sets and extending values of $C_{\alpha ,\, \beta }(x)$ as needed. It follows that for every $\alpha <\beta $ both in J, for $U_n$ -many x,

$$\begin{align*}h^\frown\langle x,\,C(x)\rangle\Vdash^*\langle\alpha,\, \xi_\alpha\rangle<_{\dot{T}}\langle\beta,\, \xi_\beta\rangle.\end{align*}$$

The rest of the proof is exactly as in Lemma 3.4, only we replace instances of $h^\frown x$ with $h^\frown \langle x,\, C(x)\rangle $ .⊣

Now we can apply Lemma 3.10 inductively, exactly as we applied Lemma 3.4 in the proof of Theorem 3.1. We construct conditions $\langle p_\alpha \mid \alpha \in J\setminus \rho \rangle $ so that for all $\alpha <\beta $ , $p_\alpha \land p_\beta \Vdash \langle \alpha ,\,\xi _\alpha \rangle <_{\dot {T}}\langle \beta ,\, \xi _\beta \rangle $ . By the $\mu $ chain condition, there are unboundedly many $\alpha $ ’s such that $p_\alpha $ is forced into a generic for $\mathbb {P}$ . This gives a branch.⊣