Fact 2.2.

1. 1. For any independent triples, $\overline {M}{\sqsubseteq }{\overline {B}}$ implies $M_1M_2\subseteq _{TV} B_1B_2$ (see Definition A.4).

2. 2. (Upward LS) For any independent triple ${\overline {A}}$ , there is an independent triple $\overline {M}\sqsupseteq {\overline {A}}$ consisting of a-models (or even, $\kappa $ -saturated for any cardinal $\kappa $ ).

3. 3. (Downward LS) For any independent triple $\overline {M}$ of models, for any infinite cardinal $\lambda $ , and for any set $X\subseteq M_1M_2$ with $|X|<\lambda $ , there is ${\overline {A}}{\sqsubseteq }\overline {M}$ with $X\subseteq A_1A_2$ and $|A_1A_2|<\lambda $ .

Fact 2.4.

1. 1. If c is V-dominated by ${\overline {A}}$ , then $\mathrm {stp}(c/A_1A_2)\vdash \mathrm {tp}(c/B_1B_2)$ for every ${\overline {B}}\sqsupseteq {\overline {A}}$ .

2. 2. If c is V-dominated by ${\overline {A}}$ , then c is V-dominated by every ${\overline {B}}\sqsupseteq {\overline {A}}$ .

3. 3. If c is V dominated by ${\overline {B}}$ and ${\overline {B}}\sqsupseteq {\overline {A}}$ with , then c is V-dominated by ${\overline {A}}$ .

4. 4. If $\overline {M}$ is an independent triple of models and if $\mathrm {tp}(c/M_1M_2)$ is $\ell $ -isolated, then c is V-dominated by $\overline {M}$ .

Proof. (1) Choose $c'$ such that $\mathrm {stp}(c'/A_1A_2)=\mathrm {stp}(c/A_1A_2)$ . Clearly, $c'$ is V-dominated by ${\overline {A}}$ as well, so we have both and , hence $\mathrm {tp}(cB_1B_2)=\mathrm {tp}(c'B_1B_2)$ .

(2) is immediate by the transitivity of ${\sqsubseteq }$ . (3) and (4) are, respectively, Lemmas 2.2 and 2.6 of [Reference Hart3].

Fact 2.5. Suppose $\overline {M}=(M_0,M_1,M_2)$ is an independent triple of a-models. Then the following are equivalent for a finite tuple c.

1. 1. c is V-dominated by $\overline {M}$ ;

2. 2. there is an independent triple ${\overline {B}}{\sqsubseteq }\overline {M}$ of finite sets with $\mathrm {tp}(c/B_1B_2)\vdash \mathrm { tp}(c/M_1M_2)$ ;

3. 3. $\mathrm {tp}(c/M_1M_2)$ is a-isolated;

4. 4. $\mathrm {tp}(c/M_1M_2)$ is $\ell $ -isolated (see Definition A.6).

Proof. $(1)\Rightarrow (2):$ By superstability, choose a finite $X_0\subseteq M_1M_2$ such that . In fact, by either Shelah’s Conclusion XII.3.5 in [Reference Shelah8] or Hart’s Relative Stationarity Lemma, Lemma 1.19 of [Reference Hart3], there is a finite X, $X_0\subseteq X\subseteq M_1M_2$ for which $\mathrm {tp}(c/X)$ is based and relatively stationary inside $M_1M_2$ . Find a finite ${\overline {A}}{\sqsubseteq }\overline {M}$ with $X\subseteq A_1A_2$ . By Fact 2.2(3), c is V-dominated by ${\overline {A}}$ , hence $\mathrm {stp}(c/A_1A_2)\vdash \mathrm {tp}(c/M_1M_2)$ by Fact 2.4(1). However, by the relative stationarity, this is strengthened to $\mathrm {tp}(c/A_1A_2)\vdash \mathrm { tp}(c/M_1M_2)$ .

$(2)\Rightarrow (3)$ is trivial, and $(3)\Rightarrow (4)$ is by Conclusion XII.2.11 of [Reference Shelah8]. Finally, $(4)\Rightarrow (1)$ is immediate from Fact 2.4(4).

Proof. As (5) and (2) are equivalent by Fact 2.5, it suffices to show the equivalence of $(1),(2),(3)$ and $(4),(5),(6)$ separately.

$(1)\Rightarrow (3)$ is trivial. $(3)\Rightarrow (2)$ : Suppose c is V-dominated by a-models ${\overline {N}}$ . Choose a finite ${\overline {B}}{\sqsubseteq }{\overline {N}}$ with $\mathrm {tp}(c/B_1B_2)\vdash \mathrm {tp}(c/N_1N_2)$ . By Fact 2.2(3), choose an independent triple $\overline {M}{\sqsubseteq }{\overline {N}}$ of countable models with $B_1B_2\subseteq M_1M_2$ . By Fact 2.4(3), c is V-dominated by $M_1M_2$ , hence $\mathrm {tp}(c/M_1M_2)$ is isolated by (3). In particular, $\mathrm {tp}(c/B_1B_2)$ is isolated, hence $\mathrm {tp}(c/N_1N_2)$ is isolated as well.

$(2)\Rightarrow (1)$ : Suppose c is V-dominated by ${\overline {A}}$ . By Fact 2.2(2), choose a triple $\overline {M}$ of a-models with ${\overline {A}}{\sqsubseteq }\overline {M}$ . Then c is V-dominated by $\overline {M}$ , hence $\mathrm {tp}(c/M_1M_2)$ is isolated. As , $\mathrm {tp}(c/A_1A_2)$ is isolated by the Open Mapping Theorem, Fact A.3.

Turning to $(4),(5),(6)$ , $(4)\Rightarrow (6)$ is trivial. $(6)\Rightarrow (5)$ : Suppose ${\overline {N}}$ is an independent triple of a-models and $\mathrm {tp}(c/N_1N_2)$ is $\ell $ -isolated. As T is countable, choose a countable $A\subseteq N_1N_2$ such that for every $\varphi (x,y)$ , $\mathrm {tp}_\varphi (c/N_1N_2)$ is isolated by some $L(A)$ -formula $\psi (x)$ . By Fact 2.2(3), choose a triple of countable models $\overline {M}{\sqsubseteq }{\overline {N}}$ with $A\subseteq M_1M_2$ . By (6), $\mathrm {tp}(c/M_1M_2)$ is isolated, hence $\mathrm {tp}(c/N_1N_2)$ is isolated as well by Fact 2.2(1) and Lemma A.5.

$(5)\Rightarrow (4)$ : Say $\overline {M}$ is an arbitrary triple of models and $\mathrm {tp}(c/M_1M_2)$ is $\ell $ -isolated. By Fact 2.2(2), choose an independent triple of a-models ${\overline {N}}\sqsupseteq \overline {M}$ . By Fact A.8, $\mathrm {tp}(c/N_1N_2)$ is also $\ell $ -isolated, hence $\mathrm {tp}(c/N_1N_2)$ is isolated by (5). Thus, $\mathrm {tp}(c/M_1M_2)$ is isolated by the Open Mapping Theorem.

Fact 3.3. Suppose $\overline {M}$ is a ${\cal P}^-(n)$ -stable system of a-models. Then the following are equivalent for a finite tuple c.

1. 1. c is ${\cal P}^-(n)$ -dominated by $\overline {M}$ ;

2. 2. there is some ${\cal P}^-(n)$ -system ${\overline {B}}{\sqsubseteq }\overline {M}$ of finite sets such that $\mathrm { tp}(c/\bigcup {\overline {B}})\vdash \mathrm {tp}(c/\bigcup \overline {M})$ ;

3. 3. $\mathrm {tp}(c/\bigcup \overline {M})$ is a-isolated;

4. 4. $\mathrm {tp}(c/\bigcup \overline {M})$ is $\ell $ -isolated.

Proof. We prove this by induction on $n\ge 2$ . That T has a-s $P^-(2)$ -DI was noted above, so fix $n\ge 2$ and assume T has a-s ${\cal P}^-(k)$ -DI for all $2\le k\le n$ . Let $\overline {M}=(M_s:s\in {\cal P}^-(n+1))$ be an atomic-special ${\cal P}^-(n+1)$ -stable system. We first note that, under these hypotheses, the standard method of blowing up $\overline {M}$ to a ${\cal P}^-(n+1)$ -system of $\aleph _1$ -saturated models preserves being atomic-special. To see that, choose an enumeration $(s_i:i<2^{n+1}-1)$ of ${\cal P}^-(n+1)$ such that $s_i\subseteq s_j$ implies $i\le j$ . We recursively construct a sequence $(N_{s_i})$ of $\aleph _1$ -saturated models satisfying and when $|s_i|\ge 2$ , $N_{s_i}$ is chosen to be $\aleph _1$ -prime over $M_{s_i}\bigcup {\overline {N}}_{<s_i}$ . It follows from Fact XII 2.6 of [Reference Shelah8] that the resulting system ${\overline {N}}=(N_s:s\in {\cal P}^-(n+1))$ is a stable system, and in fact $\overline {M}{\sqsubseteq }{\overline {N}}$ . To see that ${\overline {N}}$ is atomic-special, we argue by induction on i, that if $\{0,1\}\subseteq s_i$ , then $N_{s_i}$ is atomic over ${\overline {N}}_{<s_i}$ . Choose i for which $\{0,1\}\subseteq s_i$ and assume that this holds for all $j<i$ . Let $k=|s_i|$ . Our inductive hypothesis implies the subsystem ${\overline {N}}_{\subset s_i}$ is an atomic-special ${\cal P}^-(k)$ -system. Since $\overline {M}$ was assumed to be atomic-special, $M_{s_i}$ is atomic over $\overline {M}_{\subset s_i}$ . However, since $\bigcup \overline {M}_{\subset s_i}{\sqsubseteq } \bigcup {\overline {N}}_{\subset s_i}$ , we also have that the set $M_{s_i}$ is atomic over $\bigcup {\overline {N}}_{\subset s_i}$ by Lemma A.5. As $N_{s_i}$ was chosen to be $\aleph _1$ -atomic over $M_{s_i}{\overline {N}}_{\subset s_i}$ , it follows that $N_{s_i}$ is also $\aleph _1$ -atomic over ${\overline {N}}_{\subset s_i}$ . Since the subsequence $(N_t:t\subset s_i)$ is an a-s ${\cal P}^-(k)$ -stable system of $\aleph _1$ -saturated models, it follows from Fact 3.3 that $N_{s_i}$ is ${\cal P}^-(k)$ -dominated by $(N_t:t\subset s_i)$ . Thus, $N_{s_i}$ is atomic over ${\overline {N}}_{\subset s_i}$ since a-s ${\cal P}^-(k)$ -DI holds.

Now suppose c is ${\cal P}^-(n+1)$ -dominated by $\overline {M}$ and choose an a-s ${\cal P}^-(n+1)$ -stable system ${\overline {N}}\sqsupseteq \overline {M}$ consisting of $\aleph _1$ -saturated models. We will show that $\mathrm {tp}(c/\bigcup {\overline {N}})$ is isolated, which, since , directly implies $\mathrm {tp}(c/\overline {M})$ is isolated by the Open Mapping Theorem.

Next, we unpack ${\overline {N}}$ into three pieces. Let ${\mathcal K}_0=(N_s:s\in {\cal P}^-(n))$ and let ${\mathcal K}_1=(N_{s\cup \{n\}}:s\in {\cal P}^-(n))$ , with the third, remaining piece $N_n$ . Note that ${\mathcal K}_0{\sqsubseteq }{\mathcal K}_1$ as ${\cal P}^-(n)$ -systems; $\bigcup {\mathcal K}_0\subseteq \bigcup {\mathcal K}_1$ , hence $\bigcup {\overline {N}}=\bigcup {\mathcal K}_1\cup \{N_n\}$ ; and that ${\mathcal K}_1$ is an a-s ${\cal P}^-(n)$ -stable system.

Claim. $N_n c$ is ${\cal P}^-(n)$ -dominated by ${\mathcal K}_1$ .

Proof. Since $N_n$ is atomic over $\bigcup {\mathcal K}_0$ , $N_n$ is ${\cal P}^-(n)$ -dominated by ${\mathcal K}_0$ . As ${\mathcal K}_0{\sqsubseteq } {\mathcal K}_1$ , it follows that $N_n$ is ${\cal P}^-(n)$ -dominated by ${\mathcal K}_1$ as well. Now choose any ${\cal P}^-(n)$ -stable system ${\overline {E}}\sqsupseteq {\mathcal K}_1$ . It follows that

Form a ${\cal P}^-(n+1)$ -stable system ${\overline {E}}^*$ by piecing together ${\mathcal K}_0$ , ${\overline {E}}$ , and $N_n$ . It is readily checked that ${\overline {N}}{\sqsubseteq }{\overline {E}}^*$ as ${\cal P}^-(n+1)$ -structures, hence , so by transitivity we have , proving the Claim.

By the Claim and ${\mathcal K}_1$ being an a-s ${\cal P}^-(n)$ -DI, we have that $cN_n$ is atomic over $\bigcup {\overline {K}}_1$ . Additionally, by Fact 2.5, there is a finite $b\subseteq \bigcup {\overline {N}}$ for which

$$ \begin{align*}\mathrm{tp}(c/b)\vdash\mathrm{tp}(c/\bigcup{\overline{N}}).\end{align*} $$

Recall that $\bigcup {\overline {N}}=\bigcup {\mathcal K}_1\cup N_n$ . Thus, $\mathrm {tp}(bc/{\overline {K}}_1)$ is isolated, which implies $\mathrm {tp}(c/{\overline {K}}_1 b)$ is isolated as well. As $\mathrm { tp}(c/b)\vdash \mathrm {tp}(c/{\overline {N}})$ , the same formula isolates $\mathrm {tp}(c/{\overline {N}})$ as well.

Proof. We argue that for all infinite cardinals $\kappa $ ,

To begin, note that this holds for $\kappa =\aleph _0$ by coupling Fact A.2(2) with Proposition 3.5. Fix an uncountable cardinal $\kappa $ and assume we have the above for all $\lambda <\kappa $ . Choose $n\ge 2$ and an atomic-special ${\cal P}^-(n)$ -stable system $\overline {M}=(M_s:s\in {\cal P}^-(n))$ with $|\bigcup \overline {M}|=\kappa $ . Let $\mu =cf(\kappa )$ . By iterating the analog of Fact 2.2(3), choose an elementary chain $(\overline {M}^\alpha :\alpha <\mu )$ of ${\cal P}^-(n)$ -systems such that

1. 1. $|\bigcup \overline {M}^\alpha |<\kappa $ ;

2. 2. $\overline {M}^\alpha {\sqsubseteq } \overline {M}^{\alpha +1}$ for all $\alpha $ ; and

3. 3. $\overline {M}=\bigcup \{\overline {M}^\alpha :\alpha <\mu \}$ .

We will recursively build a sequence of sequences $\overline {c}_\alpha $ for each $\alpha $ such that

1. 1. $\overline {c}_\alpha $ is an initial segment of $\overline {c}_\beta $ whenever $\alpha \le \beta <\mu $ ;

2. 2. $\overline {c}_\alpha $ enumerates a constructible model $N_\alpha $ over $\bigcup \overline {M}^\alpha $ ;

3. 3. $\overline {c}_\alpha $ is also a construction sequence over $\overline {M}$ (hence also over any $\overline {M}^\beta $ , $\beta \ge \alpha $ ).

To begin, since $|\bigcup \overline {M}^0|<\kappa $ , apply $(**)$ to get $\overline {c}_0$ , enumerating a constructible model over $\overline {M}^0$ . As $\bigcup \overline {M}^0\subseteq _{TV} \overline {M}$ , $\overline {c}_0$ is also a construction sequence over $\bigcup \overline {M}$ . For $\gamma <\mu $ a non-zero limit, take $\overline {c}_\gamma $ to be the concatenation of all $\overline {c}_\alpha $ , $\alpha <\gamma $ .

Say $\alpha <\mu $ and $\overline {c}_\alpha $ has been found. Let $N_\alpha =\bigcup \overline {c}_\alpha $ . Note that since $\overline {M}^\alpha {\sqsubseteq }\overline {M}^{\alpha +1}$ , the three pieces $(\overline {M}^\alpha ,\overline {M}^{\alpha +1},N_\alpha )$ form a ${\cal P}^-(n+1)$ -stable system. We also claim that it is atomic-special. For this, choose $s\in {\cal P}^-(n+1)$ with $\{0,1\}\subseteq s$ . There are three cases. First, if $s=n=\{i:i\in n\}$ , then as $N_\alpha $ is constructible over $\bigcup \overline {M}^\alpha $ , it is atomic over $\bigcup \overline {M}^\alpha $ as well. Second, if $s\subset n$ , $s\neq n$ , then by $\overline {M}^\alpha {\sqsubseteq }\overline {M}$ we have . But, as $\overline {M}$ is atomic-special, every finite $e\in M_s^\alpha $ has $\mathrm {tp}(e/\bigcup \overline {M}_{<s})$ isolated. Thus, by the Open Mapping Theorem, $\mathrm {tp}(e/\overline {M}^{\alpha +1}_{<s})$ is isolated as well. Third, if $n\in s$ , then similarly, and $\mathrm {tp}(e/\bigcup \overline {M}_{<s})$ is isolated for all finite $e\in M_s^{\alpha +1}$ , so again, $\mathrm { tp}(e/\overline {M}^{\alpha +1}_{<s})$ is isolated by the Open Mapping Theorem.

Thus, by our inductive hypothesis, there is a constructible model $N_{\alpha +1}$ over $\bigcup \overline {M}^{\alpha +1}\overline {c}_\alpha $ . However, as $\overline {M}^\alpha \subseteq _{TV} \overline {M}^{\alpha +1}$ , $\overline {c}_\alpha $ is also constructible over $\bigcup \overline {M}^{\alpha +1}$ . Thus, there is a construction sequence $\overline {c}_{\alpha +1}$ end extending $\overline {c}_\alpha $ that enumerates $N_\alpha $ . For any such choice of $\overline {c}_{\alpha +1}$ , since $\overline {M}^{\alpha +1}\subseteq _{TV} \overline {M}$ , we have that $\overline {c}_{\alpha +1}$ is also a construction sequence over $\bigcup \overline {M}$ , as required.

Proof. Let M be an a-model on which p is based and let $N=M[a]$ be a-prime over M and a realization of p with $q\not \perp N$ . As $q\in \mathbf {P_e}$ choose a non-isolated weight one, stationary $r\in S(d)$ with $d\subseteq N$ finite with $r\not \perp q$ . Let b be any realization of $r|da$ . Since $da$ is dominated by a over M and $\mathrm {tp}(b/da)\perp M$ , we have that $bda$ is dominated by a over M, from which it follows that $wt(bd/M)=1$ . Choose any finite $e\subseteq M$ so that $\mathrm {tp}(bd/M)$ is based and stationary on e. Then $\mathrm {tp}(bd/e)$ is a non-isolated weight one, stationary type non-orthogonal to p, hence $p\in \mathbf {P_e}$ .

Proof. By induction on $wt(p)$ . If $wt(p)=1$ , but there were some finite d on which p is based with $p|d$ non-isolated, then let b be any realization of $p|d$ . Then $p|db$ is stationary, weight one, and non-isolated by the Open Mapping Theorem, contradicting $p\perp \mathbf {P_e}$ , so the Lemma holds for $wt(p)=1$ .

Assume the Lemma holds for all strong types of weight at most n. Choose a strong type $p\perp \mathbf {P_e}$ of weight $n+1$ and choose a finite set d on which p is based. Choose an a-model $M\supseteq d$ and let a realize $p|M$ . We will show $tp(a/d)$ is isolated. As M is an a-model, choose an M-independent set $\{b_i:i\le n\}$ such that $\mathrm {tp}(b_i/M)$ is regular and for each i. Choose an a-model $N^*=M[b_i:i\le n]$ with $a\in N^*$ and within $N^*$ , choose $N\preceq N^*$ to be a-prime over $M\cup \{b_i:i<n\}$ . Choose any finite $e\subseteq N$ on which $\mathrm {tp}(a/N)$ is based. As $e\subseteq N$ , e is dominated by $\{b_i:i<n\}$ over M, hence $\mathrm {tp}(e/M)\perp \mathbf {P_e}$ and $wt(e/M)\le n$ . Choose a finite $h\supseteq d$ on which $\mathrm {tp}(e/M)$ is based. By our inductive hypothesis $\mathrm {tp}(e/h)$ is isolated. As well, $eh\subseteq N$ and $\mathrm {tp}(a/N)$ is based on this set, hence $\mathrm {tp}(a/eh)$ is isolated as well. Putting these together, $\mathrm {tp}(a/h)$ is isolated. However, , so $\mathrm {tp}(a/d)$ is isolated by the Open Mapping Theorem.

Proof. If $p\perp \mathbf {P_e}$ , then $q\perp \mathbf {P_e}$ for every $q\triangleleft p$ , so left to right follows from Lemma 4.6. For the converse, assume $p\not \perp \mathbf {P_e}$ . Choose a regular $q\in \mathbf {P_e}$ with $q\not \perp p$ . As $q\in \mathbf {P_e}$ , choose a stationary, weight one $r\not \perp q$ that is not always isolated. As $r\triangleleft q\triangleleft p$ , we finish.

Proof. Let $\{\alpha _i(x,b,a):i\in I\}$ enumerate the formulas that fork over b and let $\{E_j:j\in J\}$ enumerate $FE(b)$ . Note that $\bigwedge _{j\in J} E_j(x,a)\vdash p(x)$ , so by the Finite Equivalence Relation theorem, q is generated by

$$ \begin{align*}\{E_j(x,a):i\in J\}\cup\{\neg\alpha_i(x,b,a):i\in I\}.\end{align*} $$

If $\varphi (x,b,a)$ isolates q, there are finite subsets $J_0\subseteq J$ and $I_0\subseteq I\}$ entailing $\varphi (x,b,a)$ . Put $E^*(x,y):=\bigwedge _{j\in J_0}E_j(x,y)$ . Then $E^*(x,y)\in FE(b)$ , and it suffices to show that $\forall x(E^*(x,a)\vdash E_j(x,a))$ for every $j\in J$ . To see this, fix any $j\in J$ and choose any $a_1$ such that $E^*(a_1,a)$ holds. As $E^*(x,a)\in FE(b)$ , it is not a forking formula over b, hence there is some $a_2\in {\frak C}$ such that $\mathrm {stp}(a_1/b)=\mathrm {stp}(a_2,b)$ , $E^*(a_1,a_2)$ , and . From above, $\varphi (a_2,b,a)$ , hence $E_j(a_2,a)$ holds. But $\mathrm {stp}(a_1/b)=\mathrm {stp}(a_2/b)$ implies $E_j(a_1,a_2)$ , so $E_j(a_1,a)$ , as required.

Proof. Choose a finite $b\subseteq M$ on which p is based. Since p is always isolated, choose $\psi (x,b)$ isolating $p_0:=\mathrm {tp}(a/b)$ . Choose $a'\in p({\frak C})$ with and let $q=\mathrm {tp}(a/ba')$ . So q is stationary and based on $ba'$ . As q is parallel to p, it is always isolated, hence there is $\varphi (x,b,a')$ isolating $q|ba'$ . Choose $E^*(x,y)\in FE(b)$ as in Lemma 4.8. Now $a'\not \in M$ , but $\psi (M,b)$ contains a complete set of representatives of the $E^*(x,y)$ -classes consistent with $\psi (x,b)$ . Choose $a^*\in \psi (M,b)$ with $E^*(a',a^*)$ . As both $a'$ and $a^*$ realize $p_0$ , it follows from Lemma 4.8 that $\mathrm {stp}(a'/b)=\mathrm {stp}(a^*/b)$ . Choose a strong automorphism $\sigma \in Aut({\frak C}/b)$ with $\sigma (a')=a^*$ and put $q^*(x,b,a^*):=\sigma (q)$ . Then $q^*$ is stationary and, since since both $a'$ and $a^*$ are independent from a over b, it follows that $\mathrm {tp}(aa'b)=\mathrm {tp}(aa^*b)$ . That is, a realizes the stationary type $q^*$ , so p is based and stationary over $ba^*$ .

Proof. By Lemma 4.9, choose any finite $d\subseteq M$ on which p is based and stationary. Since p is always isolated, there is a formula $\theta (x,d)\in p$ isolating the type $p|d$ . We first argue that p is strongly regular via $\theta (x,d)$ . To see this, choose any global type $q\in S({\frak C})$ with $\theta (x,d)\in q$ . As $\theta (x,d)$ isolates $p|d$ , q extends the regular type $p|d$ . Thus, if $q\not \perp p$ , then $q\not \perp p|d$ , so q is the non-forking extension of $p|d$ , which is also the non-forking extension of p to $S({\frak C})$ .

To see that p is na, choose any $\varphi (x,b)\in p$ . By replacing $\varphi (x,b)$ by $\varphi (x,b)\wedge \theta (x,d)$ , we may assume $d\subseteq b$ and $\varphi (x,b)\vdash \theta (x,d)$ . Since p is always isolated, $p|b$ is isolated, say by $\delta (x,b)$ . Since M is a model, choose $e\in M$ realizing $\delta (x,b)$ . Then $e\in \varphi (M,b)$ , but $\mathrm {tp}(e/b)=p|b$ , hence is non-algebraic.

Proof. Choose $\theta (x,b,m)\in \mathrm {tp}(c/Mb)$ with $R^\infty (\theta (x,b,m))=R^\infty (c/Mb)$ . By increasing m (but keeping it finite), Lemma 4.9 allows us to assume $\mathrm {tp}(bc/M)$ is based and stationary on m. As $\mathrm {tp}(bc/M)\perp \mathbf {P_e}$ , we conclude that $\mathrm {tp}(bc/m)$ is isolated, say by $\varphi (x,y,m)$ . It follows that $\varphi (x,b,m)$ isolates $\mathrm {tp}(c/bm)$ , but we argue that $\varphi (x,b,m)$ isolates $\mathrm {tp}(c/Mb)$ .

To see this, choose any $c'$ realizing $\varphi (x,b,m)$ . Note that $\varphi (x,y,m)\vdash \theta (x,y,m)$ , so it follows that . However, $\mathrm {tp}(c'b/m)=\mathrm {tp}(cb/m)$ is stationary and both $cb$ and $c'b$ are independent with M over m, hence $\mathrm { tp}(c'bM/m)=\mathrm {tp}(cbM/m)$ . As $m\subseteq M$ , this implies $\mathrm {tp}(c'/Mb)=\mathrm {tp}(c/Mb)$ , as required.

Proof. As T and M are countable, it suffices to show that the isolated types over $Mb$ are dense. Choose any consistent formula $\psi (x,a,b)$ with a from M. Among all consistent $L(Mb)$ -formulas that imply $\psi (x,a,b)$ , choose $\theta (x,a',b)\vdash \psi (x,a,b)$ of least $R^\infty $ -rank and then choose an element $c\in \theta ({\frak C},a',b)$ that is $\ell $ -isolated over $Mb$ . Thus, $cb$ is dominated by b over M and $\mathrm { tp}(c/Mb)$ is c-isolated, hence $\mathrm {tp}(c/Mb)$ is isolated by Proposition 4.11. From this density result, a constructible model over $Mb$ exists.

Proof. Choose any well ordering $I=(\eta _\alpha :\alpha <\delta )$ such that $\eta _\alpha \triangleleft \eta _\beta $ implies $\alpha <\beta $ .

This implies that for all $\alpha <\delta $ , $I_{<\alpha }=\{\eta _\beta :\beta <\alpha \}$ is a subtree of $(I,\trianglelefteq )$ , thus

$$ \begin{align*}\overline{M}_{<\alpha}:=\bigcup\{M_\gamma:\gamma<\alpha\}\subseteq_{TV} \bigcup \overline{M}_{<\beta}:=\{M_\gamma:\gamma<\beta\}\subseteq_{TV} M_I,\end{align*} $$

for all $\alpha <\beta <\delta $ . It follows that any construction sequence $\overline {c}$ over $\overline {M}_{<\alpha }$ is also a construction sequence over $\overline {M}_{<\beta }$ and over $M_I$ .

We recursively find a sequence $(\overline {c}_\alpha :\alpha \le \delta )$ of sequences satisfying:

1. 1. $\overline {c}_\alpha $ enumerates a constructible model $N_\alpha $ over $\bigcup \overline {M}_{<\alpha }$ ; and

2. 2. $\overline {c}_\alpha $ is an initial segment of $\overline {c}_\beta $ for all $\alpha \le \beta \le \delta $ .

If we succeed, then $\overline {c}_\delta $ will be a construction sequence over $M_I$ . Put $\overline {c}_0:=\langle \rangle $ and, for all non-zero limit ordinals $\gamma $ , take $\overline {c}_\gamma $ to be the concatenation of $(\overline {c}_\alpha :\alpha <\gamma )$ . Assuming $\overline {c}_\alpha $ has been chosen with $\alpha <\delta $ , note that $M_\alpha $ is a ‘leaf’ of the subtree $\overline {M}_{\le \alpha }$ . As well,

where $N_\alpha $ is the constructible model over $\overline {M}_{<\alpha }$ enumerated by $\overline {c}_\alpha $ . From above, $\overline {c}_\alpha $ is also a construction sequence over $\overline {M}_{<\alpha } M_\alpha $ . By PMOP, there is a constructible model $N_{\alpha +1}$ over $M_\alpha N_\alpha $ , from which it follows there is an enumeration $\overline {c}_{\alpha +1}$ of $N_{\alpha +1}$ in which $\overline {c}_\alpha $ is an initial segment.

Proof. We argue by induction on $|I|$ . For $|I|=1$ , this is immediate by Lemma A.14(1) (taking $A=M_{\langle \rangle }$ and $B=\emptyset $ ). So assume $(M_\eta :\eta \in I)$ is any independent tree of models with $|I|=n+1$ and we have proved the Lemma when $|I|=n$ . Fix any strong type p that is orthogonal to every $M_\eta $ . Choose any leaf $\eta \in I$ and let $J\subseteq I$ be the subtree with universe $I\setminus \{\eta \}$ . By the inductive hypothesis, $M_J$ is essentially finite with respect to p, so the result follows by Lemma A.14(2), taking $A=M_J$ and $B=M_\eta $ .

Proof. As notation, for $K\subseteq I$ , we let $M_K$ denote $\bigcup _{\nu \in K} M_\nu $ . By Lemma 4.9 there is a finite $d_0\subseteq N$ over which $\mathrm {tp}(c/N)$ is based and stationary. It suffices to show that $\mathrm {tp}(dc/M_I)$ is isolated for any finite d with $d_0\subseteq d\subseteq N$ , so choose such a d. Choose a finite $e\subseteq M_I$ with a formula $\varphi (x,e)$ isolating $\mathrm {tp}(d/M_I)$ . Choose a finite, downward closed subtree $J\subseteq I$ containing e. As $\mathrm {tp}(c/d)$ is stationary and $\perp M_\eta $ for every $\eta \in J$ , Lemma 5.3 implies that $M_J$ is essentially finite with respect to $\mathrm {tp}(c/d)$ , so there is a finite $e^*$ , $e\subseteq e^*\subseteq M_J$ for which $\mathrm {tp}(c/de^*)\vdash \mathrm {tp}(c/dM_J)$ . As $\mathrm {tp}(c/N)\perp \mathbf {P}_e$ , by Lemma 4.6, $\mathrm { tp}(c/de^*)$ is isolated. Thus, $\mathrm {tp}(c/dM_J)$ is isolated as well. Since $\mathrm {tp}(d/M_J)$ is isolated, so is $\mathrm {tp}(cd/M_J)$ . But now, since $M_J\subseteq _{TV} M_I$ , we conclude that $\mathrm {tp}(cd/M_I)$ is isolated as well.