Order-isomorphic Morass-definable $\eta_1$-orderings

TL;DR

This paper demonstrates that under certain set-theoretic assumptions, all morass-definable $ ext{eta}_1$-orderings of size $eth_3$ are order-isomorphic, extending classical results to larger cardinalities using morasses.

Contribution

It proves the order-isomorphism of morass-definable $ ext{eta}_1$-orderings of size $eth_3$ in a Cohen extension with a simplified gap-2 morass, extending previous results to larger continuum sizes.

Findings

Morass-definable $ ext{eta}_1$-orderings of size $eth_3$ are order-isomorphic.

Existence of ultrapowers of $ ext{Reals}$ over $ ext{omega}$ that are gap-2 morass-definable.

Construction techniques extend transfinite back-and-forth to size $eth_3$.

Abstract

We prove that in the Cohen extension adding $\aleph_3$ generic reals to a model of $ZFC+CH$ containing a simplified $(\omega_1,2)$-morass, gap-2 morass-definable $\eta_1$-orderings with cardinality $\aleph_3$ are order-isomorphic. Hence it is consistent that the $2^{\aleph_0}=\aleph_3$ and that morass-definable $\eta_1$-orderings with cardinality of the continuum are order-isomorphic. We prove that there are ultrapowers of $\mathbb{R}$ over $\omega$ that are gap-2 morass-definable. The constructions use a simplified gap-2 morass, and commutativity with morass-maps and morass-embeddings, to extend a transfinite back-and-forth construction of order type $\omega_1$, to a function between objects of cardinality $\aleph_3$.

Equations

$$(f\circ g{)}_{\zeta }=f_{g(\zeta )}\circ g_{\zeta }for\zeta \le {\theta }_{\alpha },$$

$$(f\circ g{)}_{\zeta \xi }=f_{g(\zeta )g(\xi )}\circ g_{\zeta \xi }for\zeta <\xi \le {\theta }_{\alpha }.$$

$$⟨f,\{f_{\zeta }\mid \zeta <{\theta }_{\alpha }\},\{f_{\zeta \xi }\mid \zeta <\xi \le {\theta }_{\alpha }\}⟩.$$

$$g_n(h_0({\sigma }_0))=\tau$$

$$g_n(h_n^{\prime }({\sigma }_n))={\tau }_n.$$

$$g_0=g\circ g_0^{\ast }.$$

$$g_0(h_0({\sigma }_0))=g(g_0^{\ast }(h_0({\sigma }_0)))=\tau .$$

$$g_n=g^{\ast }\circ g_n^{\ast }.$$

$$g^{\ast }(g_n^{\ast }(h_0({\sigma }_0)))=\tau .$$

$$g_n^{\ast }(h_0({\sigma }_0))=g_0^{\ast }(h_0({\sigma }_0))$$

$$g{\upharpoonright }_{g_0^{\ast }(h_0({\sigma }_0))+1}=g^{\ast }{\upharpoonright }_{g_0^{\ast }(h_0({\sigma }_0))+1}.$$

$${\tau }_n\in g[{\phi }_{\zeta }].$$

$$f(\bar{\zeta })=f\circ f_1^{\prime }({\bar{\zeta }}_1)=f_1(\bar{\zeta })=\zeta .$$

$$f_2^{\prime }({\bar{\zeta }}_2)=\bar{\zeta }.$$

$$f_{\bar{\zeta }{\theta }_{\beta }}(\bar{g})=f_{\bar{\zeta }{\theta }_{\beta }}((f_1^{\prime }{)}_{{\bar{\zeta }}_1\bar{\zeta }}({\bar{g}}_1))=f_1({\bar{g}}_1)=g.$$

$$\bar{g}=(f_2^{\prime }{)}_{{\bar{\zeta }}_2\bar{\zeta }}({\bar{g}}_2).$$

$$f_{N+1}({\zeta }_{N+1})=f_{N+1}\circ f^{\prime }({\zeta }_N)=f_N({\zeta }_N)=\zeta .$$

$$(f_{N+1}{)}_{{\zeta }_{N+1}{\theta }_{{\beta }_N}}(g_{N+1})=(f_{N+1}{)}_{{\zeta }_{N+1}{\theta }_{{\beta }_{N+1}}}(f_{{\zeta }_N{\zeta }_{N+1}}^{\prime }(g_N))=(f_N{)}_{{\zeta }_N{\theta }_{{\beta }_N}}(g_N)=g.$$

$$g\circ (f_n{)}_{{\zeta }_n}=(f_n{)}_{{\theta }_{{\beta }_n}}\circ g_n.$$

$$(f_n{)}_{{\theta }_{{\beta }_n}}\circ g_n=g\circ (f_n{)}_{{\zeta }_n}.$$

$$(f_n{)}_{{\zeta }_n}[{\phi }_{{\zeta }_n}]\subseteq (f_m{)}_{{\zeta }_m}[{\phi }_{{\zeta }_m}]$$

$$(f_n{)}_{{\theta }_{{\beta }_n}}\circ g_n[{\phi }_{{\zeta }_n}]\subseteq (f_m{)}_{{\theta }_{{\beta }_m}}\circ g_m[{\phi }_{{\zeta }_m}].$$

$$D\subseteq h_{\bar{\zeta }}[{\phi }_{\bar{\zeta }}]$$

$$T\subseteq h_{{\theta }_{\gamma }}[{\phi }_{{\theta }_{\gamma }}]$$

$$h_{\bar{\zeta }{\theta }_{\gamma }}(\bar{g})=g.$$

$$(f_0{)}_{{\phi }_{{\zeta }_0}}({\rho }_0)=(f_n{)}_{{\phi }_{{\zeta }_n}}({\rho }_n)=\sigma .$$

$$f_n=h_n\circ f_n^{\ast }.$$

$$f_n^{\ast }({\zeta }_n)=\bar{\zeta }.$$

$$ssup(h[\bar{\zeta }+1])=\zeta =ssup(h_n[\bar{\zeta }+1]).$$

$$h{\upharpoonright }_{\bar{\zeta }+1}=h_n{\upharpoonright }_{\bar{\zeta }+1}$$

Full text

1 Introduction

An $\eta_{1}$-ordering without endpoints, $\langle X,<\rangle$, is a linear-ordering for which countably many consistent order constraints are necessarily witnessed by an object in $X$. That is, if $L\subset X$ and $U\subset X$ are both countable, and for every $l\in L$ and $u\in U$, $l<u$, then there is $x\in X$ such that for every $l\in L$ and $u\in U$, $l<x<u$. We consider only $\eta_{1}$-orderings without endpoints, that is $U$ or $L$ above may be empty.

By the Compactness Theorem, $\eta_{1}$-orderings exist in proliferation at all infinite cardinalities, and there are many well-studied examples of $\eta_{1}$-orderings that play a significant role in logic, topology and analysis. It is an early result of Model Theory that $\eta_{1}$-orderings having cardinality $\aleph_{1}$ are order-isomorphic. This is proved with a classic back-and-forth construction of the isomorphism. The argument is by transfinite construction of length $\omega_{1}$, requiring that only countably many order commitments need be satisfied at any step of the construction. Hence it is a consequence of the Continuum Hypothesis ($CH$) that $\eta_{1}$-orderings having cardinality of the continuum, $2^{\aleph_{0}}$, are order-isomorphic.

We are particularly interested in $\eta_{1}$-orderings without endpoints bearing cardinality of the continuum. We seek to find useful conditions on $\eta_{1}$-orderings in which $CH$ fails and $\eta_{1}$-orderings bearing cardinality of the continuum are order-isomorphic. In [References] we showed that in a Cohen extension adding $\aleph_{2}$-generic reals to a model of $ZFC+CH$ containing a simplified $(\omega_{1},1)$-morass, there is a level order-isomorphism between morass-definable $\eta_{1}$-orderings with cardinality of the continuum (Theorem 5.4, [References]). The simplified-morass plays a critical role in that we reduce the element-wise construction of a function between orderings having cardinality $\aleph_{2}$ to a construction of length $\omega_{1}$.

In this paper, working in the Cohen extension adding generic reals indexed by $\omega_{3}$ to a model of $ZFC+CH$ containing a simplified $(\omega_{1},2)$-morass, we define a level order-isomorphism between gap-2 morass-definable $\eta_{1}$-orderings having cardinality $2^{\aleph_{0}}=\aleph_{3}$. Our strategy is as follows. Given a simplified gap-2 morass, $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$, we construct a sequence of level order-preserving bijections, $\langle F_{\beta}\mid\beta<\omega_{1}\rangle$, on the “fake” morasses below $\omega_{1}$ in a manner similar to the construction of [References], but with the additional provision that the construction is closed under the embeddings of $\mathcal{F}_{\beta\gamma}$, for all $\beta<\gamma<\omega_{1}$. Then $F=\bigcup_{\beta<\omega_{1}}F_{\beta}$ is a level term function that is forced to be an order-isomorphism between those elements of the $\eta_{1}$-orderings that are also in the generic extension adding reals indexed by $\omega_{1}$. Furthermore, we are able to use the embeddings of $\mathcal{F}_{\beta\omega_{1}}$, $\beta<\omega_{1}$, to extend $F$ to a level term function that is forced to be an order-isomorphism between $\eta_{1}$-orderings having cardinality of the continuum in a model of set theory with $\aleph_{3}$-generic reals.

In the next paper we show that morass-definable and gap-2 morass-definable $\eta_{1}$-ordered real-closed fields bearing cardinality of the continuum are isomorphic in the Cohen extensions adding $\aleph_{2}$ or $\aleph_{3}$ generic reals, by an $\mathbb{R}$-linear order-preserving isomorphism. The role of $\eta_{1}$-ordered real-closed fields in the subject of automatic continuity (the existence of discontinuous homomorphisms of $C(X)$, the algebra of continuous real-valued functions on an infinite compact Hausdorff space, $X$) has been well-explored in $[\ref{Dales1}]$, $[\ref{Dales2}]$, $[\ref{Esterle1}]$, $[\ref{Esterle2}]$, $[\ref{Esterle3}]$ and $[\ref{Woodin}]$. We use the techniques of this paper to show that it is consistent that the continuum has cardinality $\aleph_{3}$, and that there exists a discontinuous homomorphism of $C(X)$, for any infinite compact Hausdorff space $X$.

2 The Simplified Gap-2 Morass

For $\kappa$ a regular cardinal, we define a simplified $(\kappa,2)$-morass as in Definition 1.3 [References].

Definition 2.1

$(Simplified\>\>(\kappa,2)-morass)$* The structure $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$ is a simplified $(\kappa,2)$-morass provided it has the following properties:*

$\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}}\rangle$* is a neat simplified $(\kappa^{+},1)$-morass $[\ref{Velleman1}]$.* 2. 2.

$\forall\alpha<\beta\leq\kappa$, $\mathcal{F}_{\alpha\beta}$ is a family of embeddings (see page 172, $[\ref{Velleman2}]$) from $\langle\langle\varphi_{\zeta}\mid\zeta<\theta_{\alpha}\rangle,\langle\mathcal{G}_{\zeta\xi}\mid\zeta<\xi\leq\theta_{\alpha}\rangle\rangle$ to $\langle\langle\varphi_{\zeta}\mid\zeta<\theta_{\beta}\rangle,\langle\mathcal{G}_{\zeta\xi}\mid\zeta<\xi\leq\theta_{\beta}\rangle\rangle$. 3. 3.

$\forall\alpha<\beta<\kappa\>\>(\mid\mathcal{F}_{\alpha\beta}\mid<\kappa)$. 4. 4.

$\forall\alpha<\beta<\gamma\leq\kappa\>\>(\mathcal{F}_{\alpha\gamma}=\{f\circ g\mid f\in\mathcal{F}_{\beta\gamma},g\in\mathcal{F}_{\alpha\beta}\})$. Here $f\circ g$ is defined by:

[TABLE]

[TABLE] 5. 5.

$\forall\alpha<\kappa$, $\mathcal{F}_{\alpha\alpha+1}$ is an amalgamation (see page 173 $[\ref{Velleman2}]$). 6. 6.

If $\beta_{1},\beta_{2}<\alpha\leq\kappa$, $\alpha$ a limit ordinal, $f_{1}\in\mathcal{F}_{\beta_{1}\alpha}$ and $f_{2}\in\mathcal{F}_{\beta_{2},\alpha}$, then $\exists\beta(\beta_{1},\beta_{2}<\beta<\alpha$ and $\exists f_{1}^{\prime}\in\mathcal{F}_{\beta_{1}\beta}\>\exists f_{2}^{\prime}\in\mathcal{F}_{\beta_{2}\beta}\>\exists g\in\mathcal{F}_{\gamma\alpha}(f_{1}=g\circ f_{1}^{\prime}$ and $f_{2}=g\circ f_{2}^{\prime}))$. 7. 7.

If $\alpha\leq\kappa$ and $\alpha$ is a limit ordinal, then:

1. (a)

$\theta_{\alpha}=\bigcup\{f[\theta_{\beta}]\mid\beta<\alpha$, $f\in\mathcal{F}_{\beta\alpha}\}$. 2. (b)

$\forall\zeta\leq\theta_{\alpha}$, $\varphi_{\zeta}=\bigcup\{f_{\bar{\zeta}}[\varphi_{\bar{\zeta}}]\mid\exists\beta<\alpha(f\in\mathcal{F}_{\beta\alpha}$, $f(\bar{\zeta})=\zeta)\}$. 3. (c)

$\forall\zeta<\xi\leq\theta_{\alpha}$, $\mathcal{G}_{\zeta\xi}=\bigcup\{f_{\bar{\zeta}\bar{\xi}}[\mathcal{G}_{\bar{\zeta}\bar{\xi}}]\mid\exists\beta<\alpha\>(f\in\mathcal{F}_{\beta\alpha}$, $f(\bar{\zeta})=\zeta$, $f(\bar{\xi})=\xi)\}$.

We consider the case in which $\kappa=\omega_{1}$. If $\alpha<\beta\leq\omega_{1}$, and $f\in\mathcal{F}_{\alpha\beta}$, then following the notational simplification of Velleman, we consider $f$ to be a triple:

[TABLE]

In the triple above, $f:\theta_{\alpha}+1\to\theta_{\beta}+1$ is an order-preserving injection with $f(\theta_{\alpha})=\theta_{\beta}$. We will refer to this as the first component of the embedding. For $\zeta\leq\theta_{\alpha}$, $f_{\zeta}:\varphi_{\zeta}\to\varphi_{f(\zeta)}$ is an order-preserving injection. We refer to this as the second component of the embedding (corresponding to $\zeta$). Finally for $\zeta<\xi\leq\theta_{\alpha}$, $f_{\zeta\xi}:\mathcal{G}_{\zeta\xi}\to\mathcal{G}_{f(\zeta)f(\xi)}$ is a function between morass maps of $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}}\rangle$. We refer to this as the third component of the embedding (corresponding to $\zeta$ and $\xi$). Embeddings satisfy a number of regularity and commutativity conditions that make them a practical tool for extending our $\omega_{1}$-inductive construction to a construction of cardinality $\aleph_{3}$.

3 Morass Maps and Morass-Embeddings

Let $M$ be a model of $ZFC+CH$ containing a simplified $(\omega_{1},2)$-morass, $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$. Let $P$ be the poset $Fn(\omega_{3}\times\omega,2)$, $G$ be $P$-generic over $M$ and $M^{P}$ be the forcing language of the poset $P$ in $M$. The construction of this paper is dependent on details of the indexing set of $P$. We are required to construct a function between terms in the forcing language that is sensitive to the precise subset of $\omega_{3}$ required for the construction of carefully selected terms of $M^{P}$. In general, if $S\subset\omega_{3}$, we let $P(S)$ be the poset $Fn(S\times\omega,2)$. In order to easily associate these partial languages with constructions along the morass, for $\beta\leq\omega_{3}$, let $P_{\beta}=P(\varphi_{\theta_{\beta}}\times\omega,2)$. If $G$ is $P$-generic over $M$, let $G(S)$ be the factor of $G$ that is $P(S)$-generic over $M$ (similarly for $G_{\beta}$).

We will construct a function between sets of terms with cardinality $\aleph_{3}$ in the forcing language, $M^{P}$, by applying the second components of embeddings of a simplified gap-2 morass to a function on a set of terms in $M^{P_{\omega_{1}}}$ with cardinality $\aleph_{1}$. If $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$ is a simplified $(\omega_{1},2)$-morass, then $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}}\rangle$ is a simplified $(\omega_{2},1)$-morass. It is easy to verify that every countable subset of $\omega_{3}$ is in the image of a single morass function of $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}}\rangle$. If every countable subset of $\omega_{3}$ were in the range of a morass map from a countable vertex, then the construction of Theorem 5.4 [References] would suffice to prove that morass-definable $\eta_{1}$-orderings are order-isomorphic in $M[G]$. However, not every countable subset of $\omega_{3}$ is anticipated by a morass map from a countable vertex of $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}}\rangle$. Instead we employ the embeddings of a simplified $(\omega_{1},2)$-morass to anticipate countable subsets of $\omega_{3}$ by countable subsets of $\omega_{1}$, and thereby construct a function between sets of terms in the forcing language having cardinality $\aleph_{3}$ by an inductive construction of length $\omega_{1}$.

The basic strategy of the central construction of this paper is to define a sequence of functions on terms in the forcing language adding generic reals indexed by $\omega_{1}$ with the enhanced back-and-forth argument similar to the argument in [References]. The term functions so constructed will be level and satisfy certain closure conditions with the second components of morass-embeddings. We use the embeddings of $\mathcal{F}_{\beta\omega_{1}}$ to “lift” the construction to a function on terms in the forcing language adding generic reals indexed by $\omega_{3}$. We use the following pair of Lemmas due to Velleman.

Lemma 3.1

(Velleman). Let $\alpha<\beta\leq\omega_{2}$, $g_{1},g_{2}\in\mathcal{G}_{\alpha\beta}$, $\tau_{1},\tau_{2}\in\varphi_{\alpha}$ and $g_{1}(\tau_{1})=g_{2}(\tau_{2})$. Then $\tau_{1}=\tau_{2}$ and $g_{1}\upharpoonright_{\tau_{1}+1}=g_{2}\upharpoonright_{\tau_{2}+1}$.

The gap-2 version of this is Lemma 2.2 [References]:

Lemma 3.2

(Velleman) Suppose $\alpha<\beta\leq\kappa$, $f_{1},f_{2}\in\mathcal{F}_{\alpha\beta}$, $\zeta_{1},\zeta_{2}\leq\theta_{\alpha}$, and $ssup(f[\zeta_{1}])=ssup(f[\zeta_{2}])$. Then $\zeta_{1}=\zeta_{2}$, $f_{1}\upharpoonright_{\zeta_{1}}=f_{2}\upharpoonright_{\zeta_{2}}$ and $(f_{1})_{\xi}=(f_{2})_{\xi}$ for all $\xi<\zeta_{1}$.

We show that every countable subset of $\omega_{3}$ is in the range of a morass function, albeit not necessarily from a countable vertex.

Lemma 3.3

Let $T$ be a countable subset of $\omega_{3}$. Then there is $\zeta<\omega_{2}$ and $g\in\mathcal{G}_{\zeta\omega_{2}}$ such that $T\subseteq g[\varphi_{\zeta}]$.

Proof. Let $\tau=sup(T)$. We assume that $\tau\in T$. Enumerate the elements of $T$, $\langle\tau_{n}\mid n\in\omega\rangle$, with $\tau=\tau_{0}$. By condition P5 of the definition of a simplified gap-1 morass there is $\zeta_{0}<\omega_{2}$, $\sigma_{0}\in\varphi_{\zeta_{0}}$ and $g_{0}\in\mathcal{G}_{\zeta_{0}\omega_{2}}$ such that $g_{0}(\sigma_{0})=\tau$. For $n\in\omega$, there is $\zeta^{\prime}<\omega_{2}$, $\sigma^{\prime}\in\varphi_{\zeta^{\prime}}$ and $g^{\prime}\in\mathcal{G}_{\zeta^{\prime}\omega_{2}}$ such that $g^{\prime}(\sigma^{\prime})=\tau_{n}$. By condition P4 of the definition of a simplified gap-1 morass there is $\zeta_{n}\geq max(\zeta_{0},\zeta^{\prime})$, $h_{0}\in\mathcal{G}_{\zeta_{0}\zeta_{n}}$, $h_{n}^{\prime}\in\mathcal{G}_{\zeta^{\prime}\zeta_{n}}$  and $g_{n}\in\mathcal{G}_{\zeta_{n}\omega_{2}}$ such that $g_{0}=g_{n}\circ h_{0}$ and $g^{\prime}=g_{n}\circ h_{n}^{\prime}$. We observe that

[TABLE]

and

[TABLE]

Hence if $n\in\omega$, $\tau$ and $\tau_{n}$ are in the image of a single morass map $g_{n}\in\mathcal{G}_{\zeta_{n}\omega_{2}}$.

Let $\omega_{2}>\zeta>\zeta_{n}$ for all $n\in\omega$. By condition P2 of the simplified gap-1 morass, there is $g\in\mathcal{G}_{\zeta\omega_{2}}$ and $g_{0}^{*}\in\mathcal{G}_{\zeta_{0}\zeta}$ such that

[TABLE]

We observe that

[TABLE]

Let $n\in\mathbb{N}$, $n>0$, $g_{n}^{*}\in\mathcal{G}_{\zeta_{n}\zeta}$ and $g^{*}\in\mathcal{G}_{\zeta\omega_{2}}$ be such that

[TABLE]

Then

[TABLE]

So by the previous lemma,

[TABLE]

and

[TABLE]

In particular, $\tau_{n}<\tau$ and $g\circ g^{*}_{n}(\sigma_{n})=\tau_{n}$. Hence, for all $n\in\omega$,

[TABLE]

$\Box$

Lemma 3.4

Assume

$\zeta<\omega_{2}$* and $g\in\mathcal{G}_{\zeta\omega_{2}}$* 2. 2.

$\beta_{1},\beta_{2}<\omega_{1}$** 3. 3.

For $i=1,2$, $f_{i}\in\mathcal{F}_{\beta_{i}\omega_{1}}$ 4. 4.

For $i=1,2$, $\bar{\zeta_{i}}<\theta_{\beta_{i}}$ and $f_{i}(\bar{\zeta_{i}})=\zeta$. 5. 5.

For $i=1,2$, $\bar{g}_{i}\in\mathcal{G}_{\bar{\zeta_{i}}\theta_{\beta_{i}}}$ and $\mathcal{G}_{\bar{\zeta_{i}}\theta_{\beta_{i}}}(\bar{g}_{i})=g$.

Then there is countable $\beta\geq\beta_{i}$, $i=1,2$, $f\in\mathcal{F}_{\beta\omega_{1}}$, $\bar{\zeta}\in\theta_{\beta}$, and $\bar{g}\in\mathcal{G}_{\bar{\zeta}\theta_{\beta}}$ such that

$f(\bar{\zeta})=\zeta$** 2. 2.

$f_{\bar{\zeta}\theta_{\beta}}(\bar{g})=g$** 3. 3.

For $i=1,2$, there are $f_{i}^{\prime}\in\mathcal{F}_{\beta_{i}\beta}$ such that $f_{i}=f\circ f_{i}^{\prime}$.

Proof: Let $\zeta$, $g$, $\beta_{i}$, $f_{i}$, $\bar{\zeta}_{i}$ and $\bar{g}_{i}$ ($i=1,2$), satisfy the hypotheses. By Condition 6 of Definition 2.1, there are $\beta$, $f$ and $f_{i}^{\prime}$ ($i=1,2$) such that $f_{i}=f\circ f_{i}^{\prime}$. Let $\bar{\zeta}=f_{1}^{\prime}(\bar{\zeta}_{1})$. Then $\bar{\zeta}<\theta_{\beta}$ and

[TABLE]

Since $f$ is an injection and $f_{2}(\bar{\zeta}_{2})=f\circ f_{2}^{\prime}(\bar{\zeta}_{2})=\zeta$,

[TABLE]

Let $\bar{g}=(f_{1}^{\prime})_{\bar{\zeta}_{1}\bar{\zeta}}(\bar{g}_{1})$. Then

[TABLE]

Since $f$ is an embedding,

[TABLE]

$\Box$

Definition 3.5

(Continuation) Let $\alpha<\beta<\gamma\leq\omega_{1}$ and $f\in\mathcal{F}_{\alpha\gamma}$. If $f^{\prime}\in\mathcal{F}_{\alpha\beta}$ and $f^{*}\in\mathcal{F}_{\beta\gamma}$ are such that $f=f^{*}\circ f^{\prime}$, then we say that $f^{*}$ is the $\beta$-continuation of $f$.

Condition 4 of Definition 2.1 states that, for any $\alpha<\beta<\gamma\leq\omega_{1}$, if $f\in\mathcal{F}_{\alpha\gamma}$, then $f$ has a $\beta$-continuation. Condition 6 of Definition 2.1 states that if $\beta_{1},\beta_{2}<\gamma\leq\omega_{1}$, $\gamma$ a limit ordinal, $f_{1}\in\mathcal{F}_{\beta_{1}\gamma}$ and $f_{2}\in\mathcal{F}_{\beta_{2}\gamma}$ then there is $\beta\geq\beta_{1},\beta_{2}$ such that $f_{1}$ and $f_{2}$ have a common $\beta$-continuation.

Lemma 3.6

Let $\zeta<\omega_{2}$, $g\in\mathcal{G}_{\zeta\omega_{2}}$ and $\langle(\beta_{n}^{\prime},f^{\prime}_{n},\zeta_{n}^{\prime},g_{n}^{\prime})\mid n\in\mathbb{N}\rangle$ be a sequence satisfying:

$f^{\prime}_{n}\in\mathcal{F}_{\beta_{n}^{\prime}\omega_{1}}$** 2. 2.

For all $n$, $\zeta_{n}^{\prime}<\theta_{\beta_{n}^{\prime}}$ and $f^{\prime}_{n}(\zeta_{n}^{\prime})=\zeta$ 3. 3.

For all $n$, $g_{n}^{\prime}\in\mathcal{G}_{\zeta_{n}^{\prime}\theta_{\beta_{n}^{\prime}}}$ and $(f^{\prime}_{n})_{\zeta_{n}^{\prime}\theta_{\beta_{n}^{\prime}}}(g_{n}^{\prime})=g$.

Then there is a sequence $\langle(\beta_{n},f_{n},\zeta_{n},g_{n})\mid n\in\mathbb{N}\rangle$ satisfying:

For all $m<n$, $\beta_{n}\geq\beta_{n}^{\prime}$ and $\beta_{n}\geq\beta_{m}$ 2. 2.

For all $n$, $f_{n}\in\mathcal{F}_{\beta_{n}\omega_{1}}$ 3. 3.

For all $n$, $f_{n}(\zeta_{n})=\zeta$ 4. 4.

For all $n$, $(f_{n})_{\zeta_{n}\theta_{\beta_{n}}}(g_{n})=g$ 5. 5.

For all $n$, $f_{n}$ is the $\beta_{n}$-continuation of $f^{\prime}_{n}$ 6. 6.

For all $m<n$, $f_{n}$ is the $\beta_{n}$-continuation of $f_{m}$.

The sequence $\langle(\beta_{n},f_{n},\zeta_{n},g_{n})\mid n\in\mathbb{N}\rangle$ is called an embedding sequence of $g$.

Proof: Let $\langle(\beta_{n}^{\prime},f^{\prime}_{n},\zeta_{n}^{\prime},g_{n}^{\prime})\mid n\in\mathbb{N}\rangle$ satisfy the hypothesis of the Lemma. We construct $\langle(\beta_{n},f_{n},\zeta_{n},g_{n})\rangle$ by induction. Let $(\beta_{0}^{\prime},f^{\prime}_{0},\zeta_{0}^{\prime},g_{0}^{\prime})=(\beta_{0},f_{0},\zeta_{0},g_{0})$. Assume $N\in\mathbb{N}$ and that $\langle(\beta_{n},f_{n},\zeta_{n},g_{n})\mid n\leq N\rangle$ satisfies the conclusion of the Lemma beneath $N$. By Condition 4 and Condition 6 of Definition 2.1, there is countable $\beta_{N+1}>\beta_{N},\beta^{\prime}_{N+1}$ and $f_{N+1}\in\mathcal{F}_{\beta_{N+1}\omega_{1}}$ that is the $\beta_{N+1}$-continuation of $f_{N}$ and $f^{\prime}_{N+1}$. Let $f_{N}=f_{N+1}\circ f^{\prime}$ where $f^{\prime}\in\mathcal{F}_{\beta_{N}\beta_{N+1}}$ and $f^{*}\in\mathcal{F}_{\beta_{N+1}\omega_{1}}$. Let $\zeta_{N+1}=f^{\prime}(\zeta_{N})$. Then

[TABLE]

Let $g_{N+1}=f^{\prime}_{\zeta_{N}\zeta_{N+1}}(g_{N})$. Then

[TABLE]

For any $n<N$, $F_{N+1}$ is the $\beta_{N+1}$-continuation of $f_{n}$. $\Box$

Definition 3.7

(Complete embedding sequence) Let $\zeta<\omega_{2}$ and $g\in\mathcal{G}_{\zeta\omega_{2}}$ and $S=\langle(\beta_{n},f_{n},\zeta_{n},g_{n})\mid n\in\mathbb{N}\rangle$ be an embedding sequence of $g$ and $D\subseteq\varphi_{\zeta}$. Then $S$ is a complete embedding sequence for $D$, with respect to $g$, provided that $D\subseteq\bigcup_{n\in\mathbb{N}}(f_{n})_{\zeta_{n}}[\varphi_{\zeta_{n}}]$.

If $S$ is a complete embedding sequence of $g$, then each embedding $f_{n}$ gives partial information about $g$. That is

[TABLE]

Later embeddings in the sequence, $f_{m}$ ($m>n$) necessarily agree with earlier embeddings about $g$, on $(f_{n})_{\zeta_{n}}[\varphi_{\zeta_{n}}]$, but in general will provide information about $g$ on a larger subset of $\varphi_{\zeta}$, $(f_{m})_{\zeta_{m}}[\varphi_{\zeta_{m}}]$. A complete embedding sequence will provide full information on $D$.

The second term of the embeddings also provides information about $g[D]$. For all $n\in\mathbb{N}$,

[TABLE]

If $m>n$, then

[TABLE]

and

[TABLE]

Lemma 3.8

Let $T\subseteq\omega_{3}$ be countable, $\zeta<\omega_{2}$, $g\in\mathcal{G}_{\zeta\omega_{2}}$ and $D\subseteq\varphi_{\zeta}$ such that $g[D]=T$. Then there is $\gamma<\omega_{1}$, $\bar{\zeta}<\theta_{\gamma}$, $h\in\mathcal{F}_{\gamma\omega_{1}}$ and $\bar{g}\in\mathcal{G}_{\bar{\zeta}\theta_{\gamma}}$ such that

[TABLE]

[TABLE]

and

[TABLE]

Proof: Let $T\subseteq\omega_{3}$, $\zeta<\omega_{2}$, $D\subseteq\varphi_{\zeta}$ and $g\in\mathcal{G}_{\zeta\omega_{2}}$ satisfy the hypothesis of the lemma. By Lemma 3.6 there is a complete embedding sequence for $D$ with respect to $g$, $S=\langle(\beta_{n},f_{n},\zeta_{n},g_{n})\mid n\in\mathbb{N}\rangle$. For all $m>n\geq 0$, $f_{m}$ is the $\beta_{m}$-continuation of $f_{n}$. Therefore, for all $n\in\mathbb{N}$, there is $\rho_{n}\in\varphi_{\zeta_{n}}$ such that

[TABLE]

Let $\gamma$ be a countable and, for all $n\in\mathbb{N}$, $\beta_{n}<\gamma$. For $n\in\mathbb{N}$, let $h_{n}\in\mathcal{F}_{\gamma\omega_{1}}$ be the $\gamma$-continuation of $f_{n}$. Let $h=h_{0}$. For each $n\in\mathbb{N}$, there is $f_{n}^{*}\in\mathcal{F}_{\beta_{n}\gamma}$ such that

[TABLE]

Let $\bar{\zeta}=f_{0}^{*}(\zeta_{0})$. For all $n>0$, $f_{n}$ is the $\beta_{n}$-continuation of $f_{0}$. Hence

[TABLE]

We observe that

[TABLE]

By a Lemma 3.2,

[TABLE]

and

[TABLE]

Since $S$ is a complete embedding sequence with respect to $g$,

[TABLE]

Then

[TABLE]

Let $\bar{g}=(f^{*}_{0})_{\zeta_{0}\theta_{0}}(g_{0})\in\mathcal{G}_{\bar{\zeta}\theta_{\gamma}}$. Then

[TABLE]

So

[TABLE]

However

[TABLE]

Therefore

[TABLE]

$\Box$

Definition 3.9

(Compatible Maps) Let $\theta\leq\kappa$ be ordinals, $f:\theta\to\kappa$ and $g:\theta\to\kappa$ be injections. The ordinal maps $f$ and $g$ are compatible provided that for any $\alpha,\beta\in\theta$, $f(\alpha)=g(\beta)$ implies $\alpha=\beta$.

If $f$ and $g$ are compatible, then $(f\cup g)^{-1}$ is a well-defined function. Lemma 3.1 implies that morass-maps with the same domain and codomain are compatible.

Lemma 3.10

Let $\beta<\omega_{1}$, $\eta$ be the splitting point of the right-branching embedding of $\mathcal{F}_{\beta\beta+1}$, $f,g\in\mathcal{F}_{\beta\beta+1}$ and $\bar{f},\bar{g}\in\mathcal{G}_{\eta\theta_{\beta}}$. Then $f_{\theta_{\beta}}\circ\bar{f}$ and $g_{\theta_{\beta}}\circ\bar{g}$ are compatible. If $p\in P_{\beta}$ then $f_{\theta_{\beta}}(p)$ and $g_{\theta_{\beta}}(p)$ are compatible conditions in $P$.

Proof. If $f$ and $g$ are left-branching embeddings then $f_{\theta_{\beta}}\circ\bar{f},g_{\theta_{\beta}}\circ\bar{g}\in\mathcal{G}_{\eta\theta_{\beta+1}}$. By Lemma 3.1 they are compatible. Assume that $f$ is the right-branching embedding of $\mathcal{F}_{\beta\beta+1}$. Then

[TABLE]

Therefore, by Lemma 3.1, $f_{\theta_{\beta}}\circ\bar{f}$ and $g_{\theta_{\beta}}\circ\bar{g}$ are compatible.

Let $p\in P_{\beta}$, $g$ and $h$ be second components of embeddings of $\mathcal{F}_{\beta\beta+1}$. So $g$ and $h$ are members of $\mathcal{G}_{\theta_{\beta}\theta_{\beta+1}}$ or are equal to $f_{\theta_{\beta}}$. Let $(\alpha,k,s)\in g(p)$, $(\alpha,k,t)\in h(p)$, $g(\alpha^{\prime})=\alpha$ and $h(\alpha^{*})=\alpha$. Then

[TABLE]

and

[TABLE]

Morass maps are compatible, hence if $g,h\in\mathcal{G}_{\theta_{\beta}\theta_{\beta+1}}$ then $\alpha^{\prime}=\alpha^{*}$ and $s=t$.

Suppose that $h=f_{\theta_{\beta}}$. Then there is $\bar{\alpha}\in\varphi_{\eta}$ (where $\eta$ is the splitting point of $f$) and $\bar{h}\in\mathcal{G}_{\eta\theta_{\beta}}$ such that $\bar{h}(\bar{\alpha})=\alpha^{*}$. So

[TABLE]

Since $f_{\eta\theta_{\beta}}(\bar{h})\in\mathcal{G}_{\theta_{\beta}\theta_{\beta+1}}$,

[TABLE]

Therefore $(\alpha^{\prime},k,t)\in p$ and $s=t$. $\Box$

4 The Gap-1 Construction

In [References], we constructed an order-isomorphism between morass-definable $\eta_{1}$-orderings in the generic extension adding $\aleph_{2}$-generic reals of a model of $ZFC+CH$ containing a simplified $(\omega_{1},1)$-morass. This was accomplished by an inductive construction of length $\omega_{1}$ on terms in the forcing language for adding $\aleph_{1}$ generic reals, subject to numerous technical constraints, and using the morass maps to “lift” the construction to the generic extension adding $\aleph_{2}$ generic reals. The benefit of the gap-1 morass is that the necessary back-and-forth construction occurs strictly in the forcing language for adding only $\aleph_{1}$ generic reals, relying on the morass-maps for the completion of the overall construction.

The conditions required for the gap-1 construction were that the $\eta_{1}$-orderings be morass-definable. We present some adjustments of definitions from the gap-1 construction that will serve us in the gap-2 construction.

Let $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$ be a simplified $(\kappa,2)$-morass.

Notation 4.1

($P_{\zeta}$, $P(A)$, $M^{P_{\zeta}}$, $M^{P(A)}$) If $\zeta\leq\kappa$, $P_{\zeta}$ is the poset adding generic reals indexed by $\varphi_{\theta_{\zeta}}$ and $M^{P_{\zeta}}$ is the set of terms in the forcing language of $P_{\zeta}$. For $A\subseteq\kappa^{++}$, $P(A)$ is the poset for adding generic reals indexed by $A$, and $M^{P(A)}$ is the set of terms in the forcing language of $P(A)$.

So $P=P_{\kappa}=P(\kappa^{++})$.

Definition 4.2

(Support) Let $\tau\in M^{P}$. The support of $\tau$, $supp(\tau)$, is the minimal subset of $\omega_{3}$, $A$, such that $\tau\in M^{P(A)}$. We say that $\tau$ has countable support provided that $A$ is countable.

Let $A\subseteq B\subseteq\omega_{3}$. We consider $P(B)=P(A)\times P(B\setminus A)$. If $G$ is $P$-generic over $M$, then $M[G]=M[G(A)\times H]$ where $G(A)$ is $P(A)$-generic over $M$ and $H$ is $P(B\setminus A)$-generic over $M[G(A)]$. If $\tau$ is a term of the forcing language of $P(A)$, the value of $\tau$ in the generic extensions will be an element of $M[G(A)]$.

Definition 4.3

(Strict Support) Let $B\subseteq\kappa$ and $\tau\in M^{P(B)}$. The term $\tau$ has strict support $B$ if there is no proper subset $A\subset B$, $p\in P$ and $\sigma\in M^{P(A)}$ such that $p\Vdash\sigma=\tau$.

If a term, $\tau$, has strict support $B$, then $val_{G(B)}(\tau)$ will not be an element of $M[G(A)]$, for any proper subset $A$ of $B$.

Definition 4.4

(Discerning Set of Terms) A set of terms, $X\in M$, $X\subseteq M^{P}$, is discerning provided that every term of $X$ has strict support.

Definition 4.5

(Level Function) Let $X$ and $Y$ be discerning sets of terms and $\phi:X\to Y$. The function $\phi$ is level if for any $x\in X$, $x$ and $\phi(x)$ have identical strict support.

Definition 4.6

(Morass-Commutative) Let $\kappa$ be regular, $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}}\rangle$ be a simplified $(\kappa,1)$-morass, $\lambda\leq\kappa$ and $X\subseteq M^{P(\kappa^{+})}$. We say that $X$ is morass-commutative beneath $\lambda$ provided that for any $\zeta<\xi\leq\lambda$ and $g\in\mathcal{G}_{\zeta\xi}$, $x\in X\cap M^{P_{\zeta}}$ iff $g(x)\in X\cap M^{P_{\xi}}$. We say that $X$ is morass-commutative if it is morass-commutative beneath $\kappa$.

Definition 4.7

(Embedding-Commutative) Let $\kappa$ be regular, $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$ be a simplified $(\kappa,2)$-morass, $\lambda\leq\kappa$ and $X\subset M^{P}$. We say that $X$ is embedding-commutative beneath $\lambda$ provided that for any $\zeta<\xi\leq\lambda$ and $f\in\mathcal{F}_{\zeta\xi}$, $x\in X\cap M^{P_{\zeta}}$ iff $f_{\theta_{\zeta}}(x)\in X\cap M^{P_{\xi}}$. We say that $X$ is embedding-commutative if it is embedding-commutative beneath $\kappa$.

Morass-commutativity and embedding-commutativity extend to relations and functions on terms in the obvious way.

Definition 4.8

(Grounded Order-Support) Let $T_{X}\in M^{P}$ be forced to be a linear-ordering and $X\subseteq M^{P}$ be a discerning set of terms for the domain of $T_{X}$. $X$ has grounded order-support provided that for all $x,y\in X$ and $G$, $P$-generic over $M$, there is $z\in M\cap X$ such that $M[G]\models x<z<y$.

If $X$ has grounded order-support, then for all $x,y\in X$, $\Vdash\exists z\in M(x<z<y)$ and for all $p\in P$ with $p\Vdash x<y$, there is $z\in X\cap M$ and $q\leq p$ such that $q\Vdash x<z<y$. For instance, $\mathbb{R}$ has grounded support.

Definition 4.9

(Upward Level-Dense) Let $T_{X}\in M^{P}$ be forced to be a linear-ordering, and $X\subseteq M^{P}$ be a set of discerning terms for the domain of $T_{X}$. $T_{X}$ is upward level-dense provide that for every $x,y,z\in X$, in which $z$ has strict order support $A\subseteq\kappa$, $p\in P$ with $p\Vdash x<z<y$ and $B\supseteq A$, there is a discerning term $w$ with strict support $B$, such that $p\Vdash x<w<y$.

If a set of discerning terms has grounded order-support, and terms bear an order relation in a generic extension, then there is an element of the ground model that is between the elements. If a set of discerning terms is upward level-dense, then between any pair of elements there are elements of arbitrarily large strict support (in the sense of containment) between them.

We will require that the sequence of term functions we are constructing is closed under second components of morass-embeddings between our “fake morasses”. That is, for $\alpha<\beta\leq\omega_{1}$, and $f\in\mathcal{F}_{\alpha\beta}$, we require that the partial term function constructed at level $\alpha$, is closed under maps, $f_{\theta_{\alpha}}:\varphi_{\theta_{\alpha}}\to\varphi_{\theta_{\beta}}$.

We revise the definition of morass-definability for extension to the gap-2 construction.

Definition 4.10

(Morass-Definable) Let $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$ be a simplified $(\omega_{1},2)$-morass, $P$ be the poset that adds generic reals indexed by $\omega_{3}$, $X\in M^{P}$ be a discerning set of terms and $R\subset X\times X$. We say $\langle X,R\rangle$ is a morass-definable $\eta_{1}$-ordering (with respect to $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$) provided that

For every $A\subseteq\omega_{2}$, $\langle X,R\rangle\cap(M^{P(A)})^{3}$ is forced to be an $\eta_{1}$-ordering 2. 2.

$X$* and $R$ are morass-commutative and embedding-commutative* 3. 3.

$X$* has grounded order-support and is upward level-dense* 4. 4.

Every term of $X$ has countable support.

If $G$ is $P$-generic over $M$ and $\langle\bf{X},\bf{R}\rangle\in M[G]$ is an $\eta_{1}$-ordering we say that $\langle\bf{X},\bf{R}\rangle\in M[G]$ is morass-definable provided there is morass-definable $\langle X,R\rangle$ with $val_{G}(\langle X,R\rangle)=\langle\bf{X},\bf{R}\rangle$.

If $\langle X,R\rangle\cap{M^{P_{\theta_{\beta}}}}$ satisfies the conditions of Definition 4.10 below $\beta$, then we say that $\langle X,Y\rangle$ is morass-definable below $\beta$.

If $M$ is a c.t.m. of $ZFC+CH$ containing a simplified $(\omega_{1},1)$-morass and $M[G]$ is the Cohen extension adding $\aleph_{2}$ generic reals, then in $M[G]$, morass-definable $\eta_{1}$-orderings are order-isomorphic. Furthermore, if $P$ is the poset for adding $\aleph_{2}$ generic reals, and $(X,<_{X})$ and $(Y,<_{Y})$ are morass-definable and are forced to be $\eta_{1}$-orderings, there is a level term function, $\phi:X\to Y$, that is forced in all $P$-generic extensions to be an order-isomorphism.

5 The Gap-2 Construction

In the proof that morass-definable $\eta_{1}$-orderings are order-isomorphic in the Cohen extension adding $\aleph_{2}$-generic reals, we built a term function with cardinality $\aleph_{2}$ by an inductive construction of length $\omega_{1}$. The constraint on the length of the chain is governed by the possibility that we are unable to to satisfy uncountably many simultaneous consistent order-constraints when committing to discrete extensions. The construction depends on commutative extensions by morass maps to “lift up” the construction to exhaust the required commitments in the forcing language adding $\aleph_{2}$ generic reals. In the argument using the gap-1 morass, all commitments can be met provided that any countable subset of $\omega_{2}$ is in the range of a single morass map from a countable vertex to the $\omega_{1}$ vertex.

This strategy does not extend to higher cardinality. Instead, we work in a model of $ZFC+CH$ that contains a simplified $(\omega_{1},2)$-morass, $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$, using the morass-embeddings between “fake” morasses, that is, initial segments of the gap-two morass for vertices $\theta_{\beta}$, $\beta\leq\omega_{1}$. The maps $f_{\theta_{\beta}}:\varphi_{\theta_{\beta}}\to\omega_{3}$, for embeddings $f\in\mathcal{F}_{\beta\omega_{1}}$, “lift” a term function on a domain of cardinality $\aleph_{1}$ to a term function on a domain with cardinality $\aleph_{3}$. Provided the term function constructed at stage $\omega_{1}$ exhausts $X_{\omega_{1}}$ and $Y_{\omega_{1}}$, the technical results of Section 3 allow us to complete the construction of a term function on $X$ by closure under second components of morass-embeddings of $\mathcal{F}_{\beta\omega_{1}}$, $\beta<\omega_{1}$.

In Section 4 we adapted the key definitions from [References] to a higher cardinality gap-1 morass and a gap-2 morass. These definitions concern technical considerations for building a function between sets of terms in the forcing language for adding Cohen generic reals that may be extended by compatible injections on ordinals. In passing from the gap-1 construction to the gap-2 construction, we need to use the gap-1 morass maps of $\mathcal{G}$ and the second components of morass-embeddings $\mathcal{F}$, to extend a partial construction of a term function below $\omega_{1}$ by closure under second components of embeddings to a term function in the forcing language adding generic reals indexed by $\omega_{3}$. That is, we will make all discrete decisions extending a function in an enhanced back-and-forth construction of length $\omega_{1}$, and “lift” the entire construction by second components of embeddings of $\mathcal{F}_{\beta\omega_{1}}$ to exhaust terms with countable support in the language adding generic reals indexed by $\omega_{3}$.

Assume that $\theta\leq\phi$ are ordinals, $f:\theta\to\phi$ is an injection, and that $\tau$ is a term in the forcing language adding generic reals indexed by $\theta$. We define $f(\tau)$ to be the term in the forcing language adding generic reals indexed by $\phi$ that results from the formal substitution of every indexing ordinal in $\theta$ appearing in $\tau$ by its image under $f$. So $f(\tau)$ is a term in the forcing language adding generic reals indexed by $f[\theta]\subseteq\phi$. Morass maps of $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}}\rangle$ and the first and second components of morass embeddings of $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$, are order-preserving injections on ordinals. We treat morass maps and second components of morass-embeddings as functions between terms of a forcing language by this convention.

The morass maps of a simplified gap-1 morass have a restricted character. There are only two morass maps from a morass vertex to its successor: identity on an associated ordinal, and a single “splitting” map that translates the cofinal end segment of that ordinal. We need to consider a new collection of ordinal maps along the morass that will enrich the terms we can define by closure under ordinal injections, the second components of the morass embeddings, $f_{\theta_{\alpha}}:\varphi_{\theta_{\alpha}}\to\varphi_{\theta_{\beta}}$, for $f\in\mathcal{F}_{\alpha\beta}$ ($\alpha<\beta\leq\omega_{1})$. We are able to restrict our attention to the morass-embeddings of the amalgamations $\mathcal{F}_{\alpha\alpha+1}$. Each $f_{\theta_{\alpha}}$ is an order-preserving injection, but may split its domain into many disconnected pieces that have less predictable intersections with morass maps.

Definition 5.1

(Embedding-closure) Let $\beta<\gamma\leq\omega_{1}$ and $X$ be a set of terms in the forcing language $M^{P}_{\gamma}$. The closure of $X$ under $\mathcal{F}_{\beta\gamma}$ is the smallest set of terms of $M^{P}_{\gamma}$ such that $x\in X\cap M^{P_{\beta}}$ and $f\in\mathcal{F}_{\beta\gamma}$ implies $f_{\theta_{\beta}}(x)\in X$. The set $X$ is closed under $\mathcal{F}_{\beta\gamma}$ if the closure of $X$ is $X$. If $\beta<\omega_{1}$, and $X$ is closed under $\mathcal{F}_{\beta\beta+1}$, then we say that $X$ is embedding-closed at level $\beta$.

We will require that our sequence of functions be embedding-closed at level $\beta$ for all $\beta<\omega_{1}$. We prove a lemma that allows for a transfinite construction of a sequence of embedding-closed term functions along a simplified gap-2 morass. If $X$ is a morass-definable $\eta_{1}$-ordering, Then $X$ has grounded order-support. If $D\subset X$ is countable, then c.c.c of $P$ implies that there is a countable extension of $D$ by elements of the ground model that has grounded order-support.

Lemma 5.2

Let $\beta$ be a countable ordinal and assume

$\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$* is a simplified $(\omega_{1},2)$-morass* 2. 2.

$X$* and $Y$ are morass-definable $\eta_{1}$-orderings* 3. 3.

$D\subseteq X\cap M^{P_{\theta_{\beta}}}$* has grounded order-support and is morass-commutative* 4. 4.

$F:D\to Y$* is a morass-commutative level term injection that is forced to be order-preserving in all generic extensions.*

Then $\phi=\bigcup\{f_{\theta_{\beta}}[F]\mid f\in\mathcal{F}_{\beta\beta+1}\}$ is a level term injection that is forced to be an order-preserving injection.

Proof: Let $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$, $X$, $Y$, $D$ and $F$ satisfy the hypotheses of the lemma, and $\phi=\bigcup\{f_{\theta_{\beta}}[F]\mid f\in\mathcal{F}_{\beta\beta+1}\}\subseteq X\times Y$. The family of embeddings, $\mathcal{F}_{\beta\beta+1}$ is an amalgamation, so it is composed of a single right-branching $f\in\mathcal{F}_{\beta\beta+1}$, and all possible left-branching embeddings. For each left branching embedding of $g\in\mathcal{F}_{\beta\beta+1}$, $g_{\theta_{\beta}}\in\mathcal{G}_{\theta_{\beta}\theta_{\beta+1}}$. The second components of embeddings of $\mathcal{F}$ are injections. Therefore $\phi$ is level, and the elements of the range of $\phi$ are discerning terms in $Y$.

To see that $\phi$ is forced to be a well-defined function, suppose that

$(x_{1},y_{1}),(x_{2},y_{2})\in F$; $f,g\in\mathcal{F}_{\beta\beta+1}$; $p\in M^{P_{\theta_{\beta+1}}}$ and $p\Vdash f_{\theta_{\beta}}(x_{1})=g_{\theta_{\beta}}(x_{2})$. Since $x_{1}$ and $x_{2}$ are discerning terms, so are $f_{\theta_{\beta}}(x_{1})$ and $g_{\theta_{\beta}}(x_{2})$. Therefore $f_{\theta_{\beta}}(x_{1})$ and $g_{\theta_{\beta}}(x_{2})$ must have identical strict support, $S$, and $p$ may be replaced by a condition of $Fn(S\times\omega,2)$. We claim that $p\Vdash f_{\theta_{\beta}}(y_{1})=g_{\theta_{\beta}}(y_{2})$. We observe that for $\alpha\in S$, there are $\alpha_{1},\alpha_{2}\in\varphi_{\theta_{\beta}}$ such that $f_{\theta_{\beta}}(\alpha_{1})=g_{\theta_{\beta}}(\alpha_{2})=\alpha$. Then there is $\bar{\alpha}\in\theta_{\eta}$ and $\bar{h}\in\mathcal{G}_{\eta\theta_{\beta}}$ such that $\bar{h}(\bar{\alpha})=\alpha_{1}$. However,

[TABLE]

So $f_{\eta}(\bar{\alpha})=\alpha_{2}$ and $f_{\eta\theta_{\beta}}(\bar{h})(\alpha_{2})=g_{\theta_{\beta}}(\alpha_{2})$. By Lemma 3.1, $f_{\eta\theta_{\beta}}(\bar{h})$ and $g_{\theta_{\beta}}$ are equal up to $\alpha_{2}$. Specifically, if $\gamma\in\varphi_{\eta}$, $h\in\mathcal{G}_{\eta\theta_{\beta}}$ and $h(\gamma)\in supp(x_{1})$, then $f_{\theta_{\beta}}\circ h(\gamma)=g_{\theta_{\beta}}\circ f_{\eta}(\gamma)$.

Define a relation $G\subseteq\varphi_{\eta}\times\varphi_{\theta_{\beta}}$ by $(\gamma,\lambda)\in G$ iff there is $h\in\mathcal{G}_{\eta\theta_{\beta}}$ with $\lambda\in supp(x_{1})$ and $h(\gamma)=\lambda$. We claim that $G$ is a well-defined function and order-preserving injection. If $h_{1}$ and $h_{2}$ are witnessing functions from the definition of $G$ for $\gamma$, then $f_{\theta_{\beta}}\circ h_{1}(\gamma)=g_{\theta_{\beta}}\circ f_{\eta}(\gamma)=f_{\theta_{\beta}}\circ h_{2}(\gamma)$. Since $f_{\theta_{\beta}}$ is an injection, $h_{1}(\gamma)=h_{2}(\gamma)$, and $G$ is a well-defined function. Assume that $\gamma_{1},\gamma_{2}\in\varphi_{\eta}$ and $G(\gamma_{1})=G(\gamma_{2})=\lambda$. Then

[TABLE]

Therefore $\gamma_{1}=\gamma_{2}$ and $G$ is an injection. So

[TABLE]

Therefore $f_{\eta}\circ G^{-1}(x_{1})=x_{2}$. Since $F$ is morass-commutative, and $D$ has grounded order-support, $\phi\upharpoonright_{X_{\eta}}$ is morass-commutative. Hence

[TABLE]

Therefore $\phi$ is forced to be a well-defined function. The proof that $\phi$ is forced to be an injection is essentially identical.

We show that $\phi$ is forced to be order-preserving. Consider $\phi$ as a function on $\{f_{\theta_{\beta}}[D]\mid f\in\mathcal{F}_{\beta\beta+1}\}$. Let $f,g\in\mathcal{F}_{\beta\beta+1}$. If $x\in M\cap D$, then $\phi(x)=F(x)$. Assume $x_{1},x_{2}\in D$ and $p\Vdash x_{1}<x_{2}$. Let $y_{1}=F(x_{1})$ and $y_{2}=F(x_{2})$ and Since $D$ has a grounded order-support, there is $z$ in $M\cap D$ and $q\leq p$ such that $q\Vdash x_{1}<z<x_{2}$. Then

[TABLE]

So

[TABLE]

Therefore it is forced that $\phi$ is order-preserving. $\Box$

Theorem 5.3

Let $M$ be a model of $ZFC+CH$ containing a simplified $(\omega_{1},2)$-morass, $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$, and $P=Fn(\omega_{3}\times\omega,2)$. Let $X$ and $Y$ be sets of discerning terms in $M^{P}$ for morass-definable $\eta_{1}$-orderings (with respect to $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$). Then there is a level function from $X$ to $Y$ that is forced to be an order-isomorphism.

Proof. For $\alpha<\omega_{1}$, let $X_{\alpha}=X\cap M^{P_{\alpha}}$ and $Y_{\alpha}=Y\cap M^{P_{\alpha}}$. We consider $X_{\alpha}$ and $Y_{\alpha}$ as the restrictions of $X$ and $Y$, resp., to the forcing language adding generic reals indexed by $\varphi_{\theta_{\alpha}}$. In any $P$-generic extension of $M$, $M[G]$, the interpretation of $X_{\alpha}$ in $M[G]$ is the interpretation of $X_{\alpha}$ in $M[G_{\alpha}]$ where $G_{\alpha}$ is the factor of $G$ that is $P_{\alpha}$-generic over $M$.

We will use the previous Lemma to construct a morass-commutative level term bijection from $X_{\omega_{1}}$ to $Y_{\omega_{1}}$ that is forced to be an order-isomorphism. The closure under embeddings, $f_{\theta_{\beta}}$ where $f\in\mathcal{F}_{\beta\omega_{1}}$, of this function will be the term function we seek.

Let $\langle x_{\beta}\mid\beta<\omega_{1}\rangle$ be an enumeration of $X_{\omega_{1}}$ that satisfies the condition $x_{\alpha}\in X_{\alpha}$ for all $\alpha<\beta$. Let $\langle y_{\beta}\mid\beta<\omega_{1}\rangle$ satisfy the same condition with respect to $Y_{\omega_{1}}$.

We will inductively construct a transfinite sequence of morass-commutative term functions $\langle F_{\beta}:D_{\beta}\to E_{\beta}\mid\beta<\omega_{1}\rangle$ that satisfies the following for all $\alpha\leq\beta<\omega_{1}$,

$D_{\beta}\subseteq X_{\theta_{\beta}}$ and $E_{\beta}\subseteq Y_{\theta_{\beta}}$ are morass-commutative, countable sets of terms with grounded order support 2. 2.

$D_{\alpha}\subseteq D_{\beta}$ and $E_{\alpha}\subseteq E_{\beta}$ 3. 3.

$x_{\beta}\in D_{\beta}$ and $y_{\beta}\in E_{\beta}$ 4. 4.

$F_{\beta}$ is a morass-commutative level term function that is forced to be an order-preserving bijection 5. 5.

$f_{\theta_{\alpha}}[F_{\alpha}]\subseteq F_{\beta}$ for all $f\in\mathcal{F}_{\alpha\beta}$

We call a sequence of term functions satisfying these conditions (beneath $\beta$) and extendable sequence. We argue be induction on $\gamma<\omega_{1}$

Case 1: $\gamma=0$. Then $x_{0}$ and $y_{0}$ are elements of the ground model, $M$. Let $F_{0}=\{(x_{0},y_{0})\}$.

Case 2: $\gamma=\beta+1$. Let $\langle F_{\alpha}:D_{\alpha}\to E_{\alpha}\mid\alpha\leq\beta\rangle$ be an extendable sequence. By Lemma 5.2, $(\bigcup_{f\in\mathcal{F}_{\beta\gamma}}f_{\theta_{\beta}}[F_{\beta}]):\bigcup_{f\in\mathcal{F}_{\beta\gamma}}f_{\theta_{\beta}}[D_{\beta}]\to\bigcup_{f\in\mathcal{F}_{\beta\gamma}}f_{\theta_{\beta}}[E_{\beta}]$ is forced to be an order-preserving bijection. Let $S=supp(x_{\gamma})$ and $\alpha\leq\gamma$ be the least ordinal for which there exists $\sigma\in G_{\alpha\varphi_{\gamma}}$ and $\bar{S}\subseteq\varphi_{\alpha}$ such that $\sigma[\bar{S}]=S$. Then there is $\bar{x}\in M^{P(\varphi_{\alpha})}$ such that $\sigma(\bar{x})=x_{\gamma}$. Let $D^{*}=\{\sigma(\bar{x})\mid\sigma\in\mathcal{G}_{\alpha\theta_{\gamma}})\}\subseteq X$. Then $D^{*}$ is the smallest morass-commutative subset of $X_{\beta+1}$ that contains $x_{\gamma}$. Let $D^{\prime}\subset X$ be a countable subset of $X_{0}$ so that $D=D^{\prime}\cup D^{*}$ has grounded order-support. By repeated applications of Lemma 4.5 of [References], there is $F^{\prime}:(D^{\prime}\cup D_{\beta})\to Y$ that is a morass-commutative level term function that is forced to be order-preserving and $D^{\prime}\cup D_{\beta}$ has grounded order-support. By Lemma 4.5, [References], there is $\bar{y}\in Y$ such that $F^{\prime}\cup\{(\bar{x},\bar{y})\}$ is a level term function that is forced to be order-preserving. Let $F=F^{\prime}\cup\{f_{\theta_{\beta}}\circ g((\bar{x},\bar{y}))\mid f\in\mathcal{F}_{\beta\gamma},g\in\mathcal{G}_{\alpha\theta_{\beta}}\}$. Then $F$ is a level term function that is forced to be a bijection. To see that $F$ is order-preserving, assume that $f,h\in\mathcal{F}_{\beta\gamma}$, $g_{1},g_{2}\in\mathcal{G}_{\alpha\theta_{\beta}}$, $x_{1}=f_{\theta_{\beta}}\circ g_{1}(\bar{x})$, $x_{2}=h_{\theta_{\beta}}\circ g_{2}(\bar{x})$, $G$ is $P_{\gamma}$-generic over $M$ and $M[G]\models x_{1}<x_{2}$. Since the domain of $F$ has grounded order-support, there is $x_{0}\in D^{\prime}$ such that $M[G]\models x_{1}<x_{0}<x_{2}$. Then

[TABLE]

Therefore $F$ is forced to be order-preserving.

Let $E$ be the range of $F$. Let $E^{*}$ be the smallest morass-commutative subset of $Y_{\beta+1}$ that contains $y_{\gamma}$. Then $E$ has grounded order-support and there is a countable extension of $E\cup E^{*}$ by elements of $Y_{0}$, $E_{\gamma}$, so that $E_{\gamma}$ has grounded order-support. Again, by applications of Lemma 4.5 [References], there is a level injection, $F_{\gamma}$, such that $F^{-1}_{\gamma}:E_{\gamma}\to X$ is a level injection extending $F^{-1}$ that is forced to be an order-preserving injection. Let $D_{\gamma}$ be the range of $F^{-1}_{\gamma}$. Then $\langle F_{\alpha}\mid\alpha\leq\gamma\rangle$ is an extendable sequence.

Case 3: $\gamma$ a limit ordinal. It is routine to verify that $F=\bigcup_{\alpha<\gamma,f\in\mathcal{F}_{\alpha\gamma}}f_{\theta_{\alpha}}[F_{\alpha}]$ is a level injection that is forced to be an order-preserving injection. In a manner identical to the successor case, $F$ may be extended to a level injection, with domain containing $x_{\gamma}$ and range containing $y_{\gamma}$, both bearing grounded order-support, in which $F$ is forced to be an order-preserving injection.

Let $F=\bigcup_{\alpha<\omega_{1},f\in\mathcal{F}_{\alpha\omega_{1}}}f_{\theta_{\alpha}}[F_{\alpha}]$. Then $F:X\to Y$ is a level bijection that is forced to be an order-isomorphism. By Lemma 3.8, the domain of $F$ is $X$ and the range of $F$ is $Y$. $\Box$

6 Ultrapowers of $\mathbb{R}$ over $\omega$

We turn our attention to ${\mathbb{R}}^{\omega}/U$, an ultrapower of $\mathbb{R}$ over a non-principal ultrafilter on $\omega$, $U$. By results of G. Dales [References], J. Esterle [References] and B. Johnson [References] the existence of an $\mathbb{R}$-linear order-preserving monomorphism from the finite elements of ${\mathbb{R}}^{\omega}/U$, for some non-principal ultrafilter on $\omega$, $U$, into the Esterle algebra is sufficient to prove the existence of a discontinuous homomorphism of $C(X)$, the algebra of continuous real valued functions on $X$, where $X$ is an infinite compact Hausdorff space. In a later paper, relying on the techniques of this paper, we prove that it is consistent that such a monomorphism exists in a model of set theory in which $2^{\aleph_{0}}=\aleph_{3}$. In anticipation of such a construction, we finish this paper with a proof that in the Cohen extension adding $\aleph_{3}$-generic reals of a model of ZFC+CH containing a simplified $(\omega_{1},2)$-morass, there is an ultrapower of $\mathbb{R}$ over a standard ultrafilter on $\omega$ that is gap-2 morass-definable, and hence is order-isomorphic with other gap-2 morass-definable $\eta_{1}$-orderings. This result extends Theorem 6.15 [References], that in the Cohen extension adding $\aleph_{2}$-generic reals of a model of $ZFC+CH$ containing a simplified gap-1 morass, ultrapowers of $\mathbb{R}$ over standard ultrafilters on $\omega$ are order-isomorphic.

By Theorem 5.3, in order to prove that an ultrapower of $\mathbb{R}$, ${\mathbb{R}}^{\omega}/U$, is order-isomorphic with a gap-2 morass-definable $\eta_{1}$-ordering, it sufficient to prove that ${\mathbb{R}}^{\omega}/U$ is gap-2 morass-definable. We show that this is so, provided that $U$ is a standard non-principal ultrafilter. We adapt the definition of standard ultrafilter (Definition 6.14 [References]) to a simplified gap-2 morass.

Definition 6.1

(Standard Term for a subset of $\omega$) A standard term for a subset of $\omega$ is a term $x\in M^{P}$, such that for each $(\tau,p)\in x$, $\tau$ is a canonical term in $M^{P}$ for a natural number.

Definition 6.2

(Standard Term for an Ultrafilter) Let $\lambda\leq\omega_{3}$ and $U\in M^{P({\lambda})}$ be a morass-commutative and embedding-commutative set of standard terms for subsets of $\omega$ (below $\lambda$) such that for all $S\subseteq\lambda$, $U\cap M^{P(S)}$ is forced to be an ultrafilter in all $P(S)$-generic extensions for $M$. Then $U$ is a standard term for an ultrafilter below $\lambda$.

If $U$ is a standard term for an ultrafilter below $\omega_{3}$, we say it is a standard ultrafilter.

Definition 6.3

(Complete Standard Term for an Ultrafilter) Let $U$ be a standard term for an ultrafilter below $\lambda$. $U$ is complete provided that for every standard term for a subset of $\omega$, $u\in M^{P(\lambda)}$, $u\in U$ iff $\Vdash u\in U$.

That is, below $\lambda$, every standard term for a subset of $\omega$ that is forced to be in $U$, is a member of $U$. Every standard term for an ultrafilter has a complete extension. Let $U$ be a complete standard term for an ultrafilter and $u$ be a standard term for a subset of $\omega$. There is a standard term for a subset of $\omega$ that decides the membership of $u$ in $U$ in all $P(\lambda)$-generic extensions. Since $U$ is forced to be an ultrafilter, $\{p\in P(\lambda)\mid p\Vdash u\in U\vee p\Vdash u\notin U\}$ is dense in $P(\lambda)$. We consider $u$ as a term for a binary sequence. In this sense, $p\Vdash n\in u$ iff $p\Vdash u_{n}=1$. It is clear that there is a standard term for a binary sequence, $v$, such that $p\Vdash u_{n}=1$ iff $p\Vdash v_{n}=0$. Let $d(u)$ be the standard term for a subset of $\omega$ so that it is forced that $d(u)=u$ if $p\Vdash u\in U$ and $d(u)=v$ if $p\Vdash u\notin U$. Then it is forced in all generic extensions that $d(u)\in U$.

We wish to construct a standard term for an ultrafilter that commutes with the second components of embeddings of $\mathcal{F}_{\beta\beta+1}$, for all $\beta<\omega_{1}$. That is, if $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}},\overrightarrow{\theta},\overrightarrow{\mathcal{F}}\rangle$ is a simplified gap-2 morass, we construct a sequence of standard terms for ultrafilters, $\langle U_{\beta}\mid\beta<\omega_{1}\rangle$, where for each successor, $\beta$, $U_{\beta}$ is a set of standard terms for subsets of $\omega$ in the language adding generic reals indexed by $\varphi_{\theta_{\beta}}$, that is forced to be a non-principal ultrafilter in all $P_{\beta}$-generic extensions of $M$. Furthermore, we require that for all countable $\beta$, and all $f\in\mathcal{F}_{\beta\beta+1}$, $f_{\theta_{\beta}}[U_{\beta}]\subseteq U_{\beta+1}$. Since $\mathcal{F}_{\beta\beta+1}$ is an amalgamation, all $f\in\mathcal{F}_{\beta\beta+1}$, with a single exception, are left-branching, and hence $f_{\theta_{\beta}}\in\mathcal{G}_{\theta_{\beta}\theta_{\beta+1}}$. By results of [References], the morass closure of $U_{\theta_{\beta}}$ under left-branching embeddings have f.i.p. The single right-branching embedding of $\mathcal{F}_{\beta\beta+1}$ must be handled independently.

Theorem 6.4

There is a standard ultrafilter, $U$, that commutes with the embeddings of $\mathcal{F}_{\beta\beta+1}$, for all $\beta<\omega_{1}$. The ultrapower, ${\mathbb{R}}^{\omega}/U$, is a gap-2 morass-definable $\eta_{1}$-ordering.

Proof. By Lemma 3.10 the morass-embeddings of an amalgamation $\mathcal{F}_{\beta\beta+1}$, for $\beta<\omega_{1}$, are compatible. We construct the sequence $\langle U_{\beta}\rangle$ by induction on $\beta$. The limit case is routine, so we assume that $\langle U_{\alpha}\mid\alpha\leq\beta\rangle$ has been defined. We wish to show that the closure of $U_{\beta}$ under the embeddings of $\mathcal{F}_{\beta\beta+1}$ has f.i.p.

Lemma 6.5

Let $U_{\beta}$ be a standard ultrafilter, and $\mathcal{F}_{\beta\beta+1}$ be an amalgamation. Let $f\in\mathcal{F}_{\beta\beta+1}$ be the right-branching embedding, and $g_{1},\ldots,g_{n}\in\mathcal{G}_{\theta_{\beta}\theta_{\beta+1}}$. If $u_{1},\ldots,u_{n},v\in U_{\beta}$, then $\Vdash g_{1}(u_{1})\cap\ldots g_{n}(u_{n})\cap f_{\theta_{\beta}}(v)\neq\emptyset$.

Proof of Lemma. Let $u$ be a standard term for a subset of $\omega$ such that $\Vdash u=u_{1}\cap\ldots\cap u_{n}\cap v$. Then $\Vdash u\in U_{\theta_{\beta}}$. If $U_{\theta_{\beta}}$ is complete, then $u\in U_{\theta_{\beta}}$. If $k\in\mathbb{N}$, $p\in P_{\beta}$ and $h\in\mathcal{F}_{\beta\beta+1}$, then $p\Vdash k\in u$ iff $h_{\theta_{\beta}}(p)\Vdash k\in h_{\theta_{\beta}}(u)$. Hence if $p\Vdash k\in u$, then for all $i\leq n$,

[TABLE]

By Lemma 3.10, $f_{\theta_{\beta}}(p),g_{1}(p),\ldots,g_{n}(p)$ are compatible, as members of $P$. It follows that

[TABLE]

If there were a generic extension in which $g_{1}(u_{1})\cap\ldots\cap g_{n}(u_{n})\cap f_{\theta_{\beta}}(v)=\emptyset$, then in that generic extension,

[TABLE]

Suppose a condition, $p\in P_{\beta+1}$, forced this. Let $S\subseteq f_{\theta_{\beta}}[\varphi_{\theta_{\beta}}]$ be the support of $f_{\theta_{\beta}}(u)$. If $\pi$ were the projection onto $S$, then by a straightforward generalization of Lemma 6.2 [References],

[TABLE]

Then,

[TABLE]

However there is a condition $q\leq f^{-1}_{\theta_{\beta}}(\pi(p))$ and $k\in\mathbb{N}$ such that $q\Vdash k\in u$. Therefore there is a condition $r\leq g_{1}(q)\cdots g_{n}(q)\cdot f_{\theta_{\beta}}(q)\leq\pi(p)$ where

[TABLE]

So the intersection of $g_{1}(u_{1}),\ldots,g_{n}(u_{n}),f_{\theta_{\beta}}(v)$ is forced in all generic extensions to be nonempty. This completes the proof of the Lemma. $\Box$

Continuing the proof of the Theorem, it follows from Lemma 6.5 that the union under the embeddings of $\mathcal{F}_{\beta\beta+1}$ of a standard ultrafilter in $M^{P_{\beta}}$ has f.i.p., and may be extended to a standard term for an ultrafilter in the forcing language adding generic reals indexed by $\varphi_{\theta_{\beta+1}}$. It is routine to see that there is a standard ultrafilter, $U\subseteq M^{P_{\omega_{2}}}$, that commutes with the embeddings of $f\in\mathcal{F}_{\alpha\beta}$ for $\alpha<\beta\leq\omega_{1}$. We claim that ${\mathbb{R}}^{\omega}/U$ is a gap-2 morass-definable $\eta_{1}$-ordering. It was shown in $[\ref{Dumas}]$ that ${\mathbb{R}}^{\omega}/U$ is upward level-dense, has countable support and is morass-commutative. By an application of the proof of Lemma 6.10 [References], ${\mathbb{R}}^{\omega}/U$ is embedding-commutative. We show that ${\mathbb{R}}^{\omega}/U$ has grounded order support. We repeat the proof of Lemma 6.13 [References]. Let $x$ and $y$ be terms for sequences of reals, $p\in P$ and $p\Vdash[x]_{U}<[y]_{U}$. There are terms for sequences or reals, $\bar{x}$ and $\bar{y}$, such that $p\Vdash[x]_{U}=[\bar{x}]_{U}$, $p\Vdash[y]_{U}=[\bar{y}]_{U}$ and $p\Vdash\forall i\in\mathbb{N}(\bar{x}_{i}<\bar{y}_{i})$. Let $\theta$ be an ordinal and $\sigma:\varphi_{\theta_{\beta+1}}\to\theta$ be an injection, where $\varphi_{\theta_{\beta+1}}\cap\theta=\emptyset$. Then, since $\mathbb{R}$ is level dense,

[TABLE]

Let $a_{i}=sup\{q\in\mathbb{Q}\mid(\exists r<p)\wedge(r\Vdash q<\bar{x}_{i})\}$ and $b_{i}=inf\{q\in\mathbb{Q}\mid(\exists r<p)(\sigma(r)\Vdash\sigma(y_{i})<q)\}$. Then

[TABLE]

For $i\in\mathbb{N}$, let $c\in\mathbb{R}^{\omega}\cap M$ be such that $a_{i}\leq c_{i}\leq b_{i}$. Let $U^{\prime}$ be a standard ultrafilter extending $U\cup\sigma(U)$. Then

[TABLE]

In the language of the gap-1 argument, ${\mathbb{R}}^{\omega}/U$ has a grounded order base. Therefore, by Lemma 4.4 [References], ${\mathbb{R}}^{\omega}/U$ has a grounded order support. Hence, ${\mathbb{R}}^{\omega}/U$ is gap-2 morass-definable. $\Box$

7 Next Results

In this paper and [References] we have extended the classical result that $\eta_{1}$-orderings of cardinality $\aleph_{1}$ are order-isomorphic. In the next paper we extend the result that there is an $\mathbb{R}$-linear isomorphism between $\eta_{1}$-ordered real-closed fields of cardinalty $\aleph_{1}$. We show that in the Cohen extension adding $\aleph_{2}$-generic reals to a model of ZFC+CH containing a simplified $(\omega_{1},1)$-morass, there is a morass-definable $\mathbb{R}$-linear isomorphism between $\eta_{1}$-ordered morass-definable real-closed fields. We then show that in the Cohen extension adding $\aleph_{3}$-generic reals to a model of ZFC+CH containing a simplified $(\omega_{1},2)$-morass, there is a gap-2 morass-definable $\mathbb{R}$-linear isomorphism between gap-2 morass-definable $\eta_{1}$-orderings. With these results we are able to extend the theorem of Woodin [References] that it is consistent that $2^{\aleph_{0}}=\aleph_{2}$ and there exists a discontinuous homomorphism of $C(X)$. We show that it is consistent that $2^{\aleph_{0}}=\aleph_{3}$ and that there exists a discontinuous homomorphism of $C(X)$, for any infinite compact Hausdorff space, $X$.