Proof. We follow the proof of [Reference Jensen12, Section 2, Lemma 2]. Let G be any $\mathbb {A}$ -generic filter with $a\in G$ . Then in $\mathrm {V}[G]$ , ${\mathord {\mathbb {B}}}/G$ is subproper, as verified by $\theta $ . Let $t=\dot {t}^{G}$ , $\bar {b}=\dot {\bar {b}}^G$ , $b=\dot {b}^G$ and $\sigma _0=\dot {\sigma }_0^{G}$ . Then $\sigma _0:{\bar {N}}\prec N$ , $\sigma _0({\bar {S}})=S$ , $h(b)\in G$ , $t\in {\bar {N}}$ and ${\bar {G}}=\sigma _0^{-1}"G$ is ${\bar {N}}$ -generic for $\mathbb {A}$ . Let

$$\begin{align*}\sigma_0^*:{\bar{N}}[{\bar{G}}]\prec N[G]\end{align*}$$

be the unique embedding extending $\sigma _0$ such that $\sigma _0^*({\bar {G}})=G$ .

We have that $H_\theta ^{V[G]}=H_\theta [G]\subseteq N[G]$ , and ${\bar {N}}[{\bar {G}}]$ is full in $\mathrm {V}[G]$ . We also have that $b/G\neq 0$ . To see this, note that since $b\in \mathrm {V}$ , it makes sense to write $b=\check {b}^G$ . By [Reference Jensen12, p. 91, Fact 3], we know that . So, since $h(b)\in G$ , this means that , which means precisely that $b/G\neq 0$ .

So, since ${\mathord {\mathbb {B}}}/G$ is subproper in $\mathrm {V}[G]$ , there is a condition $d\in {\mathord {\mathbb {B}}}/G$ with $d\le b/G$ such that whenever H is generic for ${\mathord {\mathbb {B}}}/G$ over $\mathrm {V}[G]$ with $d\in H$ , then in $\mathrm {V}[G][H]$ , there is an elementary embedding $\sigma ':{\bar {N}}[{\bar {G}}]\prec N[G]$ with $\sigma '({\bar {S}})=S$ , ${\bar {H}}=(\sigma ')^{-1}"H$ is $\bar {{\mathord {\mathbb {B}}}}/{\bar {G}}$ -generic over ${\bar {N}}[{\bar {G}}]$ and for all $i\le n$ , $\sup \sigma ^*_0"{\bar {\lambda }}_i=\sup \sigma '"{\bar {\lambda }}_i$ . We may moreover insist that $\sigma '$ maps any finite list of members of ${\bar {N}}[{\bar {G}}]$ the same way $\sigma _0^*$ does. Thus, we require that $\sigma '(t)=\sigma ^*_0(t)$ , $\sigma '(\bar {b})=\sigma _0^*(\bar {b})=b$ and $\sigma '({\bar {G}})=\sigma _0^*({\bar {G}})=G$ .

Let us temporarily fix such an H, and let $I=G*H$ . Let $\sigma _1=\sigma '{\restriction }{\bar {N}}$ . It follows that $\sigma '(\tau ^{\bar {G}})=(\sigma _1(\tau ))^G$ , for $\tau \in {\bar {N}}^{\bar {\mathbb {A}}}$ , since $\sigma '({\bar {G}})=G$ .

Then $\sigma '$ , G and I clearly satisfy conditions (C1), (C2), (C3), (C4), (C5), and (C7) above.

It follows also that (C6) is satisfied, that is, ${\bar {I}}=\sigma _1^{-1}"I$ is $\bar {{\mathord {\mathbb {B}}}}$ -generic over ${\bar {N}}$ : since ${\bar {G}}$ is $\bar {\mathbb {A}}$ -generic over ${\bar {N}}$ and ${\bar {H}}$ is $\bar {\bar {\mathbb {B}}}/{\bar {G}}$ -generic over ${\bar {N}}[{\bar {G}}]$ , it follows that $\bar {G}*\bar {H}$ is $\bar {{\mathord {\mathbb {B}}}}$ -generic over ${\bar {N}}$ . But, for $\bar {c}\in \bar {{\mathord {\mathbb {B}}}}$ , we have that $\bar {c}\in \bar {G}*\bar {H}$ iff $\bar {c}/\bar {G}\in \bar {H}$ iff $\sigma '(\bar {c}/\bar {G})=\sigma _1(\bar {c})/G\in H$ iff $\sigma _1(\bar {c})\in G*H=I$ . Thus, $\bar {G}*\bar {H}=\sigma _1^{-1}"I={\bar {I}}$ is $\bar {{\mathord {\mathbb {B}}}}$ -generic over ${\bar {N}}$ , as claimed.

So there are a name $\pi $ in $\mathrm {V}[G]^{{\mathord {\mathbb {B}}}/G}$ with $\sigma _1=\pi ^H$ and a condition $d\in ({\mathord {\mathbb {B}}}/G)^+$ that forces over $\mathrm {V}[G]$ with respect to ${\mathord {\mathbb {B}}}/G$ that $\pi $ has the properties listed.

Now, all of this is true in $\mathrm {V}[G]$ whenever G is $\mathbb {A}$ -generic over $\mathrm {V}$ , with $a\in G$ , and so, there are names $\dot {d},\dot {\pi }\in \mathrm {V}^{\mathbb {A}}$ such that $d=\dot {d}^{G}$ and $\pi =\dot {\pi }^G$ , and a forces the situation described. Let $\dot {\sigma }_1$ be a ${\mathord {\mathbb {B}}}$ -name such that $\dot {\sigma }_1^{G*H}=(\dot {\pi }^G)^H=\sigma _1$ .

The only thing that’s missing is the condition $c\in {\mathord {\mathbb {B}}}$ with $h(c)=a$ such that whenever $c\in I$ , I is ${\mathord {\mathbb {B}}}$ -generic over $\mathrm {V}$ , $G=I\cap \mathbb {A}$ and $\sigma _1=\dot {\sigma }_1^I$ , and $\sigma _0=\dot {\sigma }_0^G$ , it follows that (C1)–(C7) hold. To find the desired condition, first note that we may choose the name $\dot {d}$ in such a way that $\Vdash _{\mathbb {A}}\dot {d}\in \check {{\mathord {\mathbb {B}}}}/\dot {G}_{\mathbb {A}}$ and . Namely, given the original $\dot {d}$ such that a forces that $\dot {d}\in (\check {{\mathord {\mathbb {B}}}}/\dot {G}_{\mathbb {A}})^+$ and all the other statements listed above, there are two cases: if $a= {1\mathrm {\hspace {-0.5ex}l}} _{\mathbb {A}}$ , then since , it already follows that and $\Vdash _{\mathbb {A}}\dot {d}\in \check {{\mathord {\mathbb {B}}}}/\dot {G}_{\mathbb {A}}$ . If $a< {1\mathrm {\hspace {-0.5ex}l}} _{\mathbb {A}}$ , then let $\dot {e}\in \mathrm {V}^{\mathbb {A}}$ be a name such that $\Vdash _{\mathbb {A}}\dot {e}=0_{\check {{\mathord {\mathbb {B}}}}/\dot {G}_{\mathbb {A}}}$ , and mix the names $\dot {d}$ and $\dot {e}$ to get a name $\dot {d}'$ such that $a\Vdash _{\mathbb {A}}\dot {d}'=\dot {d}$ and $\neg a\Vdash _{\mathbb {A}}\dot {d}'=\dot {e}$ . Then $\dot {d}'$ is as desired. Clearly, $\Vdash _{\mathbb {A}}\dot {d}'\in \check {{\mathord {\mathbb {B}}}}/\dot {G}_{\mathbb {A}}$ . Since $a\Vdash _{\mathbb {A}}\dot {d}'=\dot {d}$ , it follows that , and since $\neg a\Vdash _{\mathbb {A}}\dot {d'}=\dot {e}$ , it follows that , so . So we could replace $\dot {d}$ with $\dot {d}'$ .

Then, by [Reference Jensen, Chong, Feng, Slaman, Woodin and Yang13, Section 0, Fact 4],Footnote ¹ there is a unique $c\in {\mathord {\mathbb {B}}}$ such that $\Vdash _{\mathbb {A}}\check {c}/\dot {G}_{\mathbb {A}}=\dot {d}$ , and it follows by [Reference Jensen, Chong, Feng, Slaman, Woodin and Yang13, Section 0, Fact 3] that

as wished.

Fact 2.7 [Reference Jensen, Chong, Feng, Slaman, Woodin and Yang13, p. 142].

Let $\vec {{\mathord {\mathbb {B}}}}={\langle {\mathord {\mathbb {B}}}_i\;|\;}{i<\alpha \rangle }$ be an RCS iteration.

1. (1) If $\lambda <\alpha $ and $ \mathrm {cf} (\lambda )=\omega $ , then ${\mathord {\mathbb {B}}}_\lambda $ is the inverse limit of $\vec {{\mathord {\mathbb {B}}}}{\restriction }\lambda $ .

2. (2) If $\lambda <\alpha $ and for every $i<\lambda $ , $\Vdash _{{\mathord {\mathbb {B}}}_i} \mathrm {cf} (\check {\lambda })>\omega $ , then $\bigcup _{i<\lambda }{\mathord {\mathbb {B}}}_i$ is dense in ${\mathord {\mathbb {B}}}_\lambda $ (that is, ${\mathord {\mathbb {B}}}_\lambda $ is formed using only eventually constant threads, making it the direct limit).

3. (3) If $i<\lambda $ and G is ${\mathord {\mathbb {B}}}_i$ -generic, then the above are true in $\mathrm {V}[G]$ about the iteration ${\langle {\mathord {\mathbb {B}}}_{i+j}/G\;|\;}{j<\alpha -i\rangle }$ .

Fact 2.8 (Baumgartner, see [Reference Viale, Audrito and Steila18, Theorem 3.13]).

Let ${\langle {\mathord {\mathbb {B}}}_i\;|\;}{i<\lambda \rangle }$ be an iteration such that for every $\alpha <\lambda $ , ${\mathord {\mathbb {B}}}_\alpha $ is ${<}\lambda $ -c.c., and such that the set of $\alpha <\lambda $ such that ${\mathord {\mathbb {B}}}_\alpha $ is the direct limit of $\vec {{\mathord {\mathbb {B}}}}{\restriction }\alpha $ is stationary. Then the direct limit of $\vec {{\mathord {\mathbb {B}}}}$ is ${<}\lambda $ -c.c.

Observation 2.11. Every standard RCS iteration of subproper forcings is nicely subproper.

Proof. Let $i<\alpha $ , and let $G_i$ be ${\mathord {\mathbb {B}}}_i$ -generic. We have to verify that conditions (1)–(3) of Definition 2.10 hold of ${\langle {\mathord {\mathbb {B}}}_{i+j}/G_i\;|\;}{j<\alpha -i\rangle }$ in $\mathrm {V}[G_i]$ .

Condition (1) is trivial: let $j+1<\alpha -i$ . Since $\Vdash _{{\mathord {\mathbb {B}}}_{i+j}}\check {{\mathord {\mathbb {B}}}}_{i+j+1}/\dot {G}_{i+j}$ is subproper, it clearly has to be that $\Vdash _{{\mathord {\mathbb {B}}}_{i+j}/G_i}(\check {{\mathord {\mathbb {B}}}}_{i+j+1}/\check {G}_i)/\dot {G}_{{\mathord {\mathbb {B}}}_{i+j}/\check {G}_i}$ is subproper in $\mathrm {V}[G_i]$ . It can be shown similarly that ${\langle {\mathord {\mathbb {B}}}_{i+j}/G_i\;|\;}{j<\alpha -i\rangle }$ is a standard iteration in $\mathrm {V}[G_i]$ .

For condition (2), let $\lambda <\alpha -i$ be a limit ordinal that has countable cofinality in $\mathrm {V}[G_i]$ . By Fact 2.7, part (3), the first two parts of that fact apply to ${\langle {\mathord {\mathbb {B}}}_{i+j}/G_i\;|\;}{j<\alpha -i\rangle }$ in $\mathrm {V}[G_i]$ . Thus, ${\mathord {\mathbb {B}}}_\lambda /G_i$ is the inverse limit of ${\langle {\mathord {\mathbb {B}}}_{i+j}/G_i\;|\;}{j<\lambda \rangle }$ . This implies condition (2)(a). And by Theorem 2.9, ${\mathord {\mathbb {B}}}_\lambda /G_i$ is subproper in $\mathrm {V}[G_i]$ , so condition (2)(b) holds.

Finally, condition (3) hold because part (2) of Fact 2.7 applies to ${\langle {\mathord {\mathbb {B}}}_{i+j}/G_i\;|\;}{j<\alpha -i\rangle }$ in $\mathrm {V}[G_i]$ , by part (3) of that fact.

Proof. The proof is a virtual repetition of the argument of Jensen’s proof of [Reference Jensen14, Section 4, Theorem 5, pp. 3–12], incorporating the changes necessitated by working with $\delta ({\mathord {\mathbb {B}}}_i)$ (as in [Reference Jensen14, Section 3, p. 2, Theorem 2]), except that it is somewhat simpler, because the limit of countable cofinality case is vacuous now. We have checked that the proof goes through, and so has Jensen (see [Reference Jensen14, Section 4, last three lines on p. 19]).

Proof. Let $\vec {{\mathord {\mathbb {B}}}}={\langle {\mathord {\mathbb {B}}}_i\;|\;}{i<\alpha \rangle }$ be a standard RCS iteration of forcings that are subproper and Souslin-tree preserving. We prove by induction on $\delta $ : whenever $h\le \delta <\alpha $ and $G_h$ is ${\mathord {\mathbb {B}}}_h$ -generic, then in $\mathrm {V}[G_h]$ , ${\mathord {\mathbb {B}}}_\delta /G_h$ is subproper and Souslin tree preserving.

It suffices to focus on the Souslin tree preservation, since by Theorem 2.12, ${\mathord {\mathbb {B}}}_\delta /G_h$ is subproper in $\mathrm {V}[G_h]$ whenever $h\le \delta <\alpha $ and $G_h$ is ${\mathord {\mathbb {B}}}_h$ -generic.

The successor case is trivial, as is the case $h=\delta $ . So let $\delta $ be a limit ordinal, and let $h<\delta $ .

Note that by Observation 2.11, ${\vec {{\mathord {\mathbb {B}}}}}$ is nicely subproper. We will use this, rather than that ${\vec {{\mathord {\mathbb {B}}}}}$ is an RCS iteration, whenever possible. Doing so will make it easier to slightly generalize the theorem later.

Case 1: there is an $i<\delta $ such that $ \mathrm {cf} (\delta )\le \delta ({\mathord {\mathbb {B}}}_i)$ .

Fix such an i. Then whenever $i<j<\delta $ , $\Vdash _j \mathrm {cf} (\check {\delta })\le \omega _1$ . It suffices to prove

1. (A) if $i<j<\delta $ and $G_j$ is ${\mathord {\mathbb {B}}}_j$ -generic, then in $\mathrm {V}[G_j]$ , ${\mathord {\mathbb {B}}}_\delta /G_j$ is Souslin tree preserving.

For if we have done so, then the full claim follows: let $h\le i$ , and let $G_h\subseteq {\mathord {\mathbb {B}}}_h$ be generic. By Theorem 2.12, we know that ${\mathord {\mathbb {B}}}_\delta /G_h$ is subproper in $\mathrm {V}[G_h]$ . Thus, it suffices to prove that if T is some Souslin tree in $\mathrm {V}[G_h]$ and H is ${\mathord {\mathbb {B}}}_\delta /G_h$ -generic over $\mathrm {V}[G_h]$ , then T is a Souslin tree in $\mathrm {V}[G_h][H]$ . But letting $G_{i+1}=(G_h*H)\cap {\mathord {\mathbb {B}}}_{i+1}$ , we know inductively that T is Souslin in $\mathrm {V}[G_{i+1}]=\mathrm {V}[G_h][G_{i+1}/G_h]$ , and so, by $(A)$ , T is Souslin in $\mathrm {V}[G_{i+1}*((G_h*H)/G_{i+1})]=\mathrm {V}[G_h][H]$ .

To prove $(A)$ , we would now have to fix a $j<\delta $ , a $G_j\subseteq {\mathord {\mathbb {B}}}_j$ that’s generic, and a T such that in $\mathrm {V}[G_j]$ , T is Souslin. We’d have to prove in $\mathrm {V}[G_j]$ that ${\mathord {\mathbb {B}}}_\delta /G_j$ preserves T as a Souslin tree. But the iteration ${\langle {\mathord {\mathbb {B}}}_{j+i}/G_j\;|\;}{i<\delta -j\rangle }$ is RCS (see Fact 2.7) in $\mathrm {V}[G_j]$ , and it satisfies everything in $\mathrm {V}[G_j]$ that we assumed about ${\vec {{\mathord {\mathbb {B}}}}}$ in $\mathrm {V}$ , with the addition of the fact that in $\mathrm {V}[G_j]$ , $ \mathrm {cf} (\delta )\le \omega _1$ .

Thus, it suffices to show:

1. (B) if $ \mathrm {cf} (\delta )\le \omega _1$ then ${\mathord {\mathbb {B}}}_\delta $ is Souslin tree preserving.

For the argument, carried out in $\mathrm {V}[G_j]$ would prove $(A)$ .

Note that by arguing in $\mathrm {V}[G_j]$ , but pretending $\mathrm {V}[G_j]$ is $\mathrm {V}$ , we effectively absorbed T into $\mathrm {V}$ . It is this step that’s not necessary if one only wants to preserve one fixed ground model Souslin tree.

To prove (B), let us fix a Souslin tree T.

Note that if $ \mathrm {cf} (\delta )=\omega _1$ , then for every $i<\delta $ , $\Vdash _{{\mathord {\mathbb {B}}}_i} \mathrm {cf} (\check {\delta })=\check {\omega }_1$ , as ${\mathord {\mathbb {B}}}_i$ preserves $\omega _1$ . Thus, ${\mathord {\mathbb {B}}}_\delta $ is the direct limit of $\vec {{\mathord {\mathbb {B}}}}{\restriction }\delta $ , by part (3) of Definition 2.10, that is, $\bigcup _{i<\delta }{\mathord {\mathbb {B}}}_i$ is dense in ${\mathord {\mathbb {B}}}_\delta $ in this case. Let us denote this dense set by X.

If, on the other hand, $ \mathrm {cf} (\delta )=\omega $ , then since ${\vec {{\mathord {\mathbb {B}}}}}$ is an RCS iteration, then we know by Fact 2.7 that the set $\{\bigwedge _{i<\delta }t_i\;|\;{\langle t_i\;|\;}{i<\delta \rangle }\ \text {is a thread in}\ {\vec {{\mathord {\mathbb {B}}}}}{\restriction }\delta \}$ is dense in ${\mathord {\mathbb {B}}}_\delta $ . In case $ \mathrm {cf} (\delta )=\omega $ , let X be that dense subset of ${\mathord {\mathbb {B}}}_\delta $ .

Let $\pi :\omega _1\longrightarrow \delta $ be cofinal, with $\pi (0)=0$ .

Let $\dot {A}$ be a ${\mathord {\mathbb {B}}}_\delta $ -name for a maximal antichain in T, and let $a_0\in X$ be a condition. We will find a countable $\zeta $ and a condition extending $a_0$ that forces that $\dot {A}\subseteq \check {T}|\check {\zeta }$ . Here, $T|\zeta $ is the union of the levels of T below $\zeta $ .

Let N be a model of the form $L_\tau ^{\hat {A}}$ , with $H_\theta \subseteq N$ , such that $\theta $ verifies the subproperness of each ${\mathord {\mathbb {B}}}_i$ , for $i\le \delta $ and $H_\theta $ is large enough to contain the parameters we need. Specifically, letting $S= {\langle \theta ,\delta ,\vec {{\mathord {\mathbb {B}}}},X,\dot {A},T,\pi ,a_0 \rangle } $ , we want to fix $M_0\prec N$ with $S\in M_0$ , $M_0$ countable and such that, letting $\sigma _0:{\bar {N}}\longrightarrow M_0$ be the inverse of the Mostowski collapse (so that ${\bar {N}}$ is transitive), ${\bar {N}}$ is full—it is easy to see that this situation can be arranged; see the discussion following [Reference Fuchs5, Definition 2.21]. Let ${\tilde {\delta }}=\sup (M_0\cap \delta )$ , and let $\Omega =M_0\cap \omega _1= \mathrm {crit} (\sigma _0)=\omega _1^{\bar {N}}$ . Fix an enumeration ${\langle s_n\;|\;}{n<\omega \rangle }$ of $T(\Omega )$ , the $\Omega $ -th level of T.

Let $\sigma _0^{-1}(S)={\bar {S}}= {\langle {\bar {\theta }},{\bar {\delta }},\vec {\bar {{\mathord {\mathbb {B}}}}},{\bar {X}},\dot {{\bar {A}}},{\bar {T}},{\bar {\pi }},\bar {a}_0 \rangle } $ . Let ${\langle \nu _i\;|\;}{i<\omega \rangle }$ be a sequence of ordinals $\nu _i<\omega _1^{\bar {N}}$ such that if we let ${\bar {\gamma }}_i={\bar {\pi }}(\nu _i)$ , it follows that ${\langle {\bar {\gamma }}_i\;|\;}{i<\omega \rangle }$ is cofinal in ${\bar {\delta }}$ , and such that $\nu _0=0$ , so that ${\bar {\gamma }}_0=0$ . Hence, letting $\gamma _i=\sigma _0({\bar {\gamma }}_i)$ , we have that $\sup _{i<\omega }\gamma _i=\sup (M_0\cap \delta )={\tilde {\delta }}$ . Moreover, whenever $\sigma ':{\bar {N}}\prec N$ is such that $\sigma '({\bar {\pi }})=\pi $ , it follows that for every $i<\omega $ , $\sigma '({\bar {\gamma }}_i)=\gamma _i=\pi (\nu _i)$ , since $\sigma '({\bar {\gamma }}_i)=\sigma '({\bar {\pi }}(\nu _i))=\sigma '({\bar {\pi }})(\nu _i)=\pi (\nu _i)$ .

By induction on $n<\omega $ , construct sequences ${\langle \dot {\sigma }_n\;|\;}{n<\omega \rangle }$ , ${\langle c_n\;|\;}{n<\omega \rangle }$ , ${\langle \dot {b}_n\;|\;}{n<\omega \rangle }$ and ${\langle \dot {\bar {b}}_n\;|\;}{n<\omega \rangle }$ with $c_n\in {\mathord {\mathbb {B}}}_{\gamma _n}$ , $\dot {\sigma }_n,\dot {b}_n,\dot {{\bar {b}}}_n\in \mathrm {V}^{{\mathord {\mathbb {B}}}_{\gamma _n}}$ , such that for every $n<\omega $ , $c_n$ forces the following statements with respect to ${\mathord {\mathbb {B}}}_{\gamma _n}$ :

1. (1) $\dot {\sigma }_n:\check {{\bar {N}}}\prec \check {N}$ ,

2. (2) $\dot {\sigma }_n(\check {{\bar {S}}})=\check {S}$ , and for all $k<n$ , $\dot {\sigma }_n(\dot {\bar {b}}_k)=\dot {\sigma }_k(\dot {\bar {b}}_k)$ ,

3. (3) $\dot {\sigma }_n(\dot {\bar {b}}_n)=\dot {b}_n$ and $\dot {\bar {b}}_n\in \check {{\bar {X}}}$ (and so, $\dot {b}_n\in \check {X}$ ),

4. (4) $\check {h}_{\gamma _n}(\dot {b}_n)\in \dot {G}_{{\mathord {\mathbb {B}}}_{\gamma _{n}}}$ ,

5. (5) $\dot {\sigma }_n^{-1}"\dot {G}_{{\mathord {\mathbb {B}}}_{\gamma _n}}$ is $\check {{\bar {N}}}$ -generic for $\check {\bar {{\mathord {\mathbb {B}}}}}_{\gamma _n}$ ,

6. (6) $\dot {b}_n/\dot {G}_{\gamma _n}$ forces wrt. $\check {{\mathord {\mathbb {B}}}_\delta }/\dot {G}_{\gamma _n}$ that there is a node $t<s_{n-1}$ with $t\in \dot {A}$ (for $n>0$ ),

7. (7) $c_{n-1}=h_{\gamma _{n-1}}(c_{n})$ (for $n>0$ ),

8. (8) $\dot {b}_n\le _{\check {{\mathord {\mathbb {B}}}}_\delta }\dot {b}_{n-1}$ (for $n>0$ ),

To start off, in the case $n=0$ , we set $c_0= {1\mathrm {\hspace {-0.5ex}l}} $ , $\dot {\sigma }_0=\check {\sigma }_0$ and $\dot {b}_0=\check {a}_0$ and $\dot {{\bar {b}}}_0=\sigma _0^{-1}(\dot {b}_0)$ . Clearly then, (1)–(6) are satisfied for $n=0$ (and (7)–(8), as well as the second part of (2), are vacuous for $n=0$ ).

Now suppose $m=n-1$ , and $\dot {\sigma }_l$ , $c_l$ , $\dot {b}_l$ and $\dot {{\bar {b}}}_l$ have been defined so that (1)–(8) are satisfied for $l\le m$ .

An application of Lemma 2.5 (to ${\mathord {\mathbb {B}}}_{\gamma _m}\subseteq {\mathord {\mathbb {B}}}_{\gamma _n}$ ) yields a condition $c_n\in {\mathord {\mathbb {B}}}_{\gamma _n}$ and a ${\mathord {\mathbb {B}}}_{\gamma _n}$ -name $\dot {\sigma }_n$ such that $h_{\gamma _m}(c_n)=c_m$ and whenever I is ${\mathord {\mathbb {B}}}_{\gamma _n}$ -generic over $\mathrm {V}$ with $c_n\in I$ , $G=I\cap {\mathord {\mathbb {B}}}_{\gamma _m}$ and $\sigma _n=\dot {\sigma }_n^I$ it follows that $\sigma _n:{\bar {N}}\prec N$ , $\sigma _n^{-1}"I$ is $\bar {{\mathord {\mathbb {B}}}}_{{\bar {\gamma }}_n}$ -generic over ${\bar {N}}$ , $\sigma _n({\bar {S}})=\sigma _m({\bar {S}})$ , $\sigma _n({\bar {b}}_k)=\sigma _m({\bar {b}}_k)$ for $k\le m$ , where $\sigma _m=\dot {\sigma }_m^G$ and for $k\le m$ , ${\bar {b}}_k=\dot {{\bar {b}}}_k^{I\cap {\mathord {\mathbb {B}}}_{\gamma _k}}$ as well as $b_k=\dot {b}_k^{I\cap {\mathord {\mathbb {B}}}_{\gamma _k}}$ . Moreover, we can arrange that $h_{\gamma _n}(\dot {b}_m^G)\in I$ . For this last property, let $\dot {b}$ from the statement of Lemma 2.5 be a ${\mathord {\mathbb {B}}}_{\gamma _m}$ -name for $h_{\gamma _n}(\dot {b}_m)$ and $\dot {{\bar {b}}}$ a name for the preimage of $b_m$ under $\sigma _n$ . Since inductively, $c_m$ forces that $\check {h}_{\gamma _m}(\dot {b}_m)\in \dot {G}_{\gamma _m}$ , assumption (3) of the lemma is satisfied, and we get that $h_{\gamma _n}(\dot {b}_m^G)=\dot {b}^G\in I$ , as wished.

With these definitions, all the conditions (1)–(8) are satisfied, as long as they don’t concern $\dot {b}_n$ and $\dot {{\bar {b}}}_n$ . To define $\dot {b}_n$ and $\dot {{\bar {b}}}_n$ , let I and G as described in the previous paragraph. Let $\sigma _n=\dot {\sigma }_n^I$ , $b_m=\dot {b}_m^I$ , $M_n= \mathrm {ran} (\sigma _n)$ , ${\bar {I}}=\sigma _n^{-1}"I$ , $\sigma _n^*:{\bar {N}}[{\bar {I}}]\stackrel {\sim }{\longleftrightarrow } M_n[I]\prec N[I]$ with $\sigma _n\subseteq \sigma _n^*$ and $\sigma _n^*({\bar {I}})=I$ .

Working in $\mathrm {V}[I]$ , we know that $b_m/I$ forces wrt. ${\mathord {\mathbb {B}}}_\delta /I$ that $\dot {A}/I$ is a maximal antichain in T (since $b_m\le \dot {b}_0^I=a_0$ ). Thus, the set

$$\begin{align*}D=\{t\in T\;|\;\exists y\in X\exists\bar{t}\le_T t\quad y\le_\delta b_m\land h_{\gamma_n}(y)\in I\ \text{and}\ y/I\Vdash_{{\mathord{\mathbb{B}}}_\delta/I}\check{\bar{t}}\in\dot{A}/I\}\end{align*}$$

is dense in T. Note that $D\in M_n[I]$ . Working in $M_n[I]$ , let $A\subseteq D$ be a maximal antichain, so $A\in M_n[I]$ . Since ${\mathord {\mathbb {B}}}_{\gamma _n}$ preserves T as a Souslin tree, it follows that A is countable in $M_n[I]$ , and so, $A\subseteq M_n[I]$ . Since D is dense in T, A is (in $\mathrm {V}[I]$ ) a maximal antichain in T, and $A\subseteq T|\Omega $ (note that $ \mathrm {crit} (\sigma _n)=\omega _1^{\bar {N}}=\Omega $ ). Hence, there is a $t\in A$ with $t<_T s_m$ . Since $A\subseteq D$ , it follows that $t\in D$ . Working in $M_n[I]$ again, let $b_n\in X$ , $\bar {t}\le _T t$ witness that $t\in D$ , i.e., $b_n\le _{\delta }b_m$ , $h_{\gamma _n}(b_n)\in I$ and $b_n/I$ forces wrt. ${\mathord {\mathbb {B}}}_\delta /I$ that $\check {\bar {t}}\in \dot {A}/I$ . Let ${\bar {b}}_n=(\sigma ^*_m)^{-1}(b_n)$ .

Since all of this holds in $\mathrm {V}[I]$ whenever I is ${\mathord {\mathbb {B}}}_{\gamma _n}$ -generic over $\mathrm {V}$ and $c_n\in I$ , there are ${\mathord {\mathbb {B}}}_{\gamma _m}$ -names $\dot {\bar {t}}$ for $\bar {t}$ and $\dot {b}_n$ for $b_n$ such that $c_n$ forces all of this wrt. ${\mathord {\mathbb {B}}}_{\gamma _n}$ . In particular, $c_n$ forces that $\dot {\bar {t}}\in \dot {A}$ and $\dot {\bar {t}}<_Ts_{m}$ . So (1)–(8) are satisfied. This finishes the recursive construction.

By (7), the sequence ${\langle c_n\;|\;}{c<\omega \rangle }$ is a thread, and so, $c=\bigwedge _{n<\omega }c_n\in {\mathord {\mathbb {B}}}_\delta ^+$ : if $ \mathrm {cf} (\delta )=\omega $ , then this follows from part (2)(a) of Definition 2.10. And if $ \mathrm {cf} (\delta )=\omega _1$ , then $\gamma :=\sup _{n<\omega }\gamma _n<\delta $ and $ \mathrm {cf} (\gamma )=\omega $ , so again, $c\in {\mathord {\mathbb {B}}}_\gamma ^+\subseteq {\mathord {\mathbb {B}}}_\delta ^+$ , for the same reason.

We claim that c forces that $\dot {A}$ is bounded in $\check {T}$ . To see this, let G be ${\mathord {\mathbb {B}}}_\delta $ -generic over $\mathrm {V}$ , with $c\in G$ . For $n<\omega $ , let $b_n=\dot {b}_n^G$ .

1. (C) For all $n<\omega $ , $b_n\in G$ .

Proof of (C).

Let $n<\omega $ . We have that $h_{\gamma _n}(b_n)\in G\cap {\mathord {\mathbb {B}}}_{\gamma _n}$ , by (4). For every $l\ge n$ , $b_l\le b_n$ , by (8), so $h_{\gamma _l}(b_l)\le h_{\gamma _l}(b_n)$ . Since $h_{\gamma _l}(b_l)\in G\cap {\mathord {\mathbb {B}}}_{\gamma _l}$ , this implies that for $l\ge n$ ,

$$\begin{align*}h_{\gamma_l}(b_n)\in G\cap{\mathord{\mathbb{B}}}_{\gamma_l}.\end{align*}$$

This holds for $l<n$ as well, because in that case, $h_{\gamma _l}(b_n)\ge h_{\gamma _n}(b_n)\in G$ .

Recall that ${\tilde {\delta }}=\sup _{l<\omega }\gamma _l$ . Let $M_n= \mathrm {ran} (\sigma _n)$ . It follows that $\sup (\delta \cap M_n)={\tilde {\delta }}$ (since $\sigma _n({\bar {\pi }})=\pi $ ).

Now, if $ \mathrm {cf} (\delta )=\omega _1$ , then ${\tilde {\delta }}<\delta $ , and we know that $b_n\in X=\bigcup _{i<\delta }{\mathord {\mathbb {B}}}_i$ , so there is an $i<\delta $ such that $b_n\in {\mathord {\mathbb {B}}}_i$ . But since $b_n\in M_n$ , the same is true in $M_n$ , and this means that there is an $i<{\tilde {\delta }}$ such that $b_n\in {\mathord {\mathbb {B}}}_i$ . But then, letting $l>n$ be such that $\gamma _l>i$ , it follows that $b_n=h_{\gamma _l}(b_n)\in G\cap {\mathord {\mathbb {B}}}_{\gamma _l}$ , so $b_n\in G$ , as claimed.

If, on the other hand, $ \mathrm {cf} (\delta )=\omega $ , then since $b_n\in X$ , there is a thread ${\langle t_i\;|\;}{i<\delta \rangle }$ in ${\vec {{\mathord {\mathbb {B}}}}}{\restriction }\delta $ such that $b_n=\bigwedge _{i<\delta }t_i$ . But then, for every $l<\omega $ , $t_{\gamma _l}=h_{\gamma _l}(b_n)\in G\cap {\mathord {\mathbb {B}}}_{\gamma _l}\subseteq G$ . By genericity (and thus $\mathrm {V}$ -completeness) of G, this implies that $b_n=\bigwedge _{i<\delta }t_i=\bigwedge _{l<\omega }t_{\gamma _l}\in G$ . Note that we used that ${\vec {{\mathord {\mathbb {B}}}}}$ is an RCS iteration here.

Note in particular that $b_0=a_0\in G$ . Since this is true whenever $c\in G$ , $c\Vdash _{{\mathord {\mathbb {B}}}_\delta }\check {a}_0\in \dot {G}$ , which implies that $c\le a_0$ . Moreover, c forces that $\dot {A}$ is bounded in $\check {T}$ : working in $\mathrm {V}[G]$ again, where $G\ni c$ is ${\mathord {\mathbb {B}}}_\delta $ -generic, we have that for every $n<\omega $ , there is a $t_n\in A=\dot {A}^G$ with $t_n<_T s_n$ , by (6). So A cannot contain a node a at a level greater than $\Omega $ , because the predecessor of such an a at level $\Omega $ would have to be of the form $s_m$ , for some m, and $s_m>_Tt_m$ . So $a,t_m\in A$ would be comparable. Thus, c forces wrt. ${\mathord {\mathbb {B}}}_\delta $ that $\dot {A}\subseteq \check {T}|\check {\Omega }$ .

Case 2: for all $i<\delta $ , $ \mathrm {cf} (\delta )>\delta ({\mathord {\mathbb {B}}}_i)$ .

We may also assume that $ \mathrm {cf} (\delta )>\omega _1$ , for otherwise, $ \mathrm {cf} (\delta )\le \omega _1$ and the argument of case 1 goes through (recall that we proved (B)).

It follows as in [Reference Jensen, Chong, Feng, Slaman, Woodin and Yang13, p. 143, Claim (2)] that for $i<\delta $ , $\mathrm {card}(i)\le \delta ({\mathord {\mathbb {B}}}_i)$ . But then, it follows that $\delta $ is regular, for otherwise, if $i= \mathrm {cf} (\delta )<\delta $ , it would follow that $ \mathrm {cf} (\delta )=i\le \delta ({\mathord {\mathbb {B}}}_i)< \mathrm {cf} (\delta )$ .

Thus, $\delta $ is a regular cardinal, and $\delta \ge \omega _2$ . Hence, $S^\delta _{\omega _1}$ , the set of ordinals less than $\delta $ with cofinality $\omega _1$ , is stationary in $\delta $ . For $\gamma \in S^\delta _{\omega _1}$ , since ${\mathord {\mathbb {B}}}_\gamma $ , being subproper, preserves $\omega _1$ , it follows that for every $i<\gamma $ , $\Vdash _{{\mathord {\mathbb {B}}}_i} \mathrm {cf} (\check {\gamma })>\omega $ . Thus, since ${\vec {{\mathord {\mathbb {B}}}}}$ is nicely subproper, it follows by part (3) of Definition 2.10 that ${\mathord {\mathbb {B}}}_\gamma $ is the direct limit of ${\vec {{\mathord {\mathbb {B}}}}}{\restriction }\gamma $ . Moreover, since for $i<\delta $ , $\delta ({\mathord {\mathbb {B}}}_i)<\delta = \mathrm {cf} (\delta )$ , it follows by Observation 2.2 that ${\mathord {\mathbb {B}}}_i$ is ${<}\delta ({\mathord {\mathbb {B}}}_i)^+$ -c.c., and hence ${<}\delta $ -c.c.

It follows by Fact 2.8 that the direct limit of $\vec {{\mathord {\mathbb {B}}}}{\restriction }\delta $ is ${<}\delta $ -c.c.

Again, since for all $i<\delta $ , ${\mathord {\mathbb {B}}}_i$ is ${<}\delta $ -c.c., it follows that ${\mathord {\mathbb {B}}}_i$ forces that the cofinality of $\delta $ is uncountable. So since ${\vec {{\mathord {\mathbb {B}}}}}$ is nicely subproper, it follows that ${\mathord {\mathbb {B}}}_\delta $ is the direct limit of ${\vec {{\mathord {\mathbb {B}}}}}{\restriction }\delta $ , and hence that ${\mathord {\mathbb {B}}}_\delta $ is ${<}\delta $ -c.c.

Now let $h<\delta $ , let $G_h\subseteq {\mathord {\mathbb {B}}}_h$ be generic, and let $T\in \mathrm {V}[G_h]$ be a Souslin tree. We have to show that T is still Souslin in $\mathrm {V}[G_h][H]$ , whenever $H\subseteq {\mathord {\mathbb {B}}}_\delta /G_h$ is generic over $\mathrm {V}[G_h]$ . But if there were an uncountable antichain of T in $\mathrm {V}[G_h][H]=\mathrm {V}[G_h*H]$ , then since ${\mathord {\mathbb {B}}}_\delta $ is $\delta $ -c.c. and $\delta>\omega _1$ , it would follow that this antichain exists already in $\mathrm {V}[(G_h*H)\cap {\mathord {\mathbb {B}}}_i]$ , for some $i\in [h,\delta )$ . Thus, ${\mathord {\mathbb {B}}}_i/G_h$ would fail to be Souslin tree preserving in $\mathrm {V}[G_h]$ , contradicting our inductive assumption.

Proof. Letting $\vec {{\mathord {\mathbb {B}}}}={\langle {\mathord {\mathbb {B}}}_i\;|\;}{i<\alpha \rangle }$ be a standard RCS iteration of forcings that are subproper and preserve T as a Souslin tree, we prove by induction on $\delta $ : whenever $h\le \delta <\alpha $ and $G_h$ is ${\mathord {\mathbb {B}}}_h$ -generic, then in $\mathrm {V}[G_h]$ , ${\mathord {\mathbb {B}}}_\delta /G_h$ is subproper and preserves T as a Souslin tree.

For this, the proof of Theorem 2.14 goes through almost without change. Some steps of the argument are somewhat simpler in the present context, because the Souslin tree T is in the ground model.

Observation 2.17. Every standard RCS iteration of subproper and Souslin tree preserving forcings is nicely subproper and Souslin tree preserving. Similarly, if T is a fixed Souslin tree then every standard RCS iteration of subproper and Souslinness of T preserving forcings is nicely subproper and Souslinness of T preserving.

Proof. Recall that in the proof of Theorem 2.14, we used the fact that the iteration was just nicely subproper, rather than an RCS iteration, wherever possible. The only places in the argument that used that we were dealing with an RCS iteration occurred at stages of the iteration that acquired countable cofinality. But these stages are trivial in a nicely subproper T-preserving iteration.

Proof. We follow along the lines of the proof of Theorem 2.14. So let $\vec {{\mathord {\mathbb {B}}}}={\langle {\mathord {\mathbb {B}}}_i\;|\;}{i<\alpha \rangle }$ be a standard RCS iteration of forcings that are subproper and branch preserving. We prove by induction on $\delta $ : whenever $h\le \delta <\alpha $ and $G_h$ is ${\mathord {\mathbb {B}}}_h$ -generic, then in $\mathrm {V}[G_h]$ , ${\mathord {\mathbb {B}}}_\delta /G_h$ is subproper and branch preserving.

Case 1: $\delta $ is a limit ordinal and there is an $i<\delta $ such that $ \mathrm {cf} (\delta )\le \delta ({\mathord {\mathbb {B}}}_i)$ .

As before, it suffices to prove

1. (B) if $ \mathrm {cf} (\delta )\le \omega _1$ then ${\mathord {\mathbb {B}}}_\delta $ adds no cofinal branch to an $\omega _1$ -tree.

So let us fix an $\omega _1$ -tree T. Depending on whether the cofinality of $\delta $ is $\omega $ or $\omega _1$ , let $X=\{\bigwedge _{i<\delta }t_i\;|\;\vec {t}\ \text {is a thread in}\ \vec {{\mathord {\mathbb {B}}}}{\restriction }\delta \}$ or $X=\bigcup _{i<\delta }{\mathord {\mathbb {B}}}_i$ .

Toward a contradiction, let $\dot {B}$ be a ${\mathord {\mathbb {B}}}_\delta $ -name such that some condition $a_0\in X$ forces that $\dot {B}$ is a new cofinal branch, that is, that $\dot {B}\notin \mathrm {V}^{{\mathord {\mathbb {B}}}_\gamma }$ for all $\gamma <\delta $ .

Let $\pi :\omega _1\longrightarrow \delta $ be cofinal, with $\pi (0)=0$ . Let $N=L_\tau ^A$ , with $H_\theta \subseteq N$ , such that $\theta $ verifies the subproperness of each ${\mathord {\mathbb {B}}}_i$ , for $i\le \delta $ . Let $S= {\langle \theta ,\delta ,\vec {{\mathord {\mathbb {B}}}},X,\dot {B},T,\pi ,a_0 \rangle } $ . Let $M_0\prec N$ with $S\in M_0$ , $M_0$ countable and such that, letting $\sigma _0:{\bar {N}}\longrightarrow M_0$ be the inverse of the Mostowski collapse, where ${\bar {N}}$ is full. Let ${\tilde {\delta }}=\sup (M_0\cap \delta )$ , and let $\Omega =M_0\cap \omega _1= \mathrm {crit} (\sigma _0)=\omega _1^{\bar {N}}$ . Fix an enumeration ${\langle s_n\;|\;}{n<\omega \rangle }$ of $T(\Omega )$ .

Let $\sigma _0^{-1}(S)={\bar {S}}= {\langle {\bar {\theta }},{\bar {\delta }},\vec {\bar {{\mathord {\mathbb {B}}}}},{\bar {X}},\dot {{\bar {B}}},{\bar {T}},{\bar {\pi }},\bar {a}_0 \rangle } $ . Let ${\langle \nu _i\;|\;}{i<\omega \rangle }$ be a sequence of ordinals $\nu _i<\omega _1^{\bar {N}}$ such that if we let ${\bar {\gamma }}_i={\bar {\pi }}(\nu _i)$ , it follows that ${\langle {\bar {\gamma }}_i\;|\;}{i<\omega \rangle }$ is cofinal in ${\bar {N}}$ , and such that $\nu _0=0$ , so that ${\bar {\gamma }}_0=0$ . Let $\gamma _i=\sigma _0({\bar {\gamma }}_i)$ .

By induction on $n<\omega $ , construct sequences ${\langle \dot {\sigma }_n\;|\;}{n<\omega \rangle }$ , ${\langle c_n\;|\;}{n<\omega \rangle }$ , ${\langle \dot {b}_n\;|\;}{n<\omega \rangle }$ and ${\langle \dot {\bar {b}}_n\;|\;}{n<\omega \rangle }$ with $c_n\in {\mathord {\mathbb {B}}}_{\gamma _n}$ , $\dot {\sigma }_n,\dot {b}_n,\dot {{\bar {b}}}_n\in \mathrm {V}^{{\mathord {\mathbb {B}}}_{\gamma _n}}$ , such that for every $n<\omega $ , $c_n$ forces the following statements:

1. (1) $\dot {\sigma }_n:\check {{\bar {N}}}\prec \check {N}$ ,

2. (2) $\dot {\sigma }_n(\check {{\bar {S}}})=\check {S}$ , and for all $k<n$ , $\dot {\sigma }_n(\dot {\bar {b}}_k)=\dot {\sigma }_k(\dot {\bar {b}}_k)$ ,

3. (3) $\dot {\sigma }_n(\dot {\bar {b}}_n)=\dot {b}_n$ and $\dot {\bar {b}}_n\in \check {\bar {X}}$ (and so, $\dot {b}_n\in \check {X}$ ),

4. (4) $\check {h}_{\gamma _n}(\dot {b}_n)\in \dot {G}_{{\mathord {\mathbb {B}}}_{\gamma _{n}}}$ ,

5. (5) $\dot {\sigma }_n^{-1}"\dot {G}_{{\mathord {\mathbb {B}}}_{\gamma _n}}$ is $\check {{\bar {N}}}$ -generic for $\check {\bar {{\mathord {\mathbb {B}}}}}_{\gamma _n}$ ,

6. (6) for some $t\in T|\Omega $ with $t\not <s_{n-1}$ , $\dot {b}_n$ forces wrt. ${\mathord {\mathbb {B}}}_\delta $ that $\check {t}\in \dot {B}$ , and in particular, $\dot {b}_n$ forces that $\check {s}_{n-1}\notin \dot {B}$ (for $n>0$ ),

7. (7) $c_{n-1}=h_{\gamma _{n-1}}(c_{n})$ (for $n>0$ ),

8. (8) $\dot {b}_n\le _{\check {{\mathord {\mathbb {B}}}}_\delta }\dot {b}_{n-1}$ (for $n>0$ ),

To start off, in the case $n=0$ , we set $c_0= {1\mathrm {\hspace {-0.5ex}l}} $ , $\dot {\sigma }_0=\check {\sigma }_0$ and $\dot {b}_0=\check {a}_0$ and $\dot {{\bar {b}}}_0=\sigma _0^{-1}(\dot {b}_0)$ .

Now suppose $m=n-1$ , and $\dot {\sigma }_l$ , $c_l$ , $\dot {b}_l$ , and $\dot {{\bar {b}}}_l$ have been defined, so that (1)-(8) are satisfied for $l\le m$ . An application of Lemma 2.5 (to ${\mathord {\mathbb {B}}}_{\gamma _m}\subseteq {\mathord {\mathbb {B}}}_{\gamma _n}$ ) yields $c_n\in {\mathord {\mathbb {B}}}_{\gamma _n}$ and a ${\mathord {\mathbb {B}}}_{\gamma _n}$ -name $\dot {\sigma }_n$ such that (1)–(8) are satisfied at level n, as long as they don’t refer to $\dot {b}_n$ and $\dot {\bar {b}}_n$ , as before. As before, we can also arrange that $c_n$ forces that $\check {h}_{\gamma _n}(\dot {b}_m)\in \dot {G}_{\gamma _n}$ .

To define $\dot {b}_n$ and $\dot {{\bar {b}}}_n$ , let I be ${\mathord {\mathbb {B}}}_{\gamma _n}$ -generic with $c_n\in I$ , and let $G=I\cap {\mathord {\mathbb {B}}}_{\gamma _m}$ . Let $\sigma _n=\dot {\sigma }_n^I$ , $b_m=\dot {b}_m^G$ , $M_n= \mathrm {ran} (\sigma _n)$ , ${\bar {I}}=\sigma _n^{-1}"I$ , $\sigma _n^*:{\bar {N}}[{\bar {I}}]\prec M_n[I]$ with $\sigma _n\subseteq \sigma _n^*$ and $\sigma _n^*({\bar {I}})=I$ . We have that $h_{\gamma _n}(b_m)\in I$ .

Working in $\mathrm {V}[I]$ , we know that $b_m/I$ forces wrt. ${\mathord {\mathbb {B}}}_\delta /I$ that $\dot {B}/I$ is a new cofinal branch in $\check {T}$ (that is, a branch that did not exist in $\mathrm {V}[I]$ ). Consider the set of $z\in {\mathord {\mathbb {B}}}_{\gamma _n}$ such that there exist $t_1,t_2\in T$ and $y_1,y_2\le _\delta b_m$ such that

• • $t_1,t_2$ are incomparable in T,

• • $z=h_{\gamma _n}(y_1)=h_{\gamma _n}(y_2)$ ,

• • $y_1\Vdash _{{\mathord {\mathbb {B}}}_\delta }\check {t}_1\in \dot {B}$ ,

• • $y_2\Vdash _{{\mathord {\mathbb {B}}}_\delta }\check {t}_2\in \dot {B}$ .

Then $D\in M_n$ , and D is dense in ${\mathord {\mathbb {B}}}_{\gamma _n}$ below $h_{\gamma _n}(b_m)$ . So there is a $z\in I\cap D$ , and this is witnessed by some $t_1,t_2,y_1,y_2$ . Since $M_n[I]\prec N[I]$ , these objects may be chosen in $M_n[I]$ . Thus, $t_1,t_2\in T|\Omega $ . Since $t_1$ and $t_2$ are incomparable in T, at most one of them can be below $s_m$ . Say $i<2$ is such that $t_i$ is not below $s_m$ . Then we can set $b_n=y_i$ .

Since all of this holds in $\mathrm {V}[I]$ whenever I is ${\mathord {\mathbb {B}}}_{\gamma _n}$ -generic over $\mathrm {V}$ and $c_n\in I$ , there is a ${\mathord {\mathbb {B}}}_{\gamma _n}$ -name $\dot {b}_n$ for $b_n$ such that $c_n$ forces all of this wrt. ${\mathord {\mathbb {B}}}_{\gamma _n}$ . Then (1)–(8) are satisfied. This finishes the recursive construction.

As before, the sequence ${\langle c_n\;|\;}{n<\omega \rangle }$ is a thread, and so, $c=\bigwedge _{n<\omega }c_n\in {\mathord {\mathbb {B}}}_\delta ^+$ . It follows that $c\Vdash \dot {b}_n\in \dot {G}_\delta $ , for all $n<\omega $ . In particular, $c\le a_0$ . We claim that c forces that $\dot {B}$ is bounded in $\check {T}$ . To see this, let G be ${\mathord {\mathbb {B}}}_\delta $ -generic over $\mathrm {V}$ , with $c\in G$ , and let $B=\dot {B}^G$ . Let $n<\omega $ . Since $b_{n+1}:=\dot {b}_{n+1}^G\in G$ , we have by (6) that $s_n\notin B$ . Since this holds for all $n<\omega $ , B contains no node at level $\Omega $ of T. So B is bounded in T. This is a contradiction.

Case 2: for all $i<\delta $ , $ \mathrm {cf} (\delta )>\delta ({\mathord {\mathbb {B}}}_i)$ .

Exactly as in the proof of Theorem 2.14, we may reduce to the case that $\delta>\omega _1$ is a regular cardinal. It follows that ${\mathord {\mathbb {B}}}_\delta $ is ${<}\delta $ -c.c. Thus, if $G_h$ is ${\mathord {\mathbb {B}}}_h$ -generic and ${\mathord {\mathbb {B}}}_\delta /G_h$ added a new cofinal branch to some $\omega _1$ -tree $T\in \mathrm {V}[G_h]$ , then already some earlier ${\mathord {\mathbb {B}}}_\gamma /G_h$ would add this branch, contradicting our inductive hypothesis.

Proof. We can simplify the argument of the proof of Theorem 2.20, as before.

Proof. Recall that in this situation, G is a $\mathrm {V}$ -complete filter in ${\mathord {\mathbb {B}}}$ , and that the quotient ${\mathord {\mathbb {B}}}/G_0={\mathord {\mathbb {B}}}/G$ consists of equivalence classes with respect to the equivalence relation that identifies $b_0,b_1\in {\mathord {\mathbb {B}}}$ iff $(b_0\Rightarrow b_1)\land (b_1\Rightarrow b_0)\in G$ (where $b_0\Rightarrow b_1=\neg b_0\lor b_1$ ). $b/G_0$ denotes the equivalence class of b with respect to this equivalence relation. The partial order on ${\mathord {\mathbb {B}}}/G_0$ is defined by $b_0/G_0\le _{{\mathord {\mathbb {B}}}/G_0}b_1/G_0$ iff $(b_0\Rightarrow b_1)\in G$ .

Now let $g_i=\bigwedge _{j<i}(r_{j+1}\Rightarrow r_j)$ and $s_i=\bigwedge _{j\le i}r_j$ . Then clearly, each $g_i$ is in G, and the sequence $\vec {s}$ is weakly decreasing in ${\mathord {\mathbb {B}}}$ . It follows by induction on i that $r_i\wedge g_i=s_i$ , completing the proof. This is clear if $i=0$ , since $s_0=r_0$ and $g_0=1$ . For the inductive step we get that $r_{i+1}\wedge g_{i+1}=r_{i+1}\wedge (r_{i+1}\Rightarrow r_i)\wedge g_i=r_{i+1}\wedge r_i\wedge g_i$ , which inductively is $r_{i+1}\wedge s_i=s_{i+1}$ .

Proof. Let $G_0$ be ${\mathord {\mathbb {B}}}_0$ -generic over $\mathrm {V}$ , with $a\in G_0$ . Let $\sigma _0=\dot {\sigma }_0^{G_0}$ , $M= \mathrm {ran} (\sigma _0)$ , $b=\dot {b}^{G_0}$ , etc. Then $\sigma _0:{\bar {N}}\prec M$ , and $\sigma _0$ lifts to $\sigma _0^*:{\bar {N}}[{\bar {G}}]\prec M[G_0]$ , where ${\bar {G}}=\sigma _0^{-1}"G_0$ . Since $M[G_0]\prec N[G_0]$ , we will be able to apply Lemma 2.24 in $\mathrm {V}[G_0]$ .

Note that ${}^\omega \omega \cap M[G_0]={}^\omega \omega \cap {\bar {N}}[{\bar {G}}]$ . Since ${\mathord {\mathbb {B}}}_0$ is ${}^\omega \omega $ -bounding, ${\bar {N}}$ thinks that $\bar {{\mathord {\mathbb {B}}}}_0$ is ${}^\omega \omega $ -bounding, and since ${\bar {G}}$ is $\bar {{\mathord {\mathbb {B}}}}_0$ -generic over ${\bar {N}}$ , it follows that every real of ${\bar {N}}[{\bar {G}}]$ is bounded by some real of ${\bar {N}}$ , which is bounded by g. Thus, the reals of ${\bar {N}}[{\bar {G}}]$ are bounded by g, and hence, the reals of $M[G_0]$ are bounded by g.

The ${\mathord {\mathbb {B}}}_2$ -name $\dot {f}$ can be viewed as a ${\mathord {\mathbb {B}}}_2/G_0$ -name in the obvious way. Let’s write $\dot {f}/G_0$ for this name. Then $\dot {f}/G_0$ is a ${\mathord {\mathbb {B}}}_2/G_0$ -name for a real.

By assumption (A7), let ${\vec {b}}$ be a decreasing sequence of conditions in ${\mathord {\mathbb {B}}}_2$ , below b, such that ${\vec {b}}/G_0$ is decreasing in ${\mathord {\mathbb {B}}}_2/G_0$ , and such that $\mathsf {intp}(\dot {f},{\vec {b}})\le _0g$ . This means that $\mathsf {intp}(\dot {f}/G_0,{\vec {b}}/G_0)\le _0g$ .

Thus, by Lemma 2.24, applied to $\mathbb {A}={\mathord {\mathbb {B}}}_1/G_0$ , ${\mathord {\mathbb {B}}}={\mathord {\mathbb {B}}}_2/G_0$ , the name $\dot {f}/G_0$ , the model $M[G_0]$ , the condition $b/G_0$ and the sequence ${\vec {b}}/G_0$ , there is a condition $d\in {\mathord {\mathbb {B}}}^+_1\cap M[G_0]$ such that $d/G_0\le h_{{\mathord {\mathbb {B}}}_2/G_0,{\mathord {\mathbb {B}}}_1/G_0}(b/G_0)$ and such that $d/G_0$ forces with respect to ${\mathord {\mathbb {B}}}_1/G_0$ that if ${\vec {a}}=\delta _{\dot {G}_{{\mathord {\mathbb {B}}}_1/G_0}}({\vec {b}}/G_0,\dot {f}/G_0)$ , then $\vec {a}$ is below $\check {b}/G_0$ in $({\mathord {\mathbb {B}}}_2/G_0)^+$ , interprets $(\dot {f}/G_0)\check {}$ and respects g.

Let $\dot {d}$ be a ${\mathord {\mathbb {B}}}_0$ -name for d such that a forces (with respect to ${\mathord {\mathbb {B}}}_0$ ) that all of these properties hold. Let $\dot {\bar {d}}$ be a ${\mathord {\mathbb {B}}}_0$ -name such that a forces that $\dot {\sigma }_0^*(\dot {\bar {d}})=\dot {d}$ .

We can now apply Lemma 2.5 to ${\mathord {\mathbb {B}}}_0\subseteq {\mathord {\mathbb {B}}}_1$ , $\dot {\sigma }_0$ , $\dot {{\bar {d}}}$ , $\dot {d}$ getting a condition $c\in {\mathord {\mathbb {B}}}_1$ with $h_0(c)=a$ and a ${\mathord {\mathbb {B}}}_1$ -name $\dot {\sigma }_1$ such that whenever I is ${\mathord {\mathbb {B}}}_1$ -generic over $\mathrm {V}$ with $c\in I$ and $G=I\cap {\mathord {\mathbb {B}}}_0$ , it follows that

1. (1) $\sigma _1:{\bar {N}}\prec N$ ,

2. (2) $\sigma _1({\bar {S}})=S$ ,

3. (3) $\sigma _1(\dot {{\bar {b}}}^G)=\sigma _0(\dot {{\bar {b}}}^G)$ and $\sigma _1(\dot {t}^G)=\sigma _0(\dot {t}^G)$ ,

4. (4) $\sigma _1(\dot {{\bar {d}}}^G)=\sigma _0(\dot {{\bar {d}}}^G)=\dot {d}^G$ ,

5. (5) $\dot {d}^G\in I$ ,

6. (6) $\sigma _1^{-1}"I$ is $\bar {{\mathord {\mathbb {B}}}}$ -generic over ${\bar {N}}$ ,

7. (7) for all $i\le n$ , $\sup \sigma _0"{\bar {\lambda }}_i=\sup \sigma _1"{\bar {\lambda }}_i$ .

where $\delta =\delta ({\mathord {\mathbb {B}}}_1)$ . To conclude the proof, we verify that we also have: $h_1(\dot {b}^G)\in I$ , and letting $M= \mathrm {ran} (\sigma _1)$ , there is in $M[I]$ a decreasing sequence ${\vec {c}}$ of conditions in ${\mathord {\mathbb {B}}}_2$ below $\dot {b}^G$ such that ${\vec {c}}/I$ is decreasing in ${\mathord {\mathbb {B}}}_2/I$ and $\mathsf {intp}(\dot {f},{\vec {c}})\le _0g$ .