A strong failure of aleph_0-stability for atomic classes

TL;DR

This paper demonstrates that for certain classes of atomic models in countable theories, the failure of aleph_0-stability leads to a vast number of non-isomorphic models of size aleph_1, highlighting a significant instability phenomenon.

Contribution

It establishes a strong link between the non-pcl-smallness of atomic classes and the existence of many non-isomorphic models, revealing a failure of aleph_0-stability.

Findings

Uncountably many types over pcl(a) imply 2^aleph1 non-isomorphic models.

Failure of aleph_0-stability occurs in non-pcl-small atomic classes.

The result connects atomic model properties with model-theoretic stability concepts.

Abstract

We study classes of atomic models At_T of a countable, complete first-order theory T . We prove that if At_T is not pcl-small, i.e., there is an atomic model N that realizes uncountably many types over pcl(a) for some finite tuple a from N, then there are 2^aleph1 non-isomorphic atomic models of T, each of size aleph1.

Equations

$$S_{at}^{+}(A,M):=\{p|A:p\in S_{at}(M)\}.$$

$$S_{at}^{+}(A,M)=\{p|A:p\in S_{at}(M)\}$$

$$(\mathcal{B},ϵ)\models ‘There\; isM\models \Phi \; and\; |M|={\aleph }_1’$$

$$\mathcal{C}=\{(\text{SS},\bar{b}):\bar{b}\supseteq {\bar{a}}^{\ast },\text{SS}\subseteq {\mathcal{Q}}_{\bar{b}},and{S}_{at}^{+}(A_{\text{SS}},M^{\ast })uncountable\}$$

$${\alpha }^{\ast }:=\sup \{rk(r)+1:r\in {\text{SS}}^{\ast }\}$$

$${\text{SS}}^{\prime }:=\{r^{\prime }\in {\mathcal{Q}}_{{\bar{b}}^{\prime }}:r^{\prime }extends\; some{r}_{j}withj<n\}$$

$$c:N\to \omega$$

$$\mathcal{B}[G]\models ‘There\; is\; a\; rich,{\mathcal{U}}_G-colored(M_G,{\bar{b}}^{\ast })of\; size{\aleph }_1’$$

$${\mathbb{P}}_n=\{(k,\bar{p}):k\in \omega ,\bar{p}=⟨p_{\eta }:\eta \in 2^n⟩,where\; each{p}_{\nu }\in \mathbb{Q}and\; every{k}_{p_{\nu }}=k\}$$

$$G_{\nu }:=\{\mathbf{p}\in \mathbb{Q}:for\; some{\mathbf{p}}^{\ast }=(k,\bar{p})\in G^{\ast },\mathbf{p}={\mathbf{p}}_{\nu }^{\ast }\}$$

$$G_{\eta }:=\bigcup \{G_{\eta |n}:n\in \omega \}and{\mathcal{U}}_{\eta }:=\{\ell \in \omega :\ell \in {\mathcal{U}}_{\mathbf{q}}for\; some\mathbf{q}\in G_{\eta }\}$$

Full text

1 Introduction

In a series of papers [2, 3, 4], Baldwin and the authors have begun to develop a model theory for complete sentences of $L_{\omega_{1},\omega}$ that have fewer than $2^{\aleph_{1}}$ non-isomorphic models of size $\aleph_{1}$. By well known reductions, one can replace the reference to infinitary sentences by restricting to the class of atomic models of a countable, complete first-order theory.111Specifically, for every complete sentence $\Phi$ of $L_{\omega_{1},\omega}$, there is a complete first-order theory $T$ in a countable vocabulary containing the vocabulary of $\Phi$ such that the models of $\Phi$ are precisely the reducts of the class of atomic models of $T$ to the smaller vocabulary.

Fix, for the whole of this paper, a complete theory $T$ in a countable language that has at least one atomic model222A model $M$ is atomic if, for every finite tuple $\bar{a}$ from $M$, ${\rm tp}_{M}(\bar{a})$ is principal i.e., is uniquely determined by a single formula $\varphi(\overline{x})\in{\rm tp}_{M}(\bar{a})$. of size $\aleph_{1}$. By theorems of Vaught, these restrictions on $T$ are well understood. Such a $T$ has an atomic model if and only if every consistent formula can be extended to a complete formula. Furthermore, any two countable, atomic models of $T$ are isomorphic, and a model is prime if and only if it is countable and atomic. Using a well-known union of chains argument, $T$ has an atomic model of size $\aleph_{1}$ if and only if the countable atomic model is not minimal, i.e., it has a proper elementary substructure.

The analysis of ${\bf At}_{T}$, the class of atomic models of $T$, begins by restricting the notion of types to those that can be realized in an atomic model. Suppose $M$ is atomic and $A\subseteq M$. We let $S_{at}(A)$ denote the set of complete types $p$ over $A$ for which $Ab$ is an atomic set for some (equivalently, for every) realization $b$ of $p$. It is easily checked that when $A$ is countable, $S_{at}(A)$ is a $G_{\delta}$ subset of the Stone space $S(A)$, hence $S_{at}(A)$ is Polish with respect to the induced topology. We will repeatedly use the fact that any countable, atomic set $A$ is contained in a countable, atomic model $M$. However, unlike the first-order case, some types in $S_{at}(A)$ need not extend to types in $S_{at}(M)$. Indeed, there are examples where the space $S_{at}(A)$ is uncountable (hence contains a perfect set) while $S_{at}(M)$ is countable. Thus, for analyzing types over countable, atomic sets $A\subseteq M$, we are led to consider

[TABLE]

Equivalently, $S_{at}^{+}(A,M)$ is the set of $q\in S_{at}(A)$ that can be extended to a type $q^{*}\in S_{at}(M)$.

Next, we recall the notion of pseudo-algebraicity, which was introduced in [2], that is the correct analog of algebraicity in the context of atomic models. Suppose $M$ is an atomic model, and $b,\bar{a}$ are from $M$. We say $b\in{\rm pcl}_{M}(\bar{a})$ if $b\in N$ for every elementary submodel $N\preceq M$ that contains $\bar{a}$. The seeming dependence on $M$ is illusory – as is noted in [2], if $b^{\prime},\bar{a}^{\prime}$ are inside another atomic model $M^{\prime}$, and ${\rm tp}_{M^{\prime}}(b^{\prime}\bar{a}^{\prime})={\rm tp}_{M}(b\bar{a})$, then $b\in{\rm pcl}_{M}(\bar{a})$ if and only if $b^{\prime}\in{\rm pcl}_{M^{\prime}}(\bar{a}^{\prime})$. It is easily seen that inside any atomic model $M$, ${\rm pcl}_{M}(\bar{a})$ is countable for any finite tuple $\bar{a}$. Moreover, if $f:M\rightarrow M^{\prime}$ is an isomorphism of atomic models, then $f({\rm pcl}_{M}(\bar{a}))={\rm pcl}_{M^{\prime}}(f(\bar{a}))$ setwise. As an important special case, if $\bar{a}\subseteq M^{\prime}\preceq M$ and $f:M\rightarrow M^{\prime}$ fixes $\bar{a}$ pointwise, then $f$ induces an elementary permutation on $D={\rm pcl}_{M}(\bar{a})$, which in turn induces a bijection between $S_{at}^{+}(D,M)$ and $S_{at}^{+}(D,M^{\prime})$.

We now give the major new definition of this paper:

Definition 1.1

An atomic class ${\bf At}_{T}$ with an uncountable model is pcl-small if, for every atomic model $N$ and for every finite $\bar{a}$ from $N$, $N$ realizes only countably many complete types over ${\rm pcl}_{N}(\bar{a})$. **

The name of this notion is by analogy with the first-order case – A complete, first-order theory $T$ is small if and only if for every model $N$ and every finite $\bar{a}$ from $N$, $N$ realizes only countably many complete types over $\bar{a}$. The following proposition relates ${\rm pcl}$-smallness with the spaces of types $S^{+}_{at}(D,M)$.

Proposition 1.2

The atomic class ${\bf At}_{T}$ is ${\rm pcl}$-small if and only if the space of types $S^{+}_{at}({\rm pcl}_{M}(\bar{a}),M)$ is countable for every countable, atomic model $M$ and every finite $\bar{a}$ from $M$.

Proof. First, assume that some atomic model $N$ and finite sequence $\bar{a}$ from $N$ witness that ${\bf At}_{T}$ is not ${\rm pcl}$-small. Choose $\{c_{i}:i\in\omega_{1}\}\subseteq N$ realizing distinct complete types over $D={\rm pcl}_{N}(\bar{a})$. Also, choose a countable $M\preceq N$ that contains $\bar{a}$, and hence $D$. Then $\{{\rm tp}(c_{i}/D):i\in\omega_{1}\}$ witness that $S^{+}_{at}(D,M)$ is uncountable.

For the converse, choose a countable, atomic model $M$ and $\bar{a}$ from $M$ such that $S^{+}_{at}(D,M)$ is uncountable, where $D={\rm pcl}_{M}(\bar{a})$. We will inductively construct a continuous, increasing elementary chain $\langle M_{\alpha}:\alpha<\omega_{1}\rangle$ of countable, atomic models with $M=M_{0}$ and, for each ordinal $\alpha$, there is an element $c_{\alpha}\in M_{\alpha+1}$ such that ${\rm tp}(c_{\alpha}/D)$ is not realized in $M_{\alpha}$. Given such a sequence, it is evident that $N=\bigcup_{\alpha<\omega_{1}}M_{\alpha}$ and $\bar{a}$ witness that ${\bf At}_{T}$ is not ${\rm pcl}$-small. To construct such a sequence, we have defined $M_{0}$ to be $M$ and take unions at limit ordinals. For the successor step, assume $M_{\alpha}$ has been defined. As $M$ and $M_{\alpha}$ are each countable atomic models that contain $\bar{a}$, choose an isomorphism $f:M\rightarrow M_{\alpha}$ fixing $\bar{a}$ pointwise. As noted above, $f$ fixes $D$ setwise. As $M_{\alpha}$ is countable, so is the set $\{{\rm tp}(c/D):c\in M_{\alpha}\}$. As $S^{+}_{at}(D,M)$ is uncountable, choose an atomic type $p\in S_{at}(M)$, whose restriction to $D$ is distinct from $\{f^{-1}({\rm tp}(c/D)):c\in M_{\alpha}\}$. Now choose $c_{\alpha}$ to realize $f(p)$. Then, as $M_{\alpha}c_{\alpha}$ is a countable atomic set, choose a countable elementary extension $M_{\alpha+1}\succeq M_{\alpha}$ containing $c_{\alpha}$.

Recall that an atomic class ${\bf At}_{T}$ is $\aleph_{0}$-stable333Sadly, this usage of ‘$\aleph_{0}$-stability’ is analogous, but distinct from, the familiar first-order notion. if $S_{at}(M)$ is countable for all (equivalently, for some) countable atomic models $M$. As $S^{+}_{at}(A,M)$ is a set of projections of types in $S_{at}(M)$, it will be countable whenever $S_{at}(M)$ is. This observation makes the following corollary to Proposition 1.2 immediate:

Corollary 1.3

If an atomic class ${\bf At}_{T}$ is $\aleph_{0}$-stable, then ${\bf At}_{T}$ is ${\rm pcl}$-small.

The converse to Corollary 1.3 fails. For example, the theory $T=REF(bin)$ of countably many, binary splitting equivalence relations is not $\aleph_{0}$-stable, yet ${\rm pcl}_{M}(\bar{a})=\bar{a}$ for every model $M$ and $\bar{a}$ from $M$. Thus, $S_{at}({\rm pcl}_{M}(\bar{a}))$ and hence $S^{+}_{at}({\rm pcl}(\bar{a}),M)$ is countable for every finite tuple $\bar{a}$ inside any atomic model $M$. The main theorem of this paper is:

Theorem 1.4

Let $T$ be a countable, complete theory $T$ with an uncountable atomic model. If the atomic class ${\bf At}_{T}$ is not ${\rm pcl}$-small, then there are $2^{\aleph_{1}}$ non-isomorphic models in ${\bf At}_{T}$, each of size $\aleph_{1}$.

Section 2 sets the stage for the proof. It describes the spaces of types $S^{+}_{at}(A,M)$, states a transfer theorem for sentences of $L_{\omega_{1},\omega}(Q)$, and details a non-structural configuration arising from non-${\rm pcl}$-smallness. In Section 3, the non-structural configuration is exploited to give a family of $2^{\aleph_{0}}$ non-isomorphic structures $(N,\bar{b}^{*})$, where each of the reducts $N$ is in ${\bf At}_{T}$ and has size $\aleph_{1}$. Theorem 1.4 is finally proved in Section 4. It is remarkable that whereas it is a ZFC theorem, the proof is non-uniform depending on the relative sizes of the cardinals $2^{\aleph_{0}}$ and $2^{\aleph_{1}}$.

2 Preliminaries

In this section, we develop some general tools that will be used in the proof of Theorem 1.4.

2.1 On $S^{+}_{at}(A,M)$

In this subsection we explore the space of types

[TABLE]

where $A$ is a subset of a countable, atomic model $M$.

Fix a countable, atomic model $M$ and an arbitrary subset $A\subseteq M$. Let ${\cal P}$ denote the space of complete types in one free variable over finite subsets of $M$. As $M$ is atomic, ${\cal P}$ can be identified with the set of complete formulas $\varphi(x,m)$ over $M$. Implication gives a natural partial order on ${\cal P}$, namely $p\leq q$ if and only if ${\rm dom}(p)\subseteq{\rm dom}(q)$ and $q\vdash p$. One should think of elements of ${\cal P}$ as ‘finite approximations’ of types in $S_{at}^{+}(A,M)$. We describe two conditions on $p\in{\cal P}$ that identify extreme behaviors in this regard.

Definition 2.1

We say a type $p^{*}\in S^{+}_{at}(A,M)$ lies above $p\in{\cal P}$ if there is some $\bar{p}\in S_{at}(M)$ extending $p\cup p^{*}$. As every $p\in{\cal P}$ extends to a type in $S_{at}(M)$, it follows that at least one $p^{*}\in S^{+}_{at}(A,M)$ lies above $p$.

• •

An element $p\in{\cal P}$ determines a type in $S^{+}_{at}(A,M)$ if exactly one $p^{*}\in S_{at}^{+}(A,M)$ lies above $p$.

• •

An element $p\in{\cal P}$ is $A$-large if $\{p^{*}\in S^{+}_{at}(A,M):p^{*}$ lies above $p\}$ is uncountable.

To understand these extreme behaviors, we define a rank function ${\rm rk}_{A}:{\cal P}\rightarrow(\omega_{1}+1)$ as follows:

• •

${\rm rk}_{A}(p)\geq 0$ for all $p\in{\cal P}$;

• •

For $\alpha\leq\omega_{1}$, ${\rm rk}_{A}(p)\geq\alpha$ if and only if for every $\beta<\alpha$ and for all finite $F$, ${\rm dom}(p)\subseteq F\subseteq M$, there is $q\in S_{at}(F)$ with $q\geq p$ that $\beta$-$A$ splits, where:

• –

A type $q\in S_{at}(F)$ $A$-splits if, for some $\varphi(x,\bar{a})$ with $\bar{a}$ from $A$, there are $q_{1},q_{2}\geq q$ with $q\cup\varphi(x,\bar{a})\subseteq q_{1}$ and $q\cup\neg\varphi(x,\bar{a})\subseteq q_{2}$; and $q\in S_{at}(F)$ $\beta$-$A$ splits if, in addition, ${\rm rk}_{A}(q_{1}),{\rm rk}_{A}(q_{2})\geq\beta$.

• •

For $\alpha<\omega_{1}$, we say ${\rm rk}_{A}(p)=\alpha$ if ${\rm rk}_{A}(p)\geq\alpha$, but ${\rm rk}_{A}(p)\not\geq\alpha+1$.

Proposition 2.2

If $p\in{\cal P}$ and ${\rm rk}_{A}(p)=\alpha<\omega_{1}$, then some $r\geq p$ determines a type in $S^{+}_{at}(A,M)$.

Proof. We prove this by induction on $\alpha$. We begin with $\alpha=0$. Suppose ${\rm rk}_{A}(p)=0$. As ${\rm rk}_{A}(p)\not\geq 1$, there is a finite $F$, ${\rm dom}(p)\subseteq F\subseteq M$ for which there is no $q\in S_{at}(F)$ and $\varphi(x,\bar{a})$ with $\bar{a}$ from $A$ for which $q\geq p$ and both $q\cup\{\varphi(x,\bar{a})\}$ and $q\cup\{\neg\varphi(x,\bar{a})\}$ are consistent. So fix any $r\in S_{at}(F)$ with $r\geq p$. Any such $r$ determines a type in $S^{+}_{at}(A,M)$.

Next, choose $0<\alpha<\omega_{1}$ and assume the Proposition holds for all $\beta<\alpha$. Choose $p\in S_{at}(E)$ with ${\rm rk}_{A}(p)=\alpha$. As ${\rm rk}_{A}(p)\geq\alpha$, while ${\rm rk}_{A}(p)\not\geq\alpha+1$, there is a finite $F$, $E\subseteq F\subseteq M$ for which there is no $q\in S_{at}(F)$ that both extends $p$ and $\alpha$-$A$ splits. So choose any $q\in S_{at}(F)$ with $q\geq p$. If $q$ determines a type in $S^{+}_{at}(A,M)$, then we finish, so assume otherwise. Thus, there is some $\varphi(x,\bar{a})$ with $\bar{a}$ from $A$ such that both $q\cup\{\varphi(x,\bar{a})\}$ and $q\cup\{\neg\varphi(x,\bar{a})\}$ are consistent. Choose complete types $q_{1},q_{2}\in S_{at}(F\bar{a})$ extending these partial types. Clearly, both $q_{1},q_{2}\geq q$, but since $q$ does not $\alpha$-$A$ split, at least one of them has ${\rm rk}_{A}(q_{\ell})<\alpha$. But then by our inductive hypothesis, there is $r\geq q_{\ell}$ that determines a type in $S^{+}_{at}(A,M)$ and we finish.

Next, we turn our attention to $A$-large types and types of rank at least $\omega_{1}$ and see that these coincide. We begin with two lemmas, the first involving types of rank at least $\omega_{1}$ and the second involving $A$-large types.

Lemma 2.3

Assume that $E\subseteq M$ is finite and $p\in S_{at}(E)$ has ${\rm rk}_{A}(p)\geq\omega_{1}$. Then:

For every finite $F$, $E\subseteq F\subseteq M$, there is $q\in S_{at}(F)$, $q\geq p$, with ${\rm rk}_{A}(q)\geq\omega_{1}$; and 2. 2.

There is some formula $\varphi(x,\bar{a})$ with $\bar{a}$ from $A$ and $q_{1},q_{2}\in{\cal P}$ with $p\cup\{\varphi(x,\bar{a})\}\subseteq q_{1}$, $p\cup\{\neg\varphi(x,\bar{a})\}\subseteq q_{2}$, and both ${\rm rk}_{A}(q_{1}),{\rm rk}_{A}(q_{2})\geq\omega_{1}$.

Proof. (1) Fix a finite $F$ satisfying $E\subseteq F\subseteq M$. As ${\rm rk}_{A}(p)\geq\omega_{1}$, for every $\beta<\omega_{1}$ there is some $q\geq p$ with $q\in S_{at}(F)$ for which certain extensions of $q$ have rank at least $\beta$. It follows that ${\rm rk}_{A}(q)\geq\beta$ for any such witness. However, as $S_{at}(F)$ is countable, there is some $q\in S_{at}(F)$ which serves as a witness for uncountably many $\beta$. Thus, ${\rm rk}_{A}(q)\geq\omega_{1}$ for any such $q\geq p$.

(2) Assume that there were no such formula $\varphi(x,\bar{a})$. Then, for any formula $\varphi(x,\bar{a})$, since ${\cal P}$ is countable, there would be an ordinal $\beta^{*}<\omega_{1}$ such that either every $q\in{\cal P}$ extending $p\cup\{\varphi(x,\bar{a})\}$, ${\rm rk}_{A}(q)<\beta^{*}$ or every $q\in{\cal P}$ extending $p\cup\{\neg\varphi(x,\bar{a})\}$ has ${\rm rk}_{A}(q)<\beta^{*}$. Continuing, as there are only countably many formulas $\varphi(x,\bar{a})$, there would be an ordinal $\beta^{**}<\omega_{1}$ that works for all formulas $\varphi(x,\bar{a})$. Restating this, $p$ does not $\beta^{**}$-$A$ split, so no extension of $p$ could $\beta^{**}$-$A$ split either. This contradicts ${\rm rk}_{A}(p)\geq\beta^{**}+1$.

Lemma 2.4

Suppose $q\in S_{at}(F)$ is $A$-large. Then:

For every finite $F^{\prime}$, $F\subseteq F^{\prime}\subseteq M$, there is some $A$-large $r\in S_{at}(F^{\prime})$ with $r\geq q$; and 2. 2.

For some $\varphi(x,\bar{a})$, there are $A$-large extensions $r_{1}\supseteq q\cup\{\varphi(x,\bar{a})\}$ and $r_{2}\supseteq q\cup\{\neg\varphi(x,\bar{a})\}$.

Proof. Fix such a $q$ and let ${\cal S}=\{p^{*}\in S^{+}_{at}(A,M):p^{*}$ lies above $q\}$.

(1) is immediate, since ${\cal S}$ is uncountable, while $S_{at}(F^{\prime})$ is countable.

For (2), first note that if there is no such $\varphi(x,\bar{a})$, then there is at most one $p^{*}\in{\cal S}$ with the property that:

For any formula $\varphi(x,\bar{a})$ with parameters from $A$, $\varphi(x,\bar{a})\in p^{*}$ if and only if there is an $A$-large $r\in S_{at}(F\bar{a})$ extending $q\cup\{\varphi(x,\bar{a})\}$.

It follows that for any $q^{*}\in{\cal S}-\{p^{*}\}$, $q^{*}$ lies over some $r\geq q$ that is not $A$-large. That is, using the fact that there are only countably many $r\geq q$, ${\cal S}-\{p^{*}\}$ is contained in the union of countably many countable sets. But this contradicts $q$ being $A$-large.

Proposition 2.5

For $p\in{\cal P}$, ${\rm rk}_{A}(p)\geq\omega_{1}$ if and only if $p$ is $A$-large.

Proof. First, assume that ${\rm rk}_{A}(p)\geq\omega_{1}$. Fix an enumeration $\{c_{n}:n\in\omega\}$ of $M$. Using Clauses (1) and (2) of Lemma 2.3, we inductively construct a tree $\{p_{\nu}:\nu\in 2^{<\omega}\}$ of elements of ${\cal P}$ satisfying:

${\rm rk}_{A}(p_{\nu})\geq\omega_{1}$ for all $\nu\in 2^{<\omega}$; 2. 2.

If $\lg(\nu)=n$, then $\{c_{i}:i<n\}\subseteq{\rm dom}(p_{\nu})$; 3. 3.

$p_{\langle\rangle}=p$; 4. 4.

For $\nu\trianglelefteq\mu$, $p_{\nu}\leq p_{\mu}$; 5. 5.

For each $\nu$ there is a formula $\varphi(x,\bar{a})$ with $\bar{a}$ from $A$ such that $\varphi(x,\bar{a})\in p_{\nu 0}$ and $\neg\varphi(x,\bar{a})\in p_{\nu 1}$.

Given such a tree, for each $\eta\in 2^{\omega}$, let $\bar{p}_{\eta}:=\bigcup\{p_{\eta|n}:n\in\omega\}$ and let $p^{*}_{\eta}:=\bar{p}_{\eta}|A$. By Clauses (2) and (4), each $\bar{p}_{\eta}\in S_{at}(M)$, so each $p^{*}_{\eta}\in S^{+}_{at}(A,M)$. By Clause (5), $p^{*}_{\eta}\neq p^{*}_{\eta^{\prime}}$ for distinct $\eta,\eta^{\prime}\in 2^{\omega}$. Finally, each of these types lies over $p$ by Clause (3). Thus, $p$ is $A$-large.

Conversely, we argue by induction on $\alpha<\omega_{1}$ that:

$(*)_{\alpha}:$ X If $p\in{\cal P}$ is $A$-large, then ${\rm rk}_{A}(p)\geq\alpha$.

Establishing $(*)_{0}$ is trivial, and for limit $\alpha<\omega_{1}$, it is easy to establish $(*)_{\alpha}$ given that $(*)_{\beta}$ holds for all $\beta<\alpha$. So assume $(*)_{\alpha}$ holds and we will establish $(*)_{\alpha+1}$. Choose any $A$-large $p\in{\cal P}$. Towards showing ${\rm rk}_{A}(p)\geq\alpha+1$, choose any finite $F$, ${\rm dom}(p)\subseteq F\subseteq M$. As $S_{at}(F)$ is countable and uncountably many types in $S^{+}_{at}(A,M)$ lie above $p$, there is some $A$-large $q\in S_{at}(F)$ with $q\geq p$.

Next, by Lemma 2.4 choose a formula $\varphi(x,\bar{a})$ with $\bar{a}$ from $A$ such that there are $A$-large extensions $r_{1}\supseteq q\cup\{\varphi(x,\bar{a})\}$ and $r_{2}\supseteq q\cup\{\neg\varphi(x,\bar{a})\}$. Applying $(*)_{\alpha}$ to both $r_{1},r_{2}$ gives ${\rm rk}_{A}(r_{1}),{\rm rk}_{A}(r_{2})\geq\alpha$. Thus, $q$ $\alpha$-$A$ splits. Thus, by definition of the rank, ${\rm rk}_{A}(p)\geq\alpha+1$.

We obtain the following Corollary, which is analogous to the statement ‘If $T$ is small, then the isolated types are dense’ from the first-order context.

Corollary 2.6

If $S^{+}_{at}(A,M)$ is countable, then every $p\in{\cal P}$ has an extension $q\geq p$ that determines a type in $S^{+}_{at}(A,M)$.

Proof. If $S^{+}_{at}(A,M)$ is countable, then no $p\in{\cal P}$ is $A$-large. Thus, every $p\in{\cal P}$ has ${\rm rk}_{A}(p)<\omega_{1}$ by Proposition 2.5, so has an extension determining a type in $S^{+}_{at}(A,M)$ by Proposition 2.2.

We close with a complementary result about extensions of $A$-large types.

Definition 2.7

A type $r\in S_{at}(M)$ is $A$-perfect if $r\mathord{\restriction}_{A}$ is omitted in $M$ and for every finite $\overline{m}$ from $M$, the restriction $r\mathord{\restriction}_{\overline{m}}$ is $A$-large. **

The name perfect is chosen because, relative to the usual topology on $S_{at}(M)$, there are a perfect set of $A$-perfect types extending any $A$-large $p\in{\cal P}$. However, for what follows, all we need to establish is that there are uncountably many, which is notationally simpler to prove.

Proposition 2.8

Suppose $p\in{\cal P}$ is $A$-large. Then there are uncountably many $A$-perfect $r\in S_{at}(M)$ extending $p$.

Proof. Fix an $A$-large $p\in{\cal P}$. Choose a set $R\subseteq S_{at}(M)$ of representatives for $\{p^{*}\in S_{at}^{+}(A,M):p^{*}$ lies above $p\}$, i.e., for every such $p^{*}$, there is exactly one $\bar{p}\in R$ whose restriction $\bar{p}\mathord{\restriction}_{A}=p^{*}$. As $p$ is $A$-large, $R$ is uncountable. Now, for each finite $\overline{m}$ from $M$, there are only countably many complete $q\in S_{at}(\overline{m})$, and if some $q\in S_{at}(\overline{m})$ is $A$-small, then only countably many $\bar{p}\in R$ extend $q$. As $M$ is countable, there are only countably many $\overline{m}$, hence all but countably many $\bar{p}\in R$ satisfy $\bar{p}\mathord{\restriction}_{\overline{m}}$ $A$-large for every $\overline{m}$. Further, again since $M$ is countable, at most countably many $\bar{p}\in R$ have restrictions to $A$ that are realized in $M$. Thus, all but countably many $\bar{p}\in R$ are $A$-perfect.

2.2 A transfer result

In this brief subsection we state a transfer result that follows immediately by Keisler’s completeness theorem for the logic $L_{\omega_{1},\omega}(Q)$, given in [6]. Recall that $L_{\omega_{1},\omega}(Q)$ is the logic obtained by taking the (usual) set of atomic $L$ formulas and closing under boolean combinations, existential quantification, the ‘$Q$-quantifier,’ i.e., if $\theta(y,\overline{x})$ is a formula, then so is $Qy\theta(y,\overline{x})$; and countable conjunctions of formulas involving a finite set of free variables, i.e., if $\{\psi_{i}(\overline{x}):i\in\omega\}$ is a set of formulas, then so is $\bigwedge_{i\in\omega}\psi_{i}(\overline{x})$. We are only interested in standard interpretations of these formulas, i.e., $M\models\bigwedge_{i\in\omega}\psi_{i}(\bar{a})$ if and only if $M\models\psi_{i}(\bar{a})$ for every $i\in\omega$; and $M\models Qy\theta(y,\bar{a})$ if and only if the solution set $\theta(M,\bar{a})$ is uncountable.

Throughout the discussion let $ZFC^{*}$ denote a sufficiently large, finite subset of the ZFC axioms. In the notation of [8], Proposition 2.9 states that sentences of $L_{\omega_{1},\omega}(Q)$ are grounded.

Proposition 2.9

Suppose $L$ is a countable language, and $\Phi\in L_{\omega_{1},\omega}(Q)$ are given. There is a sufficiently large, finite subset $ZFC^{*}$ of $ZFC$ such that IF there is a countable, transitive model $({\cal B},\epsilon)\models ZFC^{*}$ with $L,\Phi\in{\cal B}$ and

[TABLE]

THEN (in $V$!) there is $N\models\Phi$ and $|N|=\aleph_{1}$.

Proof. This follows immediately from Keiser’s completeness theorem for $L_{\omega_{1},\omega}$, given that provability is absolute between transitive models of set theory. More modern, ‘constructive’ proofs can be found in [1] and [2]. These use the existence ${\cal B}$-normal ultrafilters. Given an arbitrary language $L^{*}\in{\cal B}$ and any countable $L^{*}$-structure $({\cal B},E,\dots)$ where the reduct $({\cal B},E)$ is an $\omega$-model of $ZFC^{*}$, for any ${\cal B}$-normal ultrafilter ${\cal U}$, the ultrapower $Ult({\cal B},{\cal U})$ is a countable, $\omega$-model that is an $L^{*}$-elementary extension of $({\cal B},E,\dots)$. It has the additional property that for any $L^{*}$-definable subset $D$, $D^{Ult({\cal B},{\cal U})}$ properly extends $D^{{\cal B}}$ if and only if $({\cal B},E,\dots)\models`D$ is uncountable’.

Using this, one constructs (in $V$!) a continuous, $L^{*}$-elementary $\omega_{1}$-sequence $\langle{\cal B}_{\alpha}:\alpha<\omega_{1}\rangle$ of $\omega$-models, where each ${\cal B}_{\alpha+1}=Ult({\cal B}_{\alpha},{\cal U}_{\alpha})$. Then the interpretation $M^{{\cal C}}$ where ${\cal C}=\bigcup_{\alpha\in\omega_{1}}{\cal B}_{\alpha}$ will be a suitable choice of $N$. More details of this construction are given in [1] or [2].

2.3 A configuration arising from non-${\rm pcl}$-smallness

The goal of this subsection is to prove the following Proposition, the data from which will be used throughout Section 3.

Proposition 2.10

Assume $T$ is a countable, complete theory for which ${\bf At}_{T}$ has an uncountable atomic model, but is not ${\rm pcl}$-small. Then there are a countable, atomic $M^{*}\in{\bf At}_{T}$, finite sequences $\bar{a}^{*}\subseteq\bar{b}^{*}\subseteq M^{*}$, and complete 1-types $\{r_{j}(x,\bar{b}^{*}):j\in\omega\}$ such that, letting $D^{*}={\rm pcl}_{M^{*}}(\bar{a}^{*})$, $A_{n}=\bigcup\{r_{j}(M^{*},\bar{b}^{*}):j<n\}$ and $A^{*}=\bigcup\{A_{n}:n\in\omega\}$ we have:

$A^{*}\subseteq D^{*}$; 2. 2.

$S^{+}_{at}(A_{n},M^{*})$* is countable for every $n\in\omega$; but* 3. 3.

$S^{+}_{at}(A^{*},M^{*})$* is uncountable.*

Proof. Fix any countable, atomic $M^{*}\in{\bf At}_{T}$. Using Proposition 1.2 and the non-${\rm pcl}$-smallness of ${\bf At}_{T}$, choose a finite tuple $\bar{a}^{*}\subseteq M^{*}$ such that $S^{+}_{at}(D^{*},M^{*})$ is uncountable, where $D^{*}={\rm pcl}_{M^{*}}(\bar{a}^{*})\subseteq M^{*}$.

Fix any finite tuple $\bar{b}\supseteq\bar{a}^{*}$ from $M^{*}$ and look at the complete 1-types ${\cal Q}_{\bar{b}}:=\{r\in S_{at}(\bar{b})$ such that $r(M^{*})\subseteq D^{*}\}$. These types visibly induce a partition $D^{*}$, and it is easily seen that if $\bar{b}^{\prime}\supseteq\bar{b}$, the partition induced by $\bar{b}^{\prime}$ refines the partition induced by $\bar{b}$. Let ${\cal Q}:=\bigcup\{{\cal Q}_{\bar{b}}:\bar{a}^{*}\subseteq\bar{b}\subseteq M^{*}\}$.

Define a rank function ${\rm rk}:{\cal Q}\rightarrow ON\cup\{\infty\}$ as follows:

• •

${\rm rk}(c/\bar{b})\geq 0$ if and only if ${\rm tp}(c/\bar{b})\in{\cal Q}$;

• •

${\rm rk}(c/\bar{b})\geq 1$ if and only if ${\rm tp}(c/\bar{b})\in{\cal Q}$ and there are infinitely many $c^{\prime}\in D^{*}$ realizing ${\rm tp}(c/D^{*})$; and

• •

for an ordinal $\alpha\geq 2$, ${\rm rk}(c/\bar{b})\geq\alpha$ if and only if for every $\beta<\alpha$ and every $\bar{b}^{\prime}$ from $M^{*}$, there is $c^{\prime}\in D^{*}$ realizing ${\rm tp}(c/\bar{b})$ such that ${\rm rk}(c^{\prime}/\bar{b}\bar{b}^{\prime})\geq\beta$.

• •

${\rm rk}(c/\bar{b})=\alpha$ if and only if ${\rm rk}(c/\bar{b})\geq\alpha$ but ${\rm rk}(c/\bar{b})\not\geq\alpha+1$.

Claim 1. For every $r\in{\cal Q}$, ${\rm rk}(r)$ is a countable ordinal.

Proof. Assume by way of contradiction that ${\rm rk}(c/\bar{b})\geq\omega_{1}$ for some type $c/\bar{b}$. Then, for any $\bar{b}^{\prime}$ from $M$, as $D^{*}$ is countable, there is an element $c^{\prime}\in D^{*}$ such that ${\rm rk}(c^{\prime}/\bar{b}\bar{b}^{\prime})\geq\beta$ for uncountably many $\beta$’s, hence ${\rm rk}(c^{\prime}/\bar{b}\bar{b}^{\prime})\geq\omega_{1}$ as well. Using this idea, if we let $\langle\bar{b}_{n}:n\in\omega\rangle$ be an increasing sequence of finite sequences from $M^{*}$ whose union is all of $M^{*}$, then we can find a sequence $\langle c_{n}:n\in\omega\rangle$ of elements from $D^{*}$ such that, for each $n$, ${\rm rk}(c_{n}/\bar{b}_{n})\geq\omega_{1}$ and ${\rm tp}(c_{n}/\bar{b}_{n})\subseteq{\rm tp}(c_{n+1}/\bar{b}_{n+1})$. The union of these 1-types yields a complete, atomic 1-type $q\in S_{at}(M^{*})$ all of whose realizations are in ${\rm pcl}_{M^{*}}(\bar{a})$. However, since the type asserting that ‘$x=c$’ has rank 0 for each $c\in D^{*}$, $q$ is omitted in $M^{*}$. To obtain a contradiction, choose a realization $e$ of $q$ and, as $M^{*}e$ is a countable, atomic set, construct a countable, elementary extension $M^{\prime}\succeq M^{*}$ with $e\in M^{\prime}$. But now, $q$ implies that $e\in{\rm pcl}_{M^{\prime}}(\bar{a})$, yet this is contradicted by the fact that $M^{*}$ contains $\bar{a}$ but not $e$.

As notation, for a subset $\SS\subseteq{\cal Q}_{\bar{b}}$, let $A_{\SS}=\bigcup\{r(M^{*}):r\in\SS\}$, which is always a subset of $D^{*}$. Define the set of ‘candidates’ as

[TABLE]

Note that ${\cal C}$ is non-empty as $(\SS_{0},\bar{a}^{*})\in{\cal C}$, where $\SS_{0}$ is an enumeration of all the complete, pseudo-algebraic types over $\bar{a}^{*}$. Among all candidates, choose $(\SS^{*},\bar{b}^{*})\in{\cal C}$ such that

[TABLE]

is as small as possible. Enumerate $\SS^{*}=\{r_{j}:j\in\omega\}$ and put $A^{*}:=A_{\SS^{*}}$ and $A_{n}:=\bigcup\{r_{j}(M^{*},\bar{b}^{*}):j<n\}$ for each $n\in\omega$. As Clauses (1) and (3) are immediate, it suffices to prove the following Claim:

Claim 2. For each $n\in\omega$, $S^{+}_{at}(A_{n},M^{*})$ is countable.

Proof. Fix any $n\in\omega$. First, note that if ${\rm rk}(r_{j})=0$ for every $j<n$, then $A_{n}$ would be finite, which would imply $S_{at}(A_{n})$ is countable. As $S_{at}(A_{n})$ contains $S^{+}_{at}(A_{n},M^{*})$, the result follows.

Now assume $rk(r_{j})>0$ for at least one $j<n$. Let $\beta:=\max\{rk(r_{j}):j<n\}$ and let $F=\{j<n:rk(r_{j})=\beta\}$. Clearly, $\beta<\alpha^{*}$. For each $j\in F$, as $\beta>0$ but $rk(r_{j})\not\geq\beta+1$, there is a finite tuple $\bar{b}_{j}$ such that ${\rm rk}(c/\bar{b}^{*}\bar{b}_{j})<\beta$ for all $c\in r_{j}(M^{*})$.

Let $\bar{b}^{\prime}$ be the concatenation of $\bar{b}^{*}$ with each $\bar{b}_{j}$ for $j\in F$ and let

[TABLE]

Subclaim. ${\rm rk}(r^{\prime})<\beta$ for every $r^{\prime}\in\SS^{\prime}$.

Proof. Fix $r^{\prime}\in\SS^{\prime}$ and choose $c\in r^{\prime}(M^{*},\bar{b}^{\prime})$. There are two cases. On one hand, if $r^{\prime}$ extends some $r_{j}$ with $j\in F$, then ${\rm rk}(c/\bar{b}^{\prime})\leq{\rm rk}(c/\bar{b}^{*}\bar{b}_{j})<\beta$. On the other hand, if $r^{\prime}$ extends some $r_{j}$ with $r_{j}\not\in F$, then as ${\rm rk}(r_{j})<\beta$, ${\rm rk}(c/\bar{b}^{\prime})\leq{\rm rk}(c/\bar{b}^{*})<\beta$.

Clearly $A_{\SS^{\prime}}=A_{n}$, so $S^{+}_{at}(A_{n},M^{*})=S^{+}_{at}(A_{\SS^{\prime}},M^{*})$. Thus, if $S^{+}_{at}(A_{n},M^{*})$ were uncountable, then $(\SS^{\prime},\bar{b}^{\prime})$ would be a candidate, i.e., an element of ${\cal C}$. But, as $\beta<\alpha^{*}$, this is impossible by the Subclaim and the minimality of $\alpha^{*}$.

3 A family of $2^{\aleph_{0}}$ atomic models of size $\aleph_{1}$

Throughout the whole of this section, we assume that $T$ is a complete theory in a countable language for which ${\bf At}_{T}$ has an uncountable atomic model, but is not ${\rm pcl}$-small. Appealing to Proposition 2.10,

Fix, for the whole of this section, a countable atomic model ${\bf M^{*}}$, tuples ${\bf\bar{a}^{*}\subseteq\bar{b}^{*}\subseteq M^{*}}$ and sets $A^{*}$ and $A_{n}$ for each $n\in\omega$ as in Proposition 2.10.

We work with this fixed configuration for the whole of this section and, in Subsection 3.3 eventually prove:

Proposition 3.1

There is a family $\{(N_{\eta},\bar{b}^{*}):\eta\in 2^{\omega}\}$ of atomic models of $T$, each of size $\aleph_{1}$, that are pairwise non-isomorphic over $\bar{b}^{*}$.

3.1 Colorings of models realizing many types over $A^{*}$

Definition 3.2

Call a structure $(N,\bar{b}^{*})$ rich if $N\in{\bf At}_{T}$ has size $\aleph_{1}$, $M^{*}\preceq N$, and $N$ realizes uncountably many 1-types over $A^{*}$. **

Lemma 3.3

For each $n\in\omega$, a rich $(N,\bar{b}^{*})$ realizes only countably many distinct 1-types over $A_{n}$.

Proof. Fix any $(N,\bar{b}^{*})$ and $n<\omega$ as above. If $\{c_{i}:i\in\omega_{1}\}$ realize distinct types over $A_{n}$, then the types $\{{\rm tp}_{N}(c_{i}/M^{*}):i\in\omega_{1}\}$ would be distinct, contradicting $S^{+}_{at}(A_{n},M^{*})$ countable.

How can we tell whether rich structures are non-isomorphic? We introduce the notion of ${\cal U}$-colorings and Corollary 3.6 gives a sufficient condition.

Definition 3.4

Fix a subset ${\cal U}\subseteq\omega$ and a rich $(N,\bar{b}^{*})$.

• •

For elements $d,d^{\prime}\in N$, define the splitting number ${\rm spl}(d,d^{\prime})\in(\omega+1)$ to be the least $k<\omega$ such that ${\rm tp}(d/A_{k})\neq{\rm tp}(d^{\prime}/A_{k})$ if such exists; and ${\rm spl}(d,d^{\prime})=\omega$ if ${\rm tp}(d/A^{*})={\rm tp}(d^{\prime}/A^{*})$.

• •

A ${\cal U}$-coloring of a rich $(N,\bar{b}^{*})$ is a function

[TABLE]

such that for all pairs $d,d^{\prime}\in N$, at least one of the following hold:

${\rm tp}(d/A^{*})={\rm tp}(d^{\prime}/A^{*})$; or 2. 2.

$c(d)\neq c(d^{\prime})$; or 3. 3.

${\rm spl}(d,d^{\prime})\in{\cal U}$.

• •

The color filter ${\cal F}(N,\bar{b}^{*}):=\{{\cal U}\subseteq\omega:$ a ${\cal U}$-coloring of $(N,\bar{b}^{*})$ exists$\}$.

Lemma 3.5

Fix a rich $(N,\bar{b}^{*})$. Then:

${\cal F}(N,\bar{b}^{*})$* is a filter;* 2. 2.

${\cal F}(N,\bar{b}^{*})$* contains the cofinite subsets of $\omega$; but* 3. 3.

No finite ${\cal U}\subseteq\omega$ is in ${\cal F}(N,\bar{b}^{*})$.

Proof. (1) First, note that if ${\cal U}\subseteq{\cal U}^{\prime}\subseteq\omega$, then every ${\cal U}$-coloring $c$ is also a ${\cal U}^{\prime}$-coloring. Thus, ${\cal F}(N,\bar{b}^{*})$ is upward closed. Next, suppose ${\cal U}_{1}\in{\cal F}(N,\bar{b}^{*})$ via the coloring $c_{1}:N\rightarrow\omega$ and ${\cal U}_{2}\in{\cal F}(N,\bar{a}^{*}\bar{b}^{*})$ via the coloring $c_{2}:N\rightarrow\omega$. Fix any bijection $t:\omega\times\omega\rightarrow\omega$. It is easily checked that $c^{*}:N\rightarrow\omega$ defined by $c^{*}(d)=t(c_{1}(d),c_{2}(d))$ is a ${\cal U}_{1}\cap{\cal U}_{2}$-coloring of $(N,\bar{b}^{*})$. Thus, ${\cal U}_{1}\cap{\cal U}_{2}\in{\cal F}(N,\bar{b}^{*})$. So ${\cal F}(N,\bar{b}^{*})$ is a filter.

(2) As ${\cal F}(N,\bar{b}^{*})$ is a filter, it suffices to show $(\omega-n)\in{\cal F}(N,\bar{b}^{*})$ for each $n\in\omega$. So fix such an $n$. By Lemma 3.3, $N$ realizes at most countably many types over $A_{n}$. Thus, we can produce a map $c:N\rightarrow\omega$ such that $c(d)=c(d^{\prime})$ if and only if ${\rm tp}(d/A_{n})={\rm tp}(d^{\prime}/A_{n})$. As any such $c$ is an $(\omega-n)$-coloring, $(\omega-n)\in{\cal F}(N,\bar{b}^{*})$.

(3) It suffices to show that no $n=\{0,\dots,n-1\}$ is in ${\cal F}(N,\bar{b}^{*})$. To see this, let $c:N\rightarrow\omega$ be an arbitrary map. We will show that $c$ is not an $\{0,\dots,n-1\}$-coloring. As $N$ realizes $\aleph_{1}$ distinct types over $A^{*}$, there is some $m^{*}\in\omega$ and an uncountable subset $\{d_{\alpha}:\alpha<\omega_{1}\}\subseteq N$ that realize distinct types over $A^{*}$, yet $c(d_{\alpha})=m^{*}$ for each $\alpha$. However, as $N$ realizes only countably many types over $A_{n}$, there are $\alpha\neq\beta$ such that $n\leq{\rm spl}(d_{\alpha},d_{\beta})<\omega$. Thus, $c$ is not an $\{0,\dots,n-1\}$-coloring.

We close with a sufficient condition for non-isomorphism of rich models.

Corollary 3.6

Suppose that for $\ell=1,2$, $(N_{\ell},\bar{b}^{*})$ is a ${\cal U}_{\ell}$-colored rich model, and ${\cal U}_{1}\cap{\cal U}_{2}$ is finite. Then there is no isomorphism $f:N_{1}\rightarrow N_{2}$ fixing $\bar{b}^{*}$ pointwise.

Proof. If there were such an isomorphism, then $(N_{2},\bar{b}^{*})$ would be both ${\cal U}_{1}$-colored and ${\cal U}_{2}$-colored. Thus, both ${\cal U}_{1},{\cal U}_{2}\in{\cal F}(N_{2},\bar{b}^{*})$, which contradicts Lemma 3.5.

3.2 Constructing a colored rich model via forcing

Arguing as in the proof of Proposition 1.2, from the data of Lemma 2.10 we can construct a rich $(N,\bar{b}^{*})$ as the union of a continuous, elementary chain $\langle M_{\alpha}:\alpha\in\omega_{1}\rangle$ of countable, atomic models with $M_{0}=M^{*}$ such that, for each $\alpha\in\omega_{1}$ there is a distinguished $b_{\alpha}\in M_{\alpha+1}$ such that ${\rm tp}(b_{\alpha}/A^{*})$ is omitted in $M_{\alpha}$.

Our goal is to construct a sufficiently generic rich $(N,\bar{b}^{*})$, along with a coloring $c:N\rightarrow(\omega+1)$ via forcing. Our forcing $({\mathbb{Q}},\leq_{{\mathbb{Q}}})$ encodes finite approximations of such an $(N,\bar{b}^{*})$ and $c$. A fundamental building block is the notion of a striated type over a finite subset $\bar{a}$ satisfying $\bar{b}^{*}\subseteq\bar{a}\subseteq M^{*}$. As an atomic type over a finite subset is generated by a complete formula, we use the terms interchangeably.

Definition 3.7

Choose a finite tuple $\bar{a}$ with $\bar{b}^{*}\subseteq\bar{a}\subseteq M^{*}$. A striated type over $\bar{a}$ is a complete formula $\theta(\overline{x})\in S_{at}(\bar{a})$ whose variables are partitioned as $\overline{x}=\langle\overline{x}_{j}:j<\ell\rangle$ where, for each $j$, $\overline{x}_{j}=\langle x_{j,n}:n<n(j)\rangle$ is an $n(j)$-tuple of variable symbols that satisfy ${\rm tp}(x_{j,0}/\bar{a}\cup\{\overline{x}_{i}:i<j\})$ is $A^{*}$-large. The integer $\ell$ is the length of the striated type.

A simple realization of a striated type $\theta(\overline{x})$ of length $\ell$ is a sequence $\bar{b}=\langle\bar{b}_{j}:j<\ell\rangle$ of tuples from $M^{*}$ such that $M^{*}\models\theta(\bar{b})$. A perfect chain realization of $\theta(\overline{x})$ is a pair $(\overline{M},\bar{b})$, consisting of a chain $M_{0}\preceq M_{1}\preceq M_{\ell-1}\preceq M^{*}$ of $\ell$ elementary submodels of $M^{*}$ and a simple realization $\bar{b}=\langle\bar{b}_{j}:j<\ell\rangle$ from $M^{*}$ that satisfy: For each $j<\ell$,

$\bar{a}\cup\{\bar{b}_{i}:i<j\}\subseteq M_{j}$; and 2. 2.

${\rm tp}(b_{j,0}/M_{j})$ is $A^{*}$-perfect (see Definition 2.7).

Lemma 3.8

Every striated type $\theta(\overline{x})\in S_{at}(\bar{a})$ has a perfect chain realization.

Proof. We argue by induction on $\ell$, the length of the striation. For striations of length zero there is nothing to prove, so assume the Lemma holds for striated types of length $\ell$ and choose an $(\ell+1)$-striation $\theta(\overline{x})\in S_{at}(\bar{a})$. Let $\theta\mathord{\restriction}_{\ell}$ be the truncation of $\theta$ to the variables $\overline{x}\mathord{\restriction}_{\ell}=\langle\overline{x}_{j}:j<\ell\rangle$. As $\theta\mathord{\restriction}_{\ell}$ is clearly an $\ell$-striation, it has a perfect chain realization, i.e., a chain $M_{0}\preceq M_{1}\preceq M_{\ell-1}\preceq M^{*}$ and a tuple $\bar{b}=\langle\bar{b}_{j}:j<\ell\rangle$ from $M^{*}$ realizing $\theta\mathord{\restriction}_{\ell}$ such that $\bar{a}\cup\{\bar{b}_{i}:i<j\}\subseteq M_{j}$ and ${\rm tp}(b_{j,0}/M_{j})$ is $A^{*}$-perfect for each $j<\ell$.

Now, since ${\rm tp}(x_{\ell,0}/\bar{a}\bar{b})$ is $A^{*}$-large, by applying Proposition 2.8 there is an $A^{*}$-perfect type $\bar{p}\in S_{at}(M^{*})$ (in a single variable $x_{\ell,0}$) extending ${\rm tp}(x_{\ell,0}/\bar{a}\bar{b})$. Choose a countable, atomic $N\succeq M^{*}$ and $e\in N$ realizing $\bar{p}$. As $N$ and $M^{*}$ are both countable and atomic, choose an isomorphism $f:N\rightarrow M^{*}$ that fixes $\bar{a}\bar{b}$ pointwise. Then $f(M_{0})\preceq f(M_{1})\preceq\dots f(M_{\ell-1})\preceq f(M^{*})\preceq M^{*}$ is a chain. Let $b_{\ell,0}:=f(e)$ and choose $\langle b_{\ell,1}\dots,b_{\ell,n(\ell)-1}\rangle$ arbitrarily from $M^{*}$ so that, letting $\bar{b}_{\ell}=\langle\bar{b}_{\ell,n}:n<n(\ell)\rangle$, $\bar{b}\frown\bar{b}_{\ell}$ realizes $\theta(\overline{x})$. This chain and this sequence form a perfect chain realization of $\theta$.

The following Lemma is immediate, and indicates the advantage of working with $A^{*}$-perfect types.

Lemma 3.9

Let $(\overline{M},\bar{b})$ be any perfect chain realization of a striated type $\theta(\overline{x})\in S_{at}(\bar{a})$. Then for every $\bar{c}\subseteq M_{0}$, ${\rm tp}(\bar{b}/\bar{a}\bar{c})\in S_{at}(\bar{a}\bar{c})$ is a striated type extending $\theta(\overline{x})$, and $(\overline{M},\bar{b})$ is a perfect chain realization of it.

The Lemma below, whose proof simply amounts to unpacking definitions, demonstrate that striated types are rather malleable.

Lemma 3.10

If ${\rm tp}(\bar{c}/\bar{a})$ is a striated type of length $k$ and ${\rm tp}(\bar{d}/\bar{a}\bar{c})$ is a striated type of length $\ell$, then ${\rm tp}(\bar{c}\bar{d}/\bar{a})$ is a striated type of length $k+\ell$. 2. 2.

Suppose ${\rm tp}(\bar{b}/\bar{a})$ is a striated type of length $\ell$ and $k<\ell$. Let $\bar{b}_{<k}$ and $\bar{b}_{\geq k}$ be the induced partition of $\bar{b}$. Then ${\rm tp}(\bar{b}_{<k}/\bar{a})$ is a striated type of length $\ell$ and ${\rm tp}(\bar{b}_{\geq k}/\bar{a}\bar{b}_{<k})$ is a striated type of length $(\ell-k)$. Moreover, if $(\overline{M},\bar{b})$ is a perfect chain realization of ${\rm tp}(\bar{b}/\bar{a})$, then $(\overline{M}_{<k},\bar{b}_{<k})$ is a perfect chain realization of ${\rm tp}(\bar{b}_{<k}/\bar{a})$ and $(\overline{M}_{\geq k},\bar{b}_{\geq k})$ is a perfect chain realization of ${\rm tp}(\bar{b}_{\geq k}/\bar{a}\bar{b}_{<k})$.

We begin by defining a partial order $({\mathbb{Q}}_{0},\leq_{{\mathbb{Q}}_{0}})$ of ‘preconditions’. Then our forcing $({\mathbb{Q}},\leq_{{\mathbb{Q}}})$ will be a dense suborder of these preconditions.

Definition 3.11

${\mathbb{Q}}_{0}$ is the set of all ${\bf p}=({\bf\overline{a}}_{{\bf p}},u_{{\bf p}},{\overline{n}}_{{\bf p}},\theta_{{\bf p}}(\overline{x}_{{\bf p}}),k_{{\bf p}},{\cal U}_{{\bf p}},c_{{\bf p}})$, where

${\bf\overline{a}}_{{\bf p}}$ is a finite subset of $M^{*}$ containing $\bar{b}^{*}$; 2. 2.

$u_{{\bf p}}$ is a finite subset of $\omega_{1}$; 3. 3.

${\overline{n}}_{{\bf p}}=\langle n_{t}:t\in u_{{\bf p}}\rangle$ is a sequence of positive integers; 4. 4.

$\overline{x}_{{\bf p}}=\langle\overline{x}_{t,{\bf p}}:t\in u_{{\bf p}}\rangle$, where each $\overline{x}_{t,{\bf p}}=\langle x_{t,n}:n<{\overline{n}}_{t}\rangle$ is a finite sequence from the set $X=\{x_{t,n}:t\in\omega_{1},n\in\omega\}$ of variable symbols; 5. 5.

$\theta_{{\bf p}}(\overline{x}_{{\bf p}})\in S_{at}({\bf\overline{a}}_{{\bf p}})$ is a striated type of length $|u_{{\bf p}}|$ (see Definition 3.7); 6. 6.

$k_{{\bf p}}\in\omega$; 7. 7.

${\cal U}_{{\bf p}}\subseteq k_{{\bf p}}=\{0,\dots,k_{{\bf p}}-1\}$; 8. 8.

$c_{{\bf p}}:\overline{x}_{{\bf p}}\rightarrow\omega$ is a function such that for all pairs $x_{t,n},x_{s,m}$ from $\overline{x}_{{\bf p}}$ with $c_{{\bf p}}(x_{t,n})=c_{{\bf p}}(x_{s,m})$

1. (a)

either ${\rm spl}(b_{t,n},b_{s,m})\geq k_{{\bf p}}$ for all perfect chain realizations $(\overline{M},\bar{b})$ of $\theta_{{\bf p}}(\overline{x}_{{\bf p}})$; 2. (b)

or there is some $k\in{\cal U}_{{\bf p}}$ such that ${\rm spl}(b_{t,n},b_{s,m})=k$ for all perfect chain realizations $(\overline{M},\bar{b})$ of $\theta_{{\bf p}}(\overline{x}_{{\bf p}})$.

We order elements of ${\mathbb{Q}}_{0}$ by: ${\bf p}\leq_{{\mathbb{Q}}_{0}}{\bf q}$ if and only if

• •

${\bf\overline{a}}_{{\bf p}}\subseteq{\bf\overline{a}}_{{\bf q}}$;

• •

$u_{{\bf p}}\subseteq u_{{\bf q}}$ and $n_{t,{\bf p}}\leq n_{t,{\bf q}}$ for all $t\in u_{{\bf p}}$, hence $\overline{x}_{{\bf p}}$ is a subsequence of $\overline{x}_{{\bf q}}$;

• •

$\theta_{{\bf q}}(\overline{x}_{{\bf q}})\vdash\theta_{{\bf p}}(\overline{x}_{{\bf p}})$;

• •

$k_{{\bf p}}\leq k_{{\bf q}}$;

• •

${\cal U}_{{\bf p}}={\cal U}_{{\bf q}}\cap k_{{\bf p}}$ (hence, for $j<k_{{\bf p}}$, $j\in{\cal U}_{{\bf p}}$ if and only if $j\in{\cal U}_{{\bf q}}$);

• •

$c_{{\bf p}}=c_{{\bf q}}\mathord{\restriction}_{\overline{x}_{{\bf p}}}$.

Visibly, $({\mathbb{Q}}_{0},\leq_{{\mathbb{Q}}_{0}})$ is a partial order. Call a precondition ${\bf p}\in{\mathbb{Q}}_{0}$ unarily decided if, for every $x_{t,n}\in\overline{x}_{{\bf p}}$, $p(\overline{x}_{{\bf p}})$ determines a type in $S^{+}_{at}(A_{k_{{\bf p}}},M^{*})$ (see Definition 2.1). That the unarily decided preconditions are dense follows easily from the fact that $S^{+}_{at}(A_{k_{{\bf p}}},M^{*})$ is countable.

Lemma 3.12

The set $\{{\bf p}\in{\mathbb{Q}}_{0}:{\bf p}$ is unarily decided$\}$ is dense in $({\mathbb{Q}}_{0},\leq_{{\mathbb{Q}}_{0}})$. Moreover, given any ${\bf p}\in{\mathbb{Q}}_{0}$, there is a unarily decided ${\bf q}\geq_{{\mathbb{Q}}_{0}}{\bf p}$ with $\overline{x}_{{\bf q}}=\overline{x}_{{\bf p}}$ and $k_{{\bf q}}=k_{{\bf p}}$ (hence ${\cal U}_{{\bf q}}={\cal U}_{{\bf p}}$).

Proof. Fix ${\bf p}\in{\mathbb{Q}}_{0}$ and let $k:=k_{{\bf p}}$. Arguing by induction on the size of the finite set $\overline{x}_{{\bf p}}$, it is enough to strengthen $p(x_{t,n})$ individually for each $x_{t,n}\in\overline{x}_{{\bf p}}$. So fix $x_{t,n}\in\overline{x}_{{\bf p}}$. By Corollary 2.6 there is an $\bar{a}^{\prime}\supseteq\bar{a}_{{\bf p}}$ and a 1-type $q_{1}(x_{t,n})\in S_{at}(\bar{a}^{\prime})$ extending ${\rm tp}(x_{t,n}/\bar{a}_{{\bf p}})$ that determines a type in $S_{at}^{+}(A_{k_{{\bf p}}},M^{*})$. Then, using Lemma 3.10(1) we can choose a striated type $p^{\prime}(\overline{x}_{{\bf p}})\in S_{at}(\bar{a}^{\prime})$ extending $p(\overline{x}_{{\bf p}})\cup q_{1}$.

We iterate the above procedure for each of the (finitely many) elements of $\overline{x}_{{\bf p}}$. We then get a unarily decided precondition ${\bf p}^{\prime}\geq_{{\mathbb{Q}}_{0}}{\bf p}$ whose type $p^{\prime}(\overline{x}_{{\bf p}})$ still has the same free variables, and each of $k_{{\bf p}}$, ${\cal U}_{{\bf p}}$, $c_{{\bf p}}$ are unchanged.

Next, call a precondition ${\bf p}\in{\mathbb{Q}}_{0}$ fully decided if, it is unarily decided and, for each pair $x_{t,n},x_{s,m}$ from $\overline{x}_{{\bf p}}$ with $c_{{\bf p}}(x_{t,n})=c_{{\bf p}}(x_{s,m})$, if ${\rm spl}(b_{t,n},b_{s,m})\geq k_{{\bf p}}$ for some perfect chain realization $(\overline{M},\bar{b})$, then ${\rm tp}(b_{t,n}/A^{*})={\rm tp}(b_{s,m}/A^{*})$ for all perfect chain realizations $(\overline{M},\bar{b})$ of $\theta_{{\bf p}}(\overline{x}_{{\bf p}})$.

Lemma 3.13

The set $\{{\bf p}\in{\mathbb{Q}}_{0}:{\bf p}$ is fully decided$\}$ is dense in $({\mathbb{Q}}_{0},\leq_{{\mathbb{Q}}_{0}})$. Moreover, given any ${\bf p}\in{\mathbb{Q}}_{0}$, there is a fully decided ${\bf q}\geq_{{\mathbb{Q}}_{0}}{\bf p}$ with $\overline{x}_{{\bf q}}=\overline{x}_{{\bf p}}$.

Proof. It suffices to handle each pair $x_{t,n},x_{s,m}$ from $\overline{x}_{{\bf p}}$ with $c(x_{t,n})=c(x_{s,m})$ separately. Given such a pair, suppose there is some perfect chain realization $(\overline{M},\bar{b})$ of $\theta(\overline{x}_{{\bf p}})\in S_{at}({\bf\overline{a}}_{{\bf p}})$ with $k_{{\bf p}}\leq{\rm spl}(b_{t,n},b_{s,m})<\omega$. Among all such perfect chain realizations, choose one that minimizes $k^{*}={\rm spl}(b_{t,n},b_{s,m})$. Choose a formula $\varphi(x,\bar{c})$ with $\bar{c}$ from $A_{k^{*}+1}$ witnessing that ${\rm tp}(b_{t,n}/A_{k^{*}+1})\neq{\rm tp}(b_{s,m}/A_{k^{*}+1})$. As $A_{k^{*}+1}\subseteq M_{0}$, by applying Lemma 3.9, let $\theta^{*}(\overline{x}_{{\bf p}})$ be a complete formula over ${\bf\overline{a}}_{{\bf p}}\bar{c}$ isolating ${\rm tp}(\bar{b}/{\bf\overline{a}}_{{\bf p}}\bar{c})$. Form the precondition ${\bf p}^{\prime}\in{\mathbb{Q}}_{0}$ by putting ${\bf\overline{a}}_{{\bf p}^{\prime}}={\bf\overline{a}}_{{\bf p}}\bar{c}$; $\theta_{{\bf p}^{\prime}}=\theta^{*}$; $k_{{\bf p}^{\prime}}=k^{*}+1$; and ${\cal U}_{{\bf p}^{\prime}}={\cal U}_{{\bf p}}\cup\{k^{*}\}$; while leaving $\overline{x}_{{\bf p}}$ and $c_{{\bf p}}$ unchanged. It is evident that ${\rm spl}(b^{\prime}_{t,n},b^{\prime}_{s,m})=k^{*}\in{\cal U}_{{\bf p}^{\prime}}$ for all perfect chain realizations $(\overline{M},\bar{b}^{\prime})$ of $\theta_{{\bf p}^{\prime}}$. Continuing this process for each of the (finitely many) relevant pairs gives us a fully decided extension of ${\bf p}$.

Definition 3.14

The forcing $({\mathbb{Q}},\leq_{\mathbb{Q}})$ is the set of fully decided ${\bf p}\in{\mathbb{Q}}_{0}$ with the inherited order.

Lemma 3.15

The forcing $({\mathbb{Q}},\leq_{{\mathbb{Q}}})$ has the countable chain condition (c.c.c.).

Proof. Suppose $\{{\bf p}_{i}:i\in\omega_{1}\}$ is an uncountable subset of ${\mathbb{Q}}$. In light of Lemma 3.13, it suffices to find $i\neq j$ for which there is some precondition ${\bf q}\in{\mathbb{Q}}_{0}$ satisfying ${\bf p}_{i}\leq_{{\mathbb{Q}}_{0}}{\bf q}$ and ${\bf p}_{j}\leq_{{\mathbb{Q}}_{0}}{\bf q}$. First, by the $\Delta$-system lemma applied to the finite sets $\{u_{{\bf p}_{i}}\}$, we may assume that $|u_{{\bf p}_{i}}|$ is constant and there is some fixed $u^{*}$ that is an initial segment of each $u_{{\bf p}_{i}}$ and, moreover, whenever $i<j$, every element of $(u_{{\bf p}_{i}}\setminus u^{*})$ is less than every element of $(u_{{\bf p}_{j}}\setminus u^{*})$. By further trimming, but preserving uncountability, we may assume that the integer $k_{{\bf p}}$, the subset ${\cal U}_{{\bf p}}\subseteq k_{{\bf p}}$, and the parameter ${\bf\overline{a}}_{{\bf p}}$ remain constant. As notation, for $i<j$, let $f:u_{{\bf p}_{i}}\rightarrow u_{{\bf p}_{j}}$ be the unique order-preserving bijection. We may additionally assume that $n_{{\bf p}_{i}}(t)=n_{{\bf p}_{j}}(f(t))$, hence $f$ has a natural extension (also called $f$)$:\overline{x}_{{\bf p}_{i}}\rightarrow\overline{x}_{{\bf p}_{j}}$ given by $f(x_{t,n})=x_{f(t),n}$. With this identification, we may assume $\theta_{{\bf p}_{i}}(\overline{x}_{{\bf p}_{i}})=\theta_{{\bf p}_{j}}(f(\overline{x}_{{\bf p}_{i}}))$. As well, we may also assume ${\rm tp}(x_{t,n}/A_{k_{{\bf p}}})={\rm tp}(x_{f(t),n}/A_{k_{{\bf p}}})$ for every $x_{t,n}\in\overline{x}_{{\bf p}_{i}}$. As well, the colorings match up as well, i.e., $c(x_{t,n})=x_{f(t),n}$.

Now fix $i<j$. Define ${\bf q}$ by $k_{{\bf q}}:=k_{{\bf p}}$; ${\cal U}_{{\bf q}}:={\cal U}_{{\bf p}}$; and ${\bf\overline{a}}_{\bf q}:={\bf\overline{a}}_{{\bf p}}$ (the common values). Let $u_{{\bf q}}:=u_{{\bf p}_{i}}\cup u_{{\bf p}_{j}}$, and, for $t\in u_{{\bf p}_{i}}$, $n_{t,{\bf q}}=n_{t,{\bf p}_{i}}$ while $n_{t,{\bf q}}=n_{t,{\bf p}_{j}}$ for $t\in u_{{\bf p}_{j}}$. To produce the striated type $\theta_{{\bf q}}\in S_{at}({\bf\overline{a}}_{{\bf q}})$, first choose a perfect chain realization $(\overline{M},\bar{b})$ of $\theta_{{\bf p}_{i}}(\overline{x}_{{\bf p}_{i}})$. Say $|u_{{\bf p}_{i}}|=\ell=|u_{{\bf p}_{j}}|$, while $|u^{*}|=k<\ell$. By Lemma 3.10(2), ${\rm tp}(\bar{b}_{<k}/{\bf\overline{a}}_{{\bf p}})$ is a striated type of length $k$ and $(\overline{M}_{\geq k},\bar{b}_{\geq k})$ is a perfect chain realization of the striated type ${\rm tp}(\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}}\bar{b}_{<k})$ of length $(\ell-k)$. Choose $\bar{d}$ from $M_{k}$ such that ${\rm tp}(\bar{d}/{\bf\overline{a}}_{{\bf p}}\bar{b}_{<k})={\rm tp}(\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}}\bar{b}_{<k})$. Then by Lemma 3.9 (with $M_{k}$ playing the role of $M_{0}$ there), $(\overline{M}_{\geq k},\bar{b}_{\geq k})$ is a perfect chain realization of the striated type ${\rm tp}(\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}}\bar{b}_{<k}\bar{d})$. So, by Lemma 3.10(1), ${\rm tp}(\bar{d}\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}}\bar{b}_{<k})$ is a striated type of length $2(\ell-k)$. Thus, a second application of Lemma 3.10(1) implies that ${\rm tp}(\bar{b}_{<k}\bar{d}\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}})$ is a striated type of length $2\ell-k$. Let $\theta_{{\bf q}}$ be a complete formula over ${\bf\overline{a}}_{{\bf p}}$ generating this type.

In order to show that ${\bf q}$ is a precondition (i.e., an element of ${\mathbb{Q}}_{0}$) only Clause (8) requires an argument. Fix any $x_{t,n},x_{s,m}$ in $\overline{x}_{{\bf q}}$ with $c_{{\bf q}}(x_{t,n})=c_{{\bf q}}(x_{s,m})$. As both ${\bf p}_{i},{\bf p}_{j}\in{\mathbb{Q}}_{0}$, the verification is immediate if $\{t,s\}$ is a subset of either $u_{{\bf p}_{i}}$ or $u_{{\bf p}_{j}}$, so assume otherwise. By symmetry, assume $t\in u_{{\bf p}_{i}}-u^{*}$ and $s\in u_{{\bf p}_{j}}-u^{*}$. The point is that by our trimming, $x_{f(t),n}\in\overline{x}_{{\bf p}_{j}}$, $c_{{\bf p}_{j}}(x_{f(t),n})=c_{{\bf p}_{i}}(x_{t,n})$, and ${\rm tp}(x_{t,n}/A_{k_{{\bf p}}})={\rm tp}(x_{f(t),n}/A_{k_{{\bf p}}})$. There are now two cases: First, if ${\rm tp}(x_{f(t),n}/A^{*})={\rm tp}(x_{s,m}/A^{*})$, then it follows that ${\rm tp}(x_{t,n}/A_{k_{{\bf p}}})={\rm tp}(x_{s,m}/A_{k_{{\bf p}}})$, hence ${\rm spl}(e_{t,n},e_{s,m})\geq k_{{\bf p}}$ for any perfect chain realization $(\overline{N},\bar{e})$ of $\theta_{{\bf q}}$. On the other hand, if $\theta_{{\bf p}_{j}}$ ‘says’ ${\rm spl}(x_{f(t),n},x_{s,m})=k\in{\cal U}_{{\bf p}}$, then $\theta_{{\bf q}}$ ‘says’ ${\rm spl}(x_{t,n},x_{s,m})=k\in{\cal U}_{{\bf q}}$ as well. Thus, ${\bf q}\in{\mathbb{Q}}_{0}$, which suffices by Lemma 3.13.

Lemma 3.16

Each of the following sets are dense and open in $({\mathbb{Q}},\leq_{{\mathbb{Q}}})$.

For every $t\in\omega_{1}$, $D_{t}=\{{\bf p}\in{\mathbb{Q}}:t\in u_{{\bf p}}\}$; 2. 2.

For every $(t,n)\in\omega_{1}\times\omega$, $D_{t,n}=\{{\bf p}\in{\mathbb{Q}}:x_{t,n}\in\overline{x}_{{\bf p}}\}$; and 3. 3.

Henkin witnesses:* For all $t\in\omega_{1}$, all $\langle x_{s_{i},n_{i}}:i<m\rangle$ with each $s_{i}\leq t$ and all $\varphi(y,v_{i}:i<m)$, $\{{\bf p}\in{\mathbb{Q}}:\ \hbox{{\bf either}}\ \theta_{{\bf p}}(\overline{x}_{{\bf p}})\vdash\forall y\neg\varphi(y,x_{s_{i},n_{i}}:i<m)\ \hbox{{\bf or}}$ for some $n^{*}$, $\theta_{{\bf p}}(\overline{x}_{{\bf p}})\vdash\varphi(x_{t,n^{*}},x_{s_{i},n_{i}}:i<m)\}$.* 4. 4.

For all $e\in M^{*}$, $D_{e}=\{{\bf p}\in{\mathbb{Q}}:e\in{\bf\overline{a}}_{{\bf p}}$ and $\theta(\overline{x}_{{\bf p}})\vdash x_{0,n}=e$ for some $n\in\omega\}$.

Proof. That each of these sets is open is immediate. As for density, in all four clauses we will show that given some ${\bf p}\in{\mathbb{Q}}$, we will find an extension ${\bf q}\geq_{{\mathbb{Q}}}{\bf p}$ with $\overline{x}_{{\bf q}}$ a one-point extension of $\overline{x}_{{\bf p}}$. In all cases, we will put $k_{{\bf q}}:=k_{{\bf p}}$, ${\cal U}_{{\bf q}}={\cal U}_{{\bf p}}$ and since $\overline{x}_{{\bf p}}$ is finite, we can choose the color $c_{{\bf q}}$ of the ‘new element’ to be distinct from the other colors. Because of that, Clause (8) for ${\bf q}$ follows immediately from the fact ${\bf p}\in{\mathbb{Q}}$. Thus, for all four clauses, all of the work is in finding a striated type $\theta_{{\bf q}}$ extending $\theta_{{\bf p}}$.

(1) Fix $t\in\omega_{1}$ and choose an arbitrary ${\bf p}\in{\mathbb{Q}}$. If $t\in u_{{\bf p}}$ then there is nothing to prove, so assume otherwise. Let $\ell=|u_{{\bf p}}|$ and let $k=|\{s\in u_{{\bf p}}:s<t\}|$. Assume that $k<\ell$, as the case of $k=\ell$ is similar, but easier. Choose a perfect chain realization $(\overline{M},\bar{b})$ of $\theta_{{\bf p}}(\overline{x}_{{\bf p}})$. By Lemma 3.10(2), ${\rm tp}(\bar{b}_{<k}/{\bf\overline{a}}_{{\bf p}})$ is a striated type of length $k$. By Lemma 2.4(1), choose an $A^{*}$-large type $r\in S_{at}({\bf\overline{a}}_{{\bf p}}\bar{b}_{<k})$ and choose a realization $e$ of $r$ in $M_{k}$. One checks immediately that ${\rm tp}(\bar{b}_{<k}e/{\bf\overline{a}}_{{\bf p}})$ is a striated type of length $(k+1)$. Now, also by Lemma 3.10(2), $(\overline{M}_{\geq k},\bar{b}_{\geq k})$ is a perfect chain realization of ${\rm tp}(\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}}\bar{b}_{<k})$. So, by Lemma 3.9, $(\overline{M}_{\geq k},\bar{b}_{\geq k})$ is also a perfect chain realization of ${\rm tp}(\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}}\bar{b}_{<k}e)$. In particular, ${\rm tp}(\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}}\bar{b}_{<k}e)$ is a striated type of length $(\ell-k)$. Thus, by Lemma 3.10(1), ${\rm tp}(\bar{b}_{<k}e\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}})$ is a striated type of length $(\ell+1)$. Take ${\bf\overline{a}}_{{\bf q}}:={\bf\overline{a}}_{{\bf p}}$, $\overline{x}_{{\bf q}}:=\overline{x}_{{\bf p}}\cup\{x_{t,0}\}$, and take $\theta_{{\bf q}}(\overline{x}_{{\bf q}})$ to be a complete formula in ${\rm tp}(\bar{b}_{<k}e\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf q}})$.

The proofs of (2) and (3) are extremely similar. We prove (2) and indicate the adjustment necessary for (3). Fix $(t,n)\in\omega_{1}\times\omega$. By (1) and an inductive argument, we may assume we are given ${\bf p}\in{\mathbb{Q}}$ with $t\in u_{{\bf p}}$ and $x_{t,n-1}\in\overline{x}_{{\bf p}}$. Say $|u_{{\bf p}}|=\ell$ and assyne $t$ is the $(k-1)$st element of $u_{p}$ in ascending order. Choose a perfect chain realization $(\overline{M},\bar{b})$ of $\theta_{{\bf p}}(\overline{x}_{{\bf p}})$. By Lemma 3.10(2), ${\rm tp}(\bar{b}_{<k}/{\bf\overline{a}}_{{\bf p}})$ is striated of length $k$. Choose an arbitrary $e\in M_{k}$444In the proof of (3), $e$ would be a realization of $\varphi(y,b_{s_{i},n_{i}}:i<m)$ in $M_{k}$, if one existed. and adjoin it to $\bar{b}_{k-1}$. More formally, let $\bar{b}_{<k}^{*}:=\langle\bar{b}_{j}^{*}:j<k\rangle$, where $\bar{b}_{j}^{*}=\bar{b}_{j}$ for $j<k-2$, while $\bar{b}^{*}_{k-1}:=\bar{b}_{k-1}e$. Note that ${\rm tp}(\bar{b}^{*}_{<k}/{\bf\overline{a}}_{{\bf p}})$ remains a striated type of length $k$. By Lemma 3.10(2), $(\overline{M}_{\geq k},\bar{b}_{\geq k})$ is a perfect chain realization of ${\rm tp}(\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}}\bar{b}_{<k})$. So, by Lemma 3.9 it is also a perfect chain realization of ${\rm tp}(\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}}\bar{b}_{<k}^{*})$. In particular, ${\rm tp}(\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}}\bar{b}_{<k}^{*})$ is a striated type of length $(\ell-k)$, so ${\rm tp}(\bar{b}^{*}_{<k}\bar{b}_{\geq k}/{\bf\overline{a}}_{{\bf p}})$ is a striated type of length $\ell$ extending $\theta_{{\bf p}}(\overline{x}_{{\bf p}})$. Put $\overline{x}_{{\bf q}}:=\overline{x}_{{\bf p}}\cup\{x_{t,n}\}$ and let $\theta_{{\bf q}}(\overline{x}_{{\bf q}})$ be a complete formula isolating this type.

(4) is also similar and is left to the reader.

The following Proposition follows immediately from the density conditions described above.

Proposition 3.17

Let $G$ be a ${\mathbb{Q}}$-generic filter. Then, in $V[G]$, a rich, ${\cal U}_{G}$-colored atomic model of $T$ exists, where ${\cal U}_{G}=\{k\in\omega:k\in{\cal U}_{{\bf p}}$ for some ${\bf p}\in G\}$.

Proof. There is a congruence $\sim_{G}$ defined on $X=\{x_{t,n}:t\in\omega_{1},n\in\omega\}$ defined by $x_{t,n}\sim_{G}x_{s,m}$ if and only if $\theta_{{\bf p}}\vdash x_{t,n}=x_{s,m}$ for some ${\bf p}\in G$. Let $M_{G}$ be the model of $T$ with universe $X/\sim_{G}$ and relations $M_{G}\models\varphi(a_{1},\dots,a_{k})$ if and only if there are $(x_{t_{1},n_{1}},\dots,x_{t_{k},n_{k}})\in X^{k}$ such that $[x_{t_{i},n_{i}}]=a_{i}$ for each $i$ and $\theta_{{\bf p}}\vdash\varphi(x_{t_{1},n_{1}},\dots,x_{t_{k},n_{k}})$ for some ${\bf p}\in G$. Since $({\mathbb{Q}},\leq_{{\mathbb{Q}}})$ has c.c.c., $M_{G}$ has size $\aleph_{1}$. As notation, for each $t\in\omega_{1}$, let $M_{\leq t}$ be the substructure of $M_{G}$ with universe $\{[x_{s,m}]:s\leq t,m\in\omega\}$. Then $M^{*}\preceq M_{0}$ and $M_{\leq s}\preceq M_{\leq t}\preceq M_{G}$ whenever $s\leq t<\omega_{1}$. The definition of a striated type implies that ${\rm tp}([x_{t,0}]/A^{*})$ is omitted in $M_{<t}$, hence the set $\{[x_{t,0}]:t\in\omega_{1}\}$ witnesses that $(M_{G},\bar{b}^{*})$ is rich. Also, define $c_{G}:=\bigcup\{c_{{\bf p}}:{\bf p}\in G\}$. Using the fact that each ${\bf p}\in{\mathbb{Q}}$ is fully decided, check that $c_{G}$ is a ${\cal U}_{G}$-coloring of $(M_{G},\bar{b}^{*})$.

Note that in the Conclusion below, such a $G\in V$ always exists, since ${\cal B}$ is countable.

Conclusion 3.18

Suppose ${\cal B}$ is a countable, transitive model of $ZFC^{*}$, with $\{M^{*},T,L\}\subseteq{\cal B}$, and let $G\in V$, $G\subseteq{\mathbb{Q}}$ be any filter meeting every dense $D\subseteq{\mathbb{Q}}$ with $D\in{\cal B}$. Then: Let ${\cal U}_{G}=\{k\in\omega:k\in{\cal U}_{{\bf p}}$ for some ${\bf p}\in G\}$. Then:

${\cal U}_{G}\in V$; and 2. 2.

In $V$, there is a ${\cal U}_{G}$-colored, rich atomic model $(N,\bar{b}^{*})$ of $T$.

Proof. That ${\cal U}_{G}\in V$ is immediate, since both ${\cal B}$ and $G$ are. As for (2), as $G$ meets every dense set in ${\cal B}$, ${\cal B}[G]$ is a countable, transitive model of $ZFC^{*}$, and by applying Proposition 3.17,

[TABLE]

Let $L^{\prime}=L\cup\{c,R\}\cup\{c_{m}:m\in M^{*}\}$ Working in ${\cal B}[G]$, expand $M_{G}$ to an $L^{\prime}$-structure $M^{\prime}$, interpreting each $c_{m}$ by $m$, interpreting the unary function $c^{M^{\prime}}$ as $c_{G}=\bigcup\{c_{{\bf p}}:{\bf p}\in G\}$, and the unary predicate $R^{M^{\prime}}=\{[x_{t,0}]:t\in\omega_{1}\}$.

Now, for each $d,d^{\prime}\in M^{\prime}$ and $k\in\omega$, the relation ${\rm tp}_{M^{\prime}}(d/A_{k})={\rm tp}_{M^{\prime}}(d^{\prime}/A_{k})$ is definable by an $L^{\prime}_{\omega_{1},\omega}$-formula. Thus, the binary function ${\rm spl}:(M^{\prime})^{2}\rightarrow(\omega+1)$ is also $L^{\prime}_{\omega_{1},\omega}$-definable, hence, using the coloring $c$, there is an $L^{\prime}_{\omega_{1},\omega}$-sentence $\Psi$ stating that ‘$c$ induces a ${\cal U}_{G}$-coloring.’ Finally, using the $Q$-quantifier to state that $R$ is uncountable, there is an $L^{\prime}_{\omega_{1},\omega}$-sentence $\Phi\in{\cal B}[G]$ stating that the $L(\bar{b}^{*})$-reduct of a given $L^{\prime}$-structure is a rich, atomic model of $T$, that is ${\cal U}_{G}$-colored via $c$. We finish by applying Proposition 2.9 to $M^{\prime}$ and $\Phi$.

3.3 Mass production

In this subsection we define a forcing $({\mathbb{P}},\leq_{{\mathbb{P}}})$ such that a ${\mathbb{P}}$-generic filter $G$ produces a perfect set $\{G_{\eta}:\eta\in 2^{\omega}\}$ of ${\mathbb{Q}}$-generic filters such that the associated subsets $\{{\cal U}_{G_{\eta}}:\eta\in 2^{\omega}\}$ of $\omega$ are almost disjoint. Although the application there is very different, the argument in this subsection is similar to one appearing in [7].

We begin with one easy density argument concerning the partial $({\mathbb{Q}},\leq_{{\mathbb{Q}}})$. Fundamentally, it allows us to ‘stall’ the construction for any fixed, finite length of time.

Lemma 3.19

For every ${\bf p}\in{\mathbb{Q}}$ and every $k^{*}>k_{{\bf p}}$, there is ${\bf q}\geq_{{\mathbb{Q}}}{\bf p}$ such that $\overline{x}_{{\bf q}}=\overline{x}_{{\bf p}}$, (hence $c_{{\bf q}}=c_{{\bf p}}$); but $k_{{\bf q}}=k^{*}$ and ${\cal U}_{{\bf q}}={\cal U}_{{\bf p}}$, i.e., ${\cal U}_{{\bf q}}\cap[k_{{\bf p}},k^{*})=\emptyset$.

Proof. Simply define ${\bf q}$ as above and then verify that ${\bf q}\in{\mathbb{Q}}$.

Definition 3.20

For $n\in\omega$, let

[TABLE]

As notation, for ${\bf p}\in{\mathbb{P}}_{n}$, we let $k({\bf p})$ denote the (integer) first coordinate of ${\bf p}$. For each $\ell<k({\bf p})$, define the trace of $\ell$, ${\rm tr}_{\ell}({\bf p})=\{\nu\in 2^{n}:\ell\in{\cal U}_{p_{\nu}}\}$.

Let ${\mathbb{P}}=\bigcup_{n\in\omega}{\mathbb{P}}_{n}$. As notation, for ${\bf p}\in{\mathbb{P}}$, $n({\bf p})$ is the unique $n$ for which ${\bf p}\in{\mathbb{P}}_{n}$. **

Definition 3.21

Define an order $\leq_{{\mathbb{P}}}$ on ${\mathbb{P}}$ by ${\bf p}\leq_{{\mathbb{P}}}{\bf q}$ if and only if

$n({\bf p})\leq n({\bf q})$, $k({\bf p})\leq k({\bf q})$; 2. 2.

$p_{\nu}\leq_{{\mathbb{Q}}}q_{\mu}$ for all pairs $\nu\in 2^{n({\bf p})},\mu\in 2^{n({\bf q})}$ satisfying $\nu\trianglelefteq\mu$; and 3. 3.

For all $\ell\in[k({\bf p}),k({\bf q}))$, the set $\{\mu\mathord{\restriction}_{n({\bf p})}:\mu\in{\rm tr}_{\ell}({\bf q})\}$ is either empty or is a singleton.

It is easily checked that $({\mathbb{P}},\leq_{{\mathbb{P}}})$ is a partial order, hence a notion of forcing. The following Lemma describes the dense subsets of ${\mathbb{P}}$.

Lemma 3.22

For each $n$ and $k$, $\{{\bf p}\in{\mathbb{P}}:n({\bf p})\geq n\}$ and $\{{\bf p}\in{\mathbb{P}}:k({\bf p})\geq k\}$ are dense; 2. 2.

Suppose $D$ is a dense, open subset of ${\mathbb{Q}}$. Then for every $n$ and every ${\bf p}\in{\mathbb{P}}_{n}$, there is ${\bf q}\in{\mathbb{P}}_{n}$ such that ${\bf q}\geq_{{\mathbb{P}}}{\bf p}$ and, for every $\nu\in 2^{n}$, ${\bf q}_{\nu}\in D$.

Proof. Arguing by induction, it suffices to prove that for any given ${\bf p}\in{\mathbb{P}}$, there is ${\bf q}\geq_{\mathbb{P}}{\bf p}$ with $n({\bf q})=n({\bf p})+1$ and an ${\bf r}\geq_{{\mathbb{P}}}{\bf p}$ with $k({\bf r})>k({\bf p})$. Fix ${\bf p}\in{\mathbb{P}}$. Say ${\bf p}\in{\mathbb{P}}_{n}$ and ${\bf p}=(k,\bar{p})$. To construct ${\bf q}$, for each $\nu\in 2^{n}$, define $q_{\nu 0}=q_{\nu 1}=p_{\nu}$. Let $\bar{q}:=\langle q_{\mu}:\mu\in 2^{n+1}\rangle$ and ${\bf q}=(k,\bar{q})$. Then ${\bf q}\in{\mathbb{P}}_{n+1}$ and ${\bf q}\geq_{{\mathbb{P}}}{\bf p}$ (note that Clause (3) in the definition of $\leq_{{\mathbb{P}}}$ is vacuously satisfied since $k({\bf p})=k({\bf q})$).

To construct ${\bf r}$, simply apply Lemma 3.19 to each $p_{\nu}$ to produce an extension $r_{\nu}\geq_{{\mathbb{Q}}}p_{\nu}$ with $k_{r_{\nu}}=k+1$, but ${\cal U}_{r_{\nu}}={\cal U}_{p_{\nu}}$. Then let $\bar{r}:=\langle r_{\nu}:\nu\in 2^{n}\rangle$ and ${\bf r}=(k+1,\bar{r})$. Then ${\bf r}\geq_{{\mathbb{P}}}{\bf p}$ as required.

(2) Fix such a $D$ and $n$. As we are working exclusively in ${\mathbb{P}}_{n}$ and because $2^{n}$ is a fixed finite set, it suffices to prove that for any chosen $\nu\in 2^{n}$,

For every ${\bf p}\in{\mathbb{P}}_{n}$ there is ${\bf q}\in{\mathbb{P}}_{n}$ with ${\bf q}\geq_{{\mathbb{P}}}{\bf p}$ and $q_{\nu}\in D$.

To verify this, fix $\nu\in 2^{n}$ and ${\bf p}\in{\mathbb{P}}_{n}$. Concentrating on $p_{\nu}$, as $D$ is dense, choose $q_{\nu}\in D\cap{\mathbb{Q}}$ with $q_{\nu}\geq_{{\mathbb{Q}}}p_{\nu}$. Let $k^{*}:=k_{q_{\nu}}$. Next, for each $\delta\in 2^{n}$ with $\delta\neq\nu$, apply Lemma 3.19 to $p_{\delta}$, obtaining some $q_{\delta}\in{\mathbb{Q}}$ satisfying $q_{\delta}\geq_{{\mathbb{Q}}}p_{\delta}$, $k_{q_{\delta}}=k^{*}$, but ${\cal U}_{q_{\delta}}={\cal U}_{p_{\delta}}$. Now, collect all of this data into a condition ${\bf q}\in{\mathbb{P}}_{n}$ defined by $k({\bf q})=k^{*}$ and $\bar{q}=\langle q_{\gamma}:\gamma\in 2^{n}\rangle$, where each $q_{\gamma}$ is as above. To see that ${\bf q}\geq_{{\mathbb{P}}}{\bf p}$, Clause (3) is verified by noting that for every $\ell\in[k({\bf p}),k^{*})$, ${\rm tr}_{\ell}({\bf q})$ is either empty, or equals $\{\nu\}$, depending on whether or not $\ell\in{\cal U}_{q_{\nu}}$.

Notation 3.23

Suppose ${\cal B}\models ZFC^{*}$ and let $G^{*}\subseteq{\mathbb{P}}$, $G^{*}\in V$ be a filter meeting every dense subset $D^{*}\subseteq{\mathbb{P}}$ with $D^{*}\in{\cal B}$. For each $n$ and $\nu\in 2^{n}$, let

[TABLE]

Then, for each $\eta\in 2^{\omega}$, let

[TABLE]

Proposition 3.24

In the notation of 3.23:

For every $\eta\in 2^{\omega}$, $G_{\eta}\subseteq{\mathbb{Q}}$ is a filter meeting every dense $D\subseteq{\mathbb{Q}}$ with $D\in{\cal B}$; 2. 2.

The sets $\{{\cal U}_{\eta}:\eta\in 2^{\omega}\}$ are an almost disjoint family of infinite subsets of $\omega$.

Proof. (1) follows immediately from Lemma 3.22(2).

(2) Choose distinct $\eta,\eta^{\prime}\in 2^{\omega}$. Choose $n_{0}$ such that $\eta|n\neq\eta^{\prime}|n$ whenever $n\geq n_{0}$. By Lemma 3.22(1), choose ${\bf p}^{*}\in G^{*}$ with $n({\bf p}^{*})\geq n_{0}$. We show that ${\cal U}_{\eta}\cap{\cal U}_{\eta^{\prime}}$ is finite by establishing that if $\ell\in{\cal U}_{\eta}\cap{\cal U}_{\eta^{\prime}}$, then $\ell\leq k({\bf p}^{*})$.

To establish this, choose $\ell\in{\cal U}_{\eta}\cap{\cal U}_{\eta^{\prime}}$. By unpacking the definitions, choose ${\bf q}^{*},{\bf r}^{*}\in G^{*}$ such that, letting $\mu:=\eta|n({\bf q}^{*})$ and $\mu^{\prime}:=\eta^{\prime}|n({\bf r}^{*})$, we have $\ell\in{\cal U}_{{\bf q}^{*}_{\mu}}\cap{\cal U}_{{\bf r}^{*}_{\mu^{\prime}}}$. As $G^{*}$ is a filter, choose ${\bf s}^{*}\in G^{*}$ with ${\bf s}^{*}\geq_{{\mathbb{P}}}{\bf p}^{*},{\bf q}^{*},{\bf r}^{*}$. As notation, let $\delta:=\eta|n({\bf s}^{*})$ and $\delta^{\prime}:=\eta^{\prime}|n({\bf s}^{*})$.

Claim: $\ell\in{\cal U}_{{\bf s}^{*}_{\delta}}\cap{\cal U}_{{\bf s}^{*}_{\delta^{\prime}}}$.

Proof. As $\ell\in{\cal U}_{{\bf q}^{*}_{\mu}}$, $\ell<k({\bf q}^{*})$. From ${\bf q}^{*}\leq_{{\mathbb{P}}}{\bf s}^{*}$ we conclude $k({\bf q}^{*})\leq k({\bf s}^{*})$, so $\ell<k({\bf s}^{*})$ as well. From ${\bf q}^{*}\leq_{{\mathbb{P}}}{\bf s}^{*}$ and $\mu\trianglelefteq\delta$ we obtain ${\bf q}^{*}_{\mu}\leq_{{\mathbb{Q}}}{\bf s}^{*}_{\delta}$. But then, as $\ell\in{\cal U}_{{\bf q}^{*}_{\mu}}$, it follows that $\ell\in{\cal U}_{{\bf s}^{*}_{\delta}}$. That $\ell\in{\cal U}_{{\bf s}^{*}_{\delta^{\prime}}}$ is analogous, using ${\bf r}^{*}$ in place of ${\bf q}^{*}$.

Finally, assume by way of contradiction that $\ell\geq k({\bf p}^{*})$. The Claim implies that $\{\delta,\delta^{\prime}\}\subseteq{\rm tr}_{\ell}({\bf s}^{*})$. As $\ell\in[k({\bf p}^{*}),k({\bf s}^{*}))$, Clause (3) of ${\bf p}^{*}\leq_{{\mathbb{P}}}{\bf s}^{*}$ implies that $\delta|n({\bf p}^{*})=\delta^{\prime}|n({\bf p}^{*})$. But, as $\eta|n({\bf p}^{*})=\delta|n({\bf p}^{*})$ and $\eta^{\prime}|n({\bf p}^{*})=\delta^{\prime}|n({\bf p}^{*})$, we contradict our choice of ${\bf p}^{*}$.

We close this section with the proof of Proposition 3.1, which we restate for convenience.

Conclusion 3.25

There is a family $\{(N_{\eta},\bar{b}^{*}):\eta\in 2^{\omega}\}$ of $2^{\aleph_{0}}$ rich, atomic models of $T$, each of size $\aleph_{1}$, that are pairwise non-isomorphic over $\bar{b}^{*}$.

Proof. Choose any countable, transitive model ${\cal B}$ of $ZFC^{*}$ and choose any $G^{*}\in V$, $G^{*}\subseteq{\mathbb{P}}$, $G^{*}$ meets every dense subset $D^{*}\in{\cal B}$ (as ${\cal B}$ is countable, such a $G^{*}$ exists). For each $\eta\in 2^{\omega}$, choose $G_{\eta}$ and ${\cal U}_{\eta}$ as in Proposition 3.24, and apply Conclusion 3.18 to get a rich ${\cal U}_{\eta}$-colored $(N_{\eta},\bar{b}^{*})$ in $V$. That this family is pairwise non-isomorphic over $\bar{b}^{*}$ follows immediately from Corollary 3.6, since the sets $\{{\cal U}_{\eta}:\eta\in 2^{\omega}\}$ are almost disjoint.

4 The proof of Theorem 1.4

Assume that the class ${\bf At}_{T}$ is not ${\rm pcl}$-small, as witnessed by an (uncountable) model $N^{*}$ containing a finite tuple $\bar{a}^{*}$. Fix a countable, elementary substructure $M^{*}\preceq N^{*}$ that contains $\bar{a}^{*}$. To aid notation, let $D^{*}:={\rm pcl}_{N^{*}}(\bar{a}^{*})$. We now split into cases, depending on the relationship between the cardinals $2^{\aleph_{0}}$ and $2^{\aleph_{1}}$.

Case 1. $2^{\aleph_{0}}<2^{\aleph_{1}}$.

In this case, expand the language of $T$ to $L(D^{*})$, adding a new constant symbol for each $d\in D^{*}$. Then, the natural expansion $N^{*}_{D^{*}}$ $N^{*}$ to an $L(D^{*})$-structure is a model of the infinitary $L(D^{*})$-sentence $\Phi$ that entails $Th(N^{*}_{D^{*}})$ and ensures that every finite tuple is $L$-atomic with respect to $T$. As $N^{*}_{D^{*}}$ is a model of $\Phi$ that realizes uncountably many types over the empty set (after fixing $D^{*}$!), it follows from [5], Theorem 45 of Keisler that there are $2^{\aleph_{1}}$ pairwise non-$L(D^{*})$-isomorphic models $\Phi$, each of size $\aleph_{1}$. As $2^{\aleph_{0}}<2^{\aleph_{1}}$, it follows that there is a subfamily of $2^{\aleph_{1}}$ pairwise non-$L$-isomorphic reducts to the original language $L$. As each of these models are $L$-atomic, we conclude that ${\bf At}_{T}$ has $2^{\aleph_{1}}$ non-isomorphic models of size $\aleph_{1}$.

Case 2. $2^{\aleph_{0}}=2^{\aleph_{1}}$.

Choose $\bar{b}^{*}$ from $M^{*}$ as in Proposition 2.10 and apply Conclusion 3.25 to get a set ${\cal F}^{*}=\{(N_{\eta},\bar{b}^{*}):\eta\in 2^{\omega}\}$ of atomic models, each of size $\aleph_{1}$, that are pairwise non-isomorphic over $\bar{b}^{*}$. Let ${\cal F}=\{N_{\eta}:\eta\in 2^{\omega}\}$ be the set of reducts of elements from ${\cal F}^{*}$. By our cardinal hypothesis, ${\cal F}$ has size $2^{\aleph_{1}}$. The relation of $L$-isomorphism is an equivalence relation on ${\cal F}$, and each $L$-isomorphism equivalence class has size at most $\aleph_{1}$ (since $\aleph_{1}^{<\omega}=\aleph_{1}$). As $\aleph_{1}<2^{\aleph_{1}}$ we conclude that ${\cal F}$ has a subset of size $2^{\aleph_{1}}$ of pairwise non-isomorphic atomic models of $T$, each of size $\aleph_{1}$.