Discontinuous Homomorphisms of $C(X)$ with $2^{\aleph_0}>\aleph_2$

TL;DR

This paper demonstrates the consistency of the existence of discontinuous homomorphisms of $C(X)$ for any infinite compact Hausdorff space $X$ under certain set-theoretic assumptions involving forcing and morasses.

Contribution

It constructs a model where $2^{eth_0}=eth_3$ and shows that discontinuous homomorphisms of $C(X)$ exist for all infinite compact Hausdorff spaces, extending previous results.

Findings

Discontinuous homomorphisms of $C(X)$ exist in the constructed model.

The existence depends on set-theoretic assumptions involving morasses and forcing.

The results hold for any infinite compact Hausdorff space $X$.

Abstract

Assume that $M$ is a c.t.m. of $ZFC+CH$ containing a simplified $(\omega_1,2)$-morass, $P\in M$ is the poset adding $\aleph_3$ generic reals and $G$ is $P$-generic over $M$. In $M$ we construct a function between sets of terms in the forcing language, that interpreted in $M[G]$ is an $\mathbb R$-linear order-preserving monomorphism from the finite elements of an ultrapower of the reals, over a non-principal ultrafilter on $\omega$, into the Esterle algebra of formal power series. Therefore it is consistent that $2^{\aleph_0}=\aleph_3$ and, for any infinite compact Hausdorff space $X$, there exists a discontinuous homomorphism of $C(X)$, the algebra of continuous real-valued functions on $X$. For $n\in \mathbb N$, If $M$ contains a simplified $(\omega_1,n)$-morass, then in the Cohen extension of $M$ adding $\aleph_n$ generic reals there exists a discontinuous homomorphism of $C(X)$, for any infinite compact Hausdorff space $X$.

Equations

$$\mathcal{M}=⟨({\theta }_{\alpha }\mid \alpha \le {\omega }_1),({\mathcal{F}}_{\alpha \beta }\mid \alpha <\beta \le {\omega }_1)⟩$$

$$(\exists {\delta }_{\alpha }<{\theta }_{\alpha })f_{\alpha }\upharpoonright {\delta }_{\alpha }=\text{id}\upharpoonright {\delta }_{\alpha }\text{ and }{f}_{\alpha }({\delta }_{\alpha })\ge {\theta }_{\alpha }.$$

$$P(A):=Fn(A\times \omega ,2),$$

$$(\forall d\in D)(d<x⟺\varphi (d)<y).$$

$$\mid x\mid <n\mid y\mid$$

$$\mid y\mid <m\mid x\mid .$$

$$C=\{c\mid (\exists \lambda <{\gamma }_1)(\exists \kappa <{\gamma }_2)c=a_{\lambda }+b_{\kappa }\}.$$

$$a\cdot b=\sum _{c\in C}((\sum _{a_{\lambda }+b_{\kappa }=c}{\alpha }_{\lambda }\cdot {\beta }_{\kappa })x^c).$$

$$\Vert a\cdot b\Vert \le \Vert a\Vert \cdot \Vert b\Vert .$$

$$a=\sum _{\lambda <{\gamma }_1}{\alpha }_{\lambda }{x}^{a_{\lambda }}$$

$$b=\sum _{\lambda <{\gamma }_2}{\beta }_{\lambda }{x}^{b_{\lambda }}.$$

$${\alpha }_0{x}^{a_0}\ne {\beta }_0{x}^{b_0}.$$

$${\alpha }_0<q<{\beta }_0.$$

$$a<q{x}^{a_0}<b.$$

$$a_0=f:S_{{\omega }_1}\to \mathbb{R}$$

$$b_0=g:S_{{\omega }_1}\to \mathbb{R}.$$

$$f(s)<g(s).$$

$$f(s)<q<g(s).$$

$$a_0<\{(s,q)\}<b_0.$$

$$s_{c_0}=\Delta \cup \{\mu \}.$$

$$s_{b_0}<s_{c_0}<s_{a_0}.$$

$$b_0(\rho )=c_0(\rho )$$

$$0<c_0(\mu )<b_0(\mu ).$$

$$a_0<c_0<b_0.$$

$$a<q{x}^{c_0}<b.$$

$$x<w<y.$$

$$\mathbf{x}=(x_1,\dots ,x_n)\in F^n\cap M[G].$$

Full text

1 Introduction

This paper addresses a problem in the theory of Banach Algebras concerning the existence of discontinuous homomorphisms of $C(X)$, the algebra of continuous real-valued functions with domain $X$, where $X$ is an infinite compact Hausdorff space. In [References], B. Johnson proved that there is a discontinuous homomorphism of $C(X)$ provided that there is a nontrivial submultiplicative norm on the finite elements of an ultrapower of $\mathbb{R}$ over $\omega$. In [References], J. Esterle constructs an algebra of formal power series, $\mathcal{E}$, and shows in [References] that the infinitesimal elements of $\mathcal{E}$ admit a nontrivial submultiplicative norm. By results of Esterle in [References], it is known that $\mathcal{E}$ is an $\eta_{1}$-ordering of cardinality $2^{\aleph_{0}}$. Furthermore, $\mathcal{E}$ is a totally ordered field by a result of Hahn in 1907 [References], and is real-closed by a result of Maclane [References].

It is a theorem of P. Erdös, L. Gillman and M. Henriksen in [References] that any pair of $\eta_{1}$-ordered real-closed fields of cardinality $\aleph_{1}$ are isomorphic as ordered fields. In fact, it is shown using a back-and-forth argument that any order-preserving field isomorphism between countable subsets of $\eta_{1}$-ordered real-closed fields may be extended to an order-isomorphism. It is a standard result of model theory that for any non-principal ultrafilter $U$ on $\omega$, ${\mathbb{R}}^{\omega}/U$ is an $\aleph_{1}$-saturated real-closed field (and hence an $\eta_{1}$-ordering). By a result of Johnson [References], between any pair of $\eta_{1}$-ordered real-closed fields of cardinality $\aleph_{1}$ there is an $\mathbb{R}$-linear order-preserving field isomorphism (hereafter $\mathbb{R}$-isomorphism). This implies, in a model of the continuum hypothesis (CH), that there is an $\mathbb{R}$-linear order-preserving monomorphism (hereafter $\mathbb{R}$-monomorphism) from the finite elements of ${\mathbb{R}}^{\omega}/U$ to $\mathcal{E}$, and hence in models of ZFC+CH there exists a discontinuous homomorphism of $C(X)$. The proof that in a model of ZFC+CH there exists a discontinuous homomorphism of $C(X)$ is due independently to Dales [References] and Esterle [References].

Shortly thereafter R. Solovay found a model of ZFC+$\neg$CH in which all homomorphisms of $C(X)$ are continuous. Later, in his Ph.D. thesis, W.H. Woodin constructed a model of ZFC+Martin’s Axiom in which all homomorphisms of $C(X)$ are continuous. This naturally gave rise to the question of whether there is a model of set theory in which CH fails and there is a discontinuous homomorphism of $C(X)$. Woodin subsequently showed that in the Cohen extension of a model of ZFC+CH by generic reals indexed by $\omega_{2}$, there is a discontinuous homomorphism of $C(X)$ [References]. Woodin shows that in this model the gaps in $\mathcal{E}$ that must be witnessed in a classical back-and-forth construction are always countable. He observes that this construction may not be extended to a Cohen extension by more than $\aleph_{2}$ generic reals. He suggests the plausibility of using morasses to construct an $\mathbb{R}$-monomorphism from the finite elements of an ultrapower of the reals to the Esterle algebra in generic extensions with more than $\aleph_{2}$ generic reals. Woodin’s argument does not extend to higher powers of the continuum and leaves open the question of whether there exists a discontinuous homomorphism of $C(X)$ in models of set theory in which $2^{\aleph_{0}}>\aleph_{2}$. In this paper we show that the existence of a simplified $(\omega_{1},2)$-morass in a model of $ZFC+CH$ is sufficient for the existence of a discontinuous homomorphism of $C(X)$ in a model in which $2^{\aleph_{0}}=\aleph_{3}$.

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 level, morass-commutative term function that, interpreted in the Cohen extension, is an $\mathbb{R}$-monomorphism of the finite elements of an ultrapower of $\mathbb{R}$ over $\omega$ into the Esterle Algebra. This is achieved with a transfinite construction of length $\omega_{1}$, utilizing the morass functions from the gap-one morass to complete the construction of size $\aleph_{2}$ by commutativity with morass maps. Using the techniques of this argument, we construct a term function with a transfinite construction of length $\omega_{1}$ and utilize morass-commutativity with the embeddings of a gap-2 morass to complete the construction of an $\mathbb{R}$-monomorphism from the finite elements of a standard ultrapower of $\mathbb{R}$ over $\omega$ to the Esterle Algebra in the Cohen extension adding $\aleph_{3}$ generic reals.

The technical obstacles to such a construction may be reduced to conditions we call morass-extendability. This paper is dependent on the results of [References] and [References], in which term functions are constructed that are forced to be order-preserving functions. The arguments here are very similar, with the additional requirement that the functions are also ring homomorphisms.

2 Preliminaries

In our initial construction we use a simplified $(\omega_{1},1)$-morass. We construct a function on terms in the forcing language for adding $\aleph_{2}$ generic reals that is forced in all generic extensions to be an $\mathbb{R}$-monomorphism from the finite elements of ${\mathbb{R}}^{\omega}/U$, where $U$ is a standard non-principal ultrafilter (see Definition 6.14 [References]) in the generic extension, into the Esterle Algebra, $\mathcal{E}$. In some sense we follow the classical route to such constructions - extension by transcendental elements in an inductive construction of length $\omega_{1}$. We will require commutativity with morass maps to construct a function on a domain of cardinality $\aleph_{2}$ making only $\aleph_{1}$ many explicit commitments. However with each commitment of the construction, there are uncountably many future commitments implied by commutativity with morass maps.

In [References] D. Velleman defines a simplified $(\omega_{1},1)$-morass.

Definition 2.1

$($Simplified $(\omega_{1},1)$-morass$)$ A simplified $(\omega,1)$-morass is a structure

[TABLE]

*that satisfies the following conditions:

(P0) (a) $\theta_{0}=1$, $\theta_{\omega_{1}}=\omega_{2}$, $(\forall\alpha<\omega_{1})\ 0<\theta_{\alpha}<\omega_{1}$.

  (b) $\mathcal{F}_{\alpha\beta}$ is a set of order-preserving functions $f:\theta_{\alpha}\to\theta_{\beta}$.

(P1) $\mid\mathcal{F_{\alpha\beta}}\mid\leq\omega$ for all $\alpha<\beta<\omega_{1}$.

(P2) If $\alpha<\beta<\gamma$, then $\mathcal{F}_{\alpha\gamma}=\{f\circ g\mid f\in\mathcal{F}_{\beta\gamma}$, $g\in\mathcal{F}_{\alpha\beta}\}$.

(P3) If $\alpha<\omega_{1}$, then $\mathcal{F}_{\alpha(\alpha+1)}=\{\textnormal{id}\upharpoonright\theta_{\alpha},f_{\alpha}\}$ where $f_{\alpha}$ satisfies:*

[TABLE]

*(P4) If $\alpha\leq\omega_{1}$ is a limit ordinal, $\beta_{1},\beta_{2}<\alpha$, $f_{1}\in\mathcal{F}_{\beta_{1}\alpha}$ and $f_{2}\in\mathcal{F}_{\beta_{2}\alpha}$, then there is $\gamma<\alpha$, $\gamma>\beta_{1},\beta_{2}$, and there is $f_{1}^{\prime}\in\mathcal{F}_{\beta_{1}\gamma}$, $f_{2}^{\prime}\in\mathcal{F}_{\beta_{2}\gamma}$, $g\in\mathcal{F}_{\gamma\alpha}$ such that $f_{1}=g\circ f_{1}^{\prime}$ and $f_{2}=g\circ f_{2}^{\prime}$.

(P5) For all $\alpha>0$, $\theta_{\alpha}=\bigcup\{f[\theta_{\beta}]\mid\beta<\alpha$, $f\in\mathcal{F}_{\beta\alpha}\}$.*

Simplified gap-1 morasses, as well as higher gap simplified morasses, are known to exist in $L$.

We will construct, by an inductive argument of length $\omega_{1}$, a function between sets of terms in the forcing language adding $\aleph_{2}$ generic reals. We interpret the morass functions on ordinals as functions between terms in the forcing language and require that the set of terms under construction satisfy certain commutativity constraints with the morass functions. It is implicit that any commitment to an ordered pair of terms in the construction is de facto a commitment to uncountably many ordered pairs in mutually generic extensions. In [References] we worked explicitly with terms in the forcing language. We wish to simplify the details of the construction by working with objects in a forcing extension. We use the notions of term complexity and strict level, from [References], and apply it to objects in a forcing extension.

Notation 2.2

$($P(A)$)$ If $A$ is a set of ordinals, we let $P(A)$ be the poset adding generic reals indexed by the ordinals of $A$. That is,

[TABLE]

the finite partial functions from $A\times\omega$ to $2$.

Let $M$ be a c.t.m of ZFC, $\beta$ be an ordinal and $P$ be the poset adding generic reals indexed by $\beta$, then $P(\beta)=Fn(\beta\times\omega,2)$. Suppose $S\subseteq\beta$. Let $\tau\in M^{P(S)}$ be a discerning term in the forcing language adding generic reals indexed by $S$. Then $\tau$ has strict level $S$ provided that, for any proper subset of $S$, $A$, and term, $\sigma$, in the forcing language adding generic reals indexed by $A$, $\Vdash\tau\neq\sigma$. Not all terms have strict levels.

Alternatively, we consider $P(\beta)$ as the product forcing $P(A)\times P(S\setminus A)\times P(\beta\setminus S)$. Suppose $G(A)$ is $P(A)$-generic over $M$ and $G$ is $P(S\setminus A)$-generic over $M[G(A)]$, and $H$ is $P(\beta\setminus S)$-generic over $M[G(A),G]$. Then if $\tau\in M[G(A),G,H]$ has strict level $S$, $\tau\notin M[G(A)]$. So an object in a forcing extension has strict level $S$ just in case there is a term for $\tau$ in the forcing language with strict level $S$, and if $\tau$ is a term of strict level $S$, then in any generic extension the value of $\tau$ has strict level $S$. Consequently in our construction we pass freely between objects of strict level $S$ in a generic extension and term of strict level $S$ in the forcing language.

Many of the constraints required for commutativity with morass maps are expressed in terms of the strict level of objects in a forcing extension (or correspondingly, terms in the forcing language). For instance, in [References] we define a term function to be level if the strict level of any term in the domain equals the strict term of its image under the function. Such maps will commute with morass maps in the manner required by our construction.

3 Constructing an $\mathbb{R}$-monomorphism

on a real-closed field

We wish to construct a set of terms in the forcing language for adding $\aleph_{2}$ generic reals that is forced in all generic extensions to be an $\mathbb{R}$-monomorphism on the finite elements of an ultrapower of $\mathbb{R}$. It is a result of B. Johnson [References] that $\eta_{1}$-ordered real-closed fields with cardinality $\aleph_{1}$ are $\mathbb{R}$-isomorphic in models of ZFC+CH. This result strengthens the classical result that $\aleph_{1}$-saturated real closed fields of cardinality $\aleph_{1}$ are isomorphic. It is conceivable that in a naive back-and-forth construction of an order-isomorphism between $\eta_{1}$-ordered real-closed fields that a choice is made for an image (or pre-image) of the function under construction that precludes satisfaction of $\mathbb{R}$-linearity.

Definition 3.1

$($Full real-closed field$)$ A real-closed field $D$ is full iff for every finite element, $r+\delta$, where $r\in\mathbb{R}$ and $\delta$ is infinitesimal, $r\in D$.

We will need to extend two results due to B. Johnson [References] to meet the requirement of $\mathbb{R}$-linearity in the context of constructing term functions using a morass.

Lemma 3.2

$(B.Johnson)$* Assume that $D$ and $I$ are full real-closed subfields of $\eta_{1}$-ordered real-closed fields $D^{*}$ and $I^{*}$ (resp.), $\phi:D\to I$ is an $\mathbb{R}$-monomorphism, and $r\in R$. Then there is an extension of $\phi$, $\phi^{*}$, that is an $R$-monomorphism of the real closure of the field generated by $D$ and $r$, $F(D,r)$, onto the real closure of the field generated by $I$ and $r$, $F(I,r)$. Furthermore $F(D,r)$ (and consequently, $F(I,r)$) is full.*

Lemma 3.3

$(B.Johnson)$* Let $D$, $D^{*}$, $I$, $I^{*}$ and $\phi$ be as in Lemma 3.2, $x\in D^{*}$ and assume that the real closure of the field generated by $D$ and $d$, $F(D,d)$, is full. Let $y\in I^{*}$ be such that*

[TABLE]

Then there is an $\mathbb{R}$-monomorphism extending $\phi$, $\phi^{*}:F(D,r)\to I^{*}$, such that $\phi^{*}(x)=y$.

Definition 3.4

$($Archimedean valuation$)$ If $x$ and $y$ are non-zero elements of a real-closed field, they have the same Archimedean valuation, $x\sim y$, provided that there are $m,n\in\mathbb{N}$ such that

[TABLE]

and

[TABLE]

If $\mid x\mid<\mid y\mid$ and $x\nsim y$, then $x$ has Archimedean valuation greater than $y$, $x\succeq y$.

Archimedean valuation is an equivalence relation on the non-zero elements of a real-closed field (RCF). The non-zero real numbers have the same valuation. Elements with the same valuation as a real number are said to have real valuation. In a nonstandard real-closed field, elements with valuation greater than a real valuation are infinitesimal. The finite elements of a real closed field are the infinitesimal elements and those with real valuation.

4 The Esterle algebra

We define the Esterle algebra [References] and review some basic properties.

Definition 4.1

$(S_{\omega_{1}})$* $S_{\omega_{1}}$ is the lexicographic linear-ordering with domain $\{s:\omega_{1}\to 2\mid s$ has countable support and the support of s has a largest element $\}$.*

Definition 4.2

$(G_{\omega_{1}})$* $G_{\omega_{1}}$ is the ordered group with domain $\{g:S_{\omega_{1}}\to\mathbb{R}\mid g$ has countable well-ordered support$\}$, lexicographic ordering, and group operation pointwise addition.*

We define an ordered algebra of formal power series, $\mathcal{E}$. The universe of $\mathcal{E}$ is the set of formal power series, $\sum_{\lambda<\gamma}\alpha_{\lambda}x^{a_{\lambda}}$, where:

$\gamma<\omega_{1}$. 2. 2.

$(\forall\lambda<\gamma)\,\alpha_{\lambda}\in\mathbb{R}$. 3. 3.

$\{a_{\lambda}\mid\lambda<\gamma\}$ is a countable well-ordered subset of $G_{\omega_{1}}$ and $\lambda_{1}<\lambda_{2}<\gamma\Rightarrow a_{\lambda_{1}}<a_{\lambda_{2}}$.

The ordered algebra, $\mathcal{E}$, is isomorphic to the set of functions, with countable well-ordered support, from $G_{\omega_{1}}$ to $\mathbb{R}$. The lexicographic order linearly-orders $\mathcal{E}$. Addition is pointwise and multiplication is defined as follows: 

Suppose $a=\sum_{\lambda<\gamma_{1}}\alpha_{\lambda}x^{a_{\lambda}}$ and $b=\sum_{\kappa<\lambda_{2}}\beta_{\kappa}x^{b_{\kappa}}$ are members of $\mathcal{E}$. Let

[TABLE]

Then

[TABLE]

Definition 4.3

$($Esterle algebra, $\mathcal{E})$ The Esterle algebra, $\mathcal{E}$, is $\{f:G_{\omega_{1}}\to\mathbb{R}\mid f$ has countable well-ordered support$\}$. $\mathcal{E}$ is lexicographically ordered, with pointwise addition, and multiplication defined above.

$\mathbb{R}$ may be embedded in $\mathcal{E}$ by $\alpha\longmapsto\alpha x^{e}$, where $e$ is the group identity in $G_{\omega_{1}}$. Exponents in $G_{\omega_{1}}$ larger than $e$ (called positive exponents) correspond to infinitesimal Archimedean valuations, and those smaller than $e$ (called negative exponents) correspond to infinite valuations. The finite elements of $\mathcal{E}$ are those with leading exponent $\geq e$. The Archimedean valuations of the Esterle algebra are represented by the group of exponents of $\mathcal{E}$.

Theorem 4.4

(J. Esterle [References]) $\mathcal{E}$ is an $\eta_{1}$-ordered real-closed field.

A norm, $\|\,\|$, on an algebra $A$ is submultiplicative if for any $a,b\in A$,

[TABLE]

Theorem 4.5

(G. Dales [References], J. Esterle [References]) The set of finite elements of $\mathcal{E}$ bears a submultiplicative norm.

It is a standard result of model theory that if $U$ is a non-principal ultrafilter on $\omega$, the ultrapower ${\mathbb{R}}^{\omega}/U$ is an $\aleph_{1}$-saturated real-closed field. Any two $\aleph_{1}$-saturated, or $\eta_{1}$-ordered, real-closed fields with cardinality of the continuum are isomorphic in models of ZFC+CH. Hence CH implies that there is a discontinuous homomorphism of $C(X)$.

Theorem 4.6

(B.Johnson [References]) (CH) If $U$ is a non-principal ultrafilter, there is an $\mathbb{R}$-monomorphism from the finite elements of ${\mathbb{R}}^{\omega}/U$ into $\mathcal{E}$.

We turn our attention to terms in a forcing language $M^{P}$ that are forced to be members of the Esterle algebra. In [References] and [References], we found sufficient conditions for morass constructions. The aggregate of these conditions were characterized as morass-definability and gap-2 morass-definability. The central theorem of the papers were that morass-definable $\eta_{1}$-orderings are order-isomorphic in the Cohen extension adding $\aleph_{2}$ generic reals of a model of $ZFC+CH$ containing a simplified $(\omega_{1},1)$-morass; and gap-2 morass definable $\eta_{1}$-orderings are order-isomorphic in the Cohen extension adding $\aleph_{3}$ generic reals of a model of $ZFC+CH$ containing a simplified $(\omega_{1},2)$-morass.

Definition 4.7

(Level-dense) Let $T_{X}\in M^{P}$ be forced to be a linear-ordering and $X\subseteq M^{P}$ be a set of terms of strict level for the domain of $T_{X}$. $X$ is level-dense provided that for $x,y\in X$, where the support of $x$ and the support of $y$ are disjoint, and $G$, $P$-generic over $M$, if $M[G]\models x<y$ then there is $z\in X\cap M$ such that $M[G]\models x<z<y$.

Definition 4.8

(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 terms of strict level for the domain of $T_{X}$. $X$ is upward level-dense provide that for every $x,y,z\in X$, in which $z$ has strict level $A\subseteq\kappa$, $p\in P$ with $p\Vdash x<z<y$ and $B\supseteq A$, there is a term of strict level $B$, $w\in X$, such that $p\Vdash x<w<y$.

Definition 4.9

$($Morass-commutative$\,)$ Suppose $\langle(\theta_{\alpha}\mid\alpha\leq\omega_{1}),(\mathcal{F}_{\alpha\beta}\mid\alpha<\beta\leq\omega_{1})\rangle$ is a simplified $(\omega_{1},1)$-morass, $\lambda\leq\omega_{1}$ and $X\subseteq M^{P(\omega_{1})}$. We say that $X$ is morass-commutative beneath $\lambda$ if for any $\zeta<\xi\leq\lambda$, $f\in\mathcal{F}_{\zeta\xi}$ and $x\in X\cap M^{P_{\zeta}}$, $f(x)\in X$. We say that $X$ is morass-commutative if $X$ is morass-commutative beneath $\omega_{1}$.

Definition 4.10

$($Morass-definable$\,)$ Let $\langle X,<\rangle\in M[G]$ be a linear ordering. $X$ is morass-definable if there is a set of terms $T\subseteq M^{P}$

$T$* is a morass-commutative set of terms of strict level, and $val_{G}(T)=X$* 2. 2.

$T$* is level dense and upward level dense* 3. 3.

Every term of $T$ has countable support (that is, any term in $T$ is a term in $M^{P(A)}$, for some $A$ adding generic reals indexed by a countable subset of $\omega_{2}$)

Let $E\subseteq M^{P}$ be the set of terms of strict level for elements in the Esterle algebra in the forcing language of the poset $P$. It is routine to check that $\mathcal{E}$ is morass-commutative and that every element of the Esterle algebra is the interpretation of a term with countable support.

Lemma 4.11

The Esterle algebra is level-dense.

Proof: Let $a,b\in M^{P}$ be discerning terms for elements of the Esterle algebra bearing disjoint supports. Let $G$ be $P$-generic over $M$ and $M[G]\models a<b$. We wish to show that $a$ and $b$ are separated in $M[G]$ by an element of $\mathcal{E}\cap M$. We work in $M[G]$. Let $\gamma_{1}$ and $\gamma_{2}$ be countable ordinals and

[TABLE]

and

[TABLE]

If $a$ and $b$ are equal on a partial sum, then that partial sum is in $M$, so subtracting the largest common partial sum of $a$ and $b$, we may assume that $a$ and $b$ differ on the first term of the sums, and

[TABLE]

If $a_{0}=b_{0}$, then $a_{0}\in M$ and there is $q\in\mathbb{Q}$ such that

[TABLE]

Then $qx^{a_{0}}\in M$ and

[TABLE]

Hence we assume that $a$ and $b$ have distinct Archimedean valuations.

We consider the case $a_{0}<b_{0}$. Then $\alpha_{0}<0$. It is sufficient to prove that there is an element of $\mathcal{E}\cap M$ that has valuation between $a_{0}$ and $b_{0}$. Let

[TABLE]

and

[TABLE]

If $a_{0}$ and $b_{0}$ are equal on an initial segment of their supports, then this initial segment is in the ground model. We can therefore assume that $a_{0}$ and $b_{0}$ either have distinct least members, or have the same least member of their supports, $s\in M$, and

[TABLE]

In the latter case there is $q\in\mathbb{Q}$ such that

[TABLE]

Then $\{(s,q)\}\in M$ and

[TABLE]

Hence we have left to consider the case in which $s_{a_{0}}$ is the least member of the support of $a_{0}$, $s_{b_{0}}$ is the least member of the support of $b_{0}$ and $s_{a_{0}}\neq s_{b_{0}}$. If either $s_{a_{0}}$ or $s_{b_{0}}$ are in $M$ we can find a member of $G_{\omega_{1}}\cap M$ that is a valuation between $a_{0}$ and $b_{0}$. So we may assume that neither $s_{a_{0}}$ nor $s_{b_{0}}$ are in $M$. As we shall see, it is sufficient to show that between any distinct elements of $S_{\omega_{1}}$ from mutually generic extensions, there is a member of $S_{\omega_{1}}$ in the ground model. We assume without loss of generality that $s_{b_{0}}<s_{a_{0}}$ (the case $s_{a_{0}}<s_{b_{0}}$ is altogether similar). Treating $s_{b_{0}}$ and $s_{a_{0}}$ as countable subsets of $\omega_{1}$, let $\mu$ be the least element of $s_{b_{0}}$ that is not a member of $s_{a_{0}}$. Let $\Delta=s_{a_{0}}\cap\mu$. Then $\Delta=s_{b_{0}}\cap\mu\in M$. We note that $b_{0}(\mu)>0$, otherwise $b_{0}<a_{0}$, contrary to assumption. Let

[TABLE]

Then $s_{c_{0}}\in M$ and

[TABLE]

Let $c_{0}\in M$ be defined so that, for $\rho<\mu$,

[TABLE]

and

[TABLE]

Then $c_{0}\in M$ and

[TABLE]

Since $a<b$, the coefficient of $a_{0}>0$. Let $q\in\mathbb{Q}$ and $0<q<a_{0}$. Then $qx^{c_{0}}\in M$ and

[TABLE]

Therefore $\mathcal{E}$ is level dense. $\Box$

Lemma 4.12

The Esterle algebra is upward level-dense.

Let $A\subseteq B$ be countable subsets of $\omega_{1}$. Suppose $x,y,z\in M^{P}$, with $x<z<y$, and $z$ has level $A$. It is sufficient to show that there is an element $w\in\mathcal{E}$ of strict level $B$ such that

[TABLE]

We may assume without loss of generality that $z\in M$. If $x$ and $y$ have an identical partial sum, then $z$ must share that partial sum, and it is therefore in $M$. We may subtract the partial sum from all three formal power series and pass to $x$ and $y$ that disagree on the first term of the formal power series. If $x$ and $y$ have the same Archimedean valuation, $a$, then the first term of $z$ has valuation $a$. Let $\alpha\in\mathbb{R}$ be of strict level $B$ and lie between the initial coefficients of $x$ and $y$. Let $w=\alpha x^{a}$. Then $w$ has strict level $B$ and

[TABLE]

So we assume that $x$ and $y$ have distinct valuation. If $x$ and $z$ have the same valuation, $a$, and the initial coefficient of $x$ is negative, let $\alpha\in\mathbb{R}$ be negative and greater than the the initial coefficient of $x$ and have strict level $B$. If the initial coefficient of $x$ is positive, let $\alpha$ be positive and greater then the initial coefficient of $x$ and have strict level $B$. In either case, let $w=\alpha x^{a}$. Then $w$ has strict level $B$ and

[TABLE]

The cases for $x$ and $y$ having the same valuation are similar.

If $x$ and $y$ have distinct valuation, let $\alpha\in\mathbb{R}$ be positive and have strict level $B$. Let $w=\alpha\cdot z$. Then $w$ has strict level $B$ and

[TABLE]

Therefore $\mathcal{E}$ is upward level-dense. $\Box$

Theorem 4.13

The Esterle algebra is morass-definable.

5 Extendable functions

In order to extend an order-monomorphism by commutativity with a splitting map we must satisfy both algebraic conditions and order constraints. In the next section we show that the morass-commutative extension of an $\mathbb{R}$-monomorphism, satisfying certain technical constraints (extendability), may be extended to an $\mathbb{R}$-monomorphism satisfying those same constraints. The technical constraints are those required for an inductive construction along the vertices of a simplified morasses.

We state the central technical condition that permits the inductive construction of the following sections. We restrict our attention to functions from terms for a standard ultrapower of $\mathbb{R}$ to terms for elements of the Esterle algebra.

Definition 5.1

$($Extendable Function$)$ Let $M$ be a c.t.m.of ZFC, $P$ be the poset adding generic reals indexed by a countable ordinal and $G$ be $P$-generic over $M$. The term function, $\phi:X\to Y$, is extendable provided that the following are satisfied:

$X$* is a set of terms of strict level for a subring of a standard ultrapower of $\mathbb{R}$ (over $\omega$) in $M[G]$ that has countable transcendence degree over $\mathbb{R}$* 2. 2.

$Y$* is a set of terms of strict level for elements of the Esterle algebra that is closed under partial sums and contains coefficients* 3. 3.

$\phi:X\to Y$* is a level term function that is forced to be an $\mathbb{R}$-isomorphism (with respect to $\mathbb{R}\cap X$).*

We interpret a splitting map on a set of countable ordinals as a function on terms in the forcing language as in [References]. If $\nu$ is a vertex of a morass, $\theta_{\nu}$ is the ordinal associated with $\nu$, $X\cap M^{P(\theta_{\nu})}$ is morass-commutative beneath $\nu$ and $\sigma\in\mathcal{F}_{\nu\nu+1}$ is the splitting function on $\theta_{\nu}$ then $X\cup\sigma[X]$ is morass-commutative beneath $\nu+1$. We will show that an extendable function may be commutatively extended by a splitting function to an extendable function.

6 Commutative extensions of term functions

In the inductive construction of the following sections we will need technical lemmas of two types: those insuring that commutativity with morass maps may be used to extend extendable functions to extendable functions, and those allowing the extension of the domain of an extendable function by a specified element to an extendable function. Throughout this section we assume:

$\phi:X\to Y$ is an extendable function. 2. 2.

$\sigma$ is a splitting map with splitting point $\delta=0$. That is, $\theta<\omega_{1}$, $\sigma:\theta\to\omega_{1}$ and $\sigma[\theta]\cap\theta=\emptyset$. 3. 3.

$P$ is the poset adding generic reals indexed by $\theta$. 4. 4.

$G$ is $P$-generic over $M$. 5. 5.

$H$ is $\sigma[P]$-generic over $M[G]$.

We wish to show that the morass-commutative extension of $\phi$ to the ring generated by $X\cup\sigma[X]$ is extendable.

6.1 Splitting maps and algebraic independence

The central result of this subsection states, roughly, that a subset of a field in a generic extension that is algebraically independent (AI) over the restriction of the field to the ground model, will be AI over the restriction of an extension field to a mutually generic extension. It will follow that the union of an AI subset of a field in a generic extension with its morass “split” in a mutually generic extension will be AI over the restriction of the field to the ground model.

Lemma 6.1

Let $P$ be the poset adding generic reals indexed by $\omega_{1}$ and $G$ be $P$-generic over $M$. Let $F$ be a morass-definable real-closed field in $M[G]$. Let $\theta<\omega_{1}$, $P(\theta)$ be the poset adding generic reals indexed by $\theta$, $G(\theta)$ be the $P(\theta)$-generic factor of $G$, and $\sigma$ be a splitting map on $\theta$ (with splitting point $\delta=0$). If $\chi=\{x_{1},\ldots,x_{n}\}\subseteq(F\cap M[G])$ is linearly independent (LI) over $M\cap F$ and $H$ is $\sigma[P]$-generic over $M[G]$, then $\chi$ is LI over $M[H]\cap F$.

Proof: Let

[TABLE]

Suppose that there is a non-zero vector of $\mathbf{y}\in F^{n}\cap M[H]$ which is orthogonal to $\mathbf{x}$. Without loss of generality we may assume that the components of $\mathbf{y}$ are LI over $M\cap F$. We consider terms for $\mathbf{x}$ and $\mathbf{y}$, $x\in M^{P}$ and $y\in M^{\sigma[P]}$ respectively. Let $p\in P$, $q\in\sigma[P]$, be such that

[TABLE]

We work below $p\cdot q$ and assume that $x\cdot y=0$ in all $P(\theta\cup\sigma[\theta])$-generic extensions. Let $\langle\sigma_{i}\mid i\in\omega\rangle$ be a sequence of splitting functions such that

$\sigma_{0}=\sigma$. 2. 2.

For any $i\in\omega$, $\sigma_{i}$ has splitting point $\delta=0$. 3. 3.

For any $i\in\omega$, the range of $\sigma_{i}$, $\theta_{i}$, is contained in $\omega_{1}$. 4. 4.

If $i\neq j$, then the range of $\sigma_{i}$ is disjoint from the range of $\sigma_{j}$.

Let $P_{i}$ be the poset adding generic reals indexed by $\theta_{i}$ and $Q_{1}$ be the poset adding generic reals indexed by $\bigcup_{i\leq n}\theta_{i}$. Let $q_{1}\in Q_{1}$ be a condition which forces that the span of terms $\{\sigma_{i}(y)\mid i\leq n\}$ has maximal possible dimension, $m\leq n$, and let $\tau$ be a term for the row-reduced echelon form of the matrix with rows $\sigma_{1}(y),\ldots,\sigma_{n}(y)$. We consider $q_{1}$ as a condition in $\Pi_{i=1}^{n}P_{i}$, $p_{1}\cdots p_{n}$, in which $p_{i}\leq q$ for all $i\leq n$. Hence $p\cdot q_{1}$ forces that $x$ is in the null space of $\tau$.

For $0<i\leq n$, let

[TABLE]

Then

[TABLE]

Furthermore, through the definitions of [References], $\sigma_{i}^{*}$ is naturally interpreted as a function from $P_{i}$ to $P_{i+n}$ and as well as a function from $M^{P_{i}}$ to $M^{P_{i+n}}$. Let

[TABLE]

If $Q_{2}$ is the poset adding generic reals indexed by $\bigcup_{i=1}^{n}\theta_{i+n}$ then $\sigma^{*}:Q_{1}\to Q_{2}$. Furthermore, since $F$ is morass-definable, $\sigma^{*}(\tau)$ is forced in all $Q_{2}$-generic extensions to be a row-reduced matrix with dimension $m$.

Let $H_{1}$ be $Q_{1}$ generic over $M$, with $q_{1}\in H_{1}$ and $H_{2}$ be $Q_{2}$-generic over $M[H_{1}]$, with $\sigma^{*}(q_{1})\in H_{2}$. Let $A_{1}$ be the value of $\tau$ in $M[H_{1}]$ and $A_{2}$ be the value of $\sigma(\tau)$ in $M[H_{2}]$. Then $A_{1}$ and $A_{2}$ are row-reduced matrices with dimension $m$. Furthermore, since $m$ is the maximum possible rank for $\tau$ in any generic extension,

[TABLE]

The row reduced echelon form is canonical, so

[TABLE]

However $M[H_{1}]$ and $M[H_{2}]$ are mutually generic extensions, $A_{1}\in M[H_{1}]$ and $A_{2}\in M[H_{2}]$. Therefore

[TABLE]

In $M[G,H_{1},H_{2}]$, $x$ is in the null-space of $A_{1}\in M$. This contradicts the assumption that the components of $x$ are LI over $M\cap F$. $\Box$

Lemma 6.2

Let $F$, $G$, $H$, $\chi$ satisfy the hypotheses of Lemma 6.1. Assume $n\in\mathbb{N}$, $\bar{x}=(x_{1},\ldots,x_{n})\in F^{n}\cap M[G]$, $\bar{y}=(y_{1},\ldots,y_{n})\in F^{n}\cap M[H]$ and

[TABLE]

Then $\bar{y}$ is in the span of vectors in $F^{n}\cap M$, all of which are orthogonal to $\bar{x}$.

Proof: Let

[TABLE]

and

[TABLE]

Similar to the previous argument, we work beneath a condition that forces $\bar{x}\perp\bar{y}$. Let $\sigma_{1},\ldots,\sigma_{m}$ be splitting functions with pairwise disjoint ranges and $\sigma_{1}(\bar{x}),\ldots\sigma_{m}(\bar{x})$ be a maximal LI set of vectors in mutually generic extensions. It is a consequence of this maximal linear independence that in any mutually generic extension of $M$, the interpretation of $\bar{x}$ is in the span of $\sigma_{1}(\bar{x}),\ldots,\sigma_{m}(\bar{x})$. For $i\leq n$, let

[TABLE]

and

[TABLE]

Let $A_{1}$ be the rank $m$ matrix with row vectors $\sigma_{1}(\bar{x}),\ldots,\sigma_{m}(\bar{x})$ as the first $m$ rows, and the the zero vector for the remaining rows. The nullity of $A_{1}$ is $n-m$. Let $B_{1}$ be the row-reduced echelon form of the basis matrix of the null space of $A_{1}$. We note that, in a generic extension including generic reals indexed by $\sigma(P)$, $\bar{y}$ is in the null space of $A_{1}$ and hence in the span of the rows of $B_{1}$. Let $\tau_{A}\in M^{P^{*}}$ be a term for $A_{1}$, and $\tau_{B}$ be a term for $B_{1}$. We work beneath a condition of $P^{*}$ that forces that $\tau_{B}$ is the row-reduced echelon form of the basis matrix of the null space of $\tau_{A}$. Let $G^{*}$ be $P^{*}$-generic and $H^{*}$ be $P^{*}$-generic over $M[G^{*}][H]$. As in the previous Lemma, if $A_{2}$ [resp. $B_{2}$] is the interpretation of $\tau_{A}$ [resp. $\tau_{B}$] in $M[H^{*}]$, then $A_{2}$, is a row-reduced echelon matrix of rank $n$ and, in $M[H^{*}][H]$, $\bar{y}$ is in the null space of $A_{2}$. Additionally $B_{2}$ is the row-reduced echelon form of the basis matrix for the null space of $A_{2}$. Every row of $A_{2}$ is in the span of $A_{1}$, and both have dimension $m$, so the range of $A_{1}$ and $A_{2}$ are identical. Therefore

[TABLE]

However $B_{1}\in M[G^{*}]$ and $B_{2}\in M[H^{*}]$, with $G^{*}$ and $H^{*}$ mutually generic. Hence

[TABLE]

Therefore $\bar{y}$ is in the span of vectors in $F^{n}\cap M$. $\Box$

Corollary 6.3

Let $F$, $G$ and $H$ satisfy the hypotheses of Lemma 6.1. If $\chi\in M[G]\cap F$ is algebraically independent (AI) over $M\cap F$, then $\chi$ is AI over $M[H]\cap F$.

Proof: Let $\chi^{*}$ be the multiplicative semi-group generated by the elements of $\chi$. Then $\chi^{*}$ is LI over $M\cap F$. By Lemma 6.1, $\chi^{*}$ is LI over $M[H]\cap F$. Therefore $\chi$ is AI over $M[H]\cap F$. $\Box$

Corollary 6.4

If $\chi$ is $AI$ over $M\cap F$, then $\chi\cup\sigma[\chi]$ is AI over $M\cap F$.

Proof: Let $G$ be $P$-generic over $M$, and $H$ be $\sigma[P]$-generic over $M[G]$. $F$ is morass-commutative so $\sigma[\chi]$ is AI over $M$. Suppose there is a nontrivial linear combination (over $M\cap F$) of distinct elements of the semigroup generated by $\chi\cup\sigma[\chi]$ that equals 0. By Lemma 6.4, $\chi$ is AI over $M[H]\cap F$, so there must be a nontrivial linear combination (over $M\cap F$) of elements of the semigroup generated by $\sigma[\chi]$ that equals 0. However $F$ is morass commutative, and $\sigma[\chi]$ is AI over $M\cap F$. $\Box$

Lemma 6.5

Let $X$ be a subring of a standard ultrapower of $\mathbb{R}$, ${\mathbb{R}}^{\omega}/U$. Let $X^{*}$ be the ring generated by $X\cup\sigma[X]$, where $\sigma$ is a splitting function. If $\phi:X\to\mathcal{E}$ is an extendable $\mathbb{R}$-monomorphism, then there is an $\mathbb{R}$-monomorphism, $\phi^{*}:X^{*}\to\mathcal{E}$, extending $\phi$ and $\sigma(\phi)$.

Proof: An element of $X^{*}$ may be expressed as $\sum_{i=1}^{n}x_{i}\cdot\sigma(y_{i})$, for some $n\in\mathbb{N}$, and $x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}\in X$. Let $S$ be the set of expressions of this form. We define a function $\psi:S\to\mathcal{E}$ where

[TABLE]

Let $\iota:S\to X^{*}$ be the natural quotient map from the expressions of $S$ to $X^{*}$. The kernel of $\iota$ is the set of expressions of $S$ that sum to [math] in $X^{*}$. $\psi$ defines a ring homomorphism on $X^{*}$ if and only if for any $s$ in the kernel of $\iota$,

[TABLE]

For $i\leq n$ let $z_{i}=\sigma(y_{i})$ and $s=\sum_{i=1}^{n}x_{i}\cdot z_{i}$ be in the kernel of $\iota$. Then in $X^{*}$,

[TABLE]

Let

[TABLE]

[TABLE]

and

[TABLE]

By Lemma 6.2, $\bar{z}$ is in the span of elements of $X^{n}\cap M$ that are orthogonal to $\bar{x}$. Let $\{b_{1},\ldots,b_{m}\}$ be an LI set of vectors of $X^{n}\cap M$ orthogonal to $\bar{x}$ that contains $\bar{z}$ in its span. Let $\langle\cdot,\cdot\rangle$ be the dot product and $(\alpha_{1},\ldots,\alpha_{m})\in\sigma([X])^{n}\cap M[H]$ be such that

[TABLE]

Let $\bar{\phi}:X^{n}\to\mathcal{E}^{n}$ be defined by

[TABLE]

Recall that for a splitting map $\sigma$, $\sigma(\phi):\sigma[X]\to\mathcal{E}$ is defined so that $\sigma$ and $\phi$ commute. Then

[TABLE]

However $b_{i}\in X^{n}\cap M$ for all $i\leq n$, so

[TABLE]

Hence

[TABLE]

However, for all $i\leq m$, $b_{i}\perp\bar{x}$. So for all $i\leq m$,

[TABLE]

and $\psi(\langle\bar{x},\bar{z}\rangle)=0$. Therefore $\psi$ defines a ring homomorphism on $X^{*}$, $\phi^{*}$, that extends $\phi\cup\sigma(\phi)$ to $X^{*}$.

Assume that $\bar{x},\bar{y}\in M[G]\cap X^{n}$, $\bar{z}=\sigma(\bar{y})$ and

[TABLE]

then there are

[TABLE]

and

[TABLE]

such that $\{x_{1}^{\prime},\ldots,x_{m}^{\prime}\}$ is LI over $M\cap X$ and

[TABLE]

Then $\{\phi(x_{1}^{\prime}),\ldots,\phi(x_{m}^{\prime})\}$ is LI over $\phi[X]\cap M$. By Lemma 6.1, $\{\phi(x_{1}^{\prime}),\ldots,\phi(x_{m}^{\prime})\}$ is LI over $\mathcal{E}\cap M[H]$. Therefore

[TABLE]

Thus $\phi^{*}$ is a monomorphism.

Let $r\in\mathbb{R}\cap X^{*}$. Then $r$ is an element of the ring generated by the reals of $\bar{X}\cup\sigma[\bar{X}]$. Hence, for all $r\in\mathbb{R}\cap X^{*}$,

[TABLE]

and $\phi^{*}$ is $\mathbb{R}$-linear. We show that $\phi^{*}$ is order-preserving. If $x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}\in X$, we show that $\sum_{i=1}^{n}x_{i}\cdot\sigma(y_{i})>0$ iff $\sum_{i=1}^{n}\phi^{*}(x_{i})\cdot\phi^{*}(\sigma(y_{i}))>0$.

If $n=1$, then the sign of $x_{1}\cdot\sigma(y_{1})$ is the sign of the product of the leading coefficients of $x_{1}$ and $\sigma(y_{1})$, which are preserved by $\phi$ and $\sigma(\phi)$ respectively. So

[TABLE]

Assume that $n\geq 2$, and

[TABLE]

Let

[TABLE]

By assumption $\phi[X]$ is closed under partial sums, so there are elements

$u_{1},\ldots,u_{j},u_{j+1},\ldots u_{k},v_{1},\ldots,v_{k}\in Y$ such that

$y=\sum_{i=1}^{k}u_{i}\cdot\sigma(v_{i})$. 2. 2.

For $i\leq j$, every term in the power series expansion of $u_{i}\cdot\sigma(v_{i})$ has power (valuation) less than $a_{0}$. 3. 3.

For $j<i\leq k$, every term of the power series expansion of $u_{i}\cdot\sigma(v_{i})$ has power (valuation) at least $a_{0}$.

Every term of $y$ expressed as a power series has valuation no less than $a_{0}$, therefore

[TABLE]

and

[TABLE]

If $j<i\leq k$, then $s_{i}=u_{i}\cdot\sigma(v_{i})$ has valuation no less than $a_{0}$ in $G_{\omega_{1}}$. If $s_{i}$ has valuation greater than $a_{0}$, then $s_{i}$ has Archimedean valuation greater than $y$. Let $S$ be the set of indexes of the $u_{i}\cdot\sigma(v_{i})$ with valuation $a_{0}$ and $T$ be the set of indexes of the $u_{i}\cdot\sigma(v_{i})$ with valuation greater than $a_{0}$. $S$ is nonempty and every element of $T$ has Archimedean valuation greater than every element of $S$.

Since $\phi$ is extendable, there are $b_{0}\in M[G]\cap G_{\omega_{1}}$ and $c_{0}\in M[H]\cap G_{\omega_{1}}$ such that $x^{b_{0}}\in\phi[X]$, $x^{c_{0}}\in\sigma[\phi[X]]$, $b_{0},c_{0}\geq e$ (in $G_{\omega_{1}}$) and

[TABLE]

Let $u,v\in X$ be such that

[TABLE]

and

[TABLE]

We let $\bar{X}$ be the field generated by $X$ and $\psi:\bar{X}\to\mathcal{E}$ be the unique order-preserving field-monomorphism extending $\phi$. We have previously observed that $\phi^{*}$ is a ring monomorphism and $\psi\cup\sigma(\psi)$ is an order-preserving injection. So

[TABLE]

and

[TABLE]

The real number $\alpha_{0}$ is in the domain and range of $\phi^{*}$, $\phi^{*}$ is $\mathbb{R}$-linear and $\phi^{*}\subseteq\psi$, so

[TABLE]

Let $z=\sum_{0<\lambda<\gamma}\alpha_{\lambda}x^{a_{\lambda}-a_{0}}$.

The range of $\phi^{*}$ is closed under partial sums, so $\sum_{0<\lambda<\gamma}\alpha_{\lambda}x^{a}_{\lambda}$ is in the range of $\phi^{*}$ and $u\cdot v$ is in the range of $\psi$. Thus $z\in\psi[\bar{X}]$. Every term of $z$ is infinitesimal. Therefore $\psi^{-1}(z)$ is infinitesimal and $\sum_{i=1}^{n}x_{i}\cdot\sigma(y_{i})$, $\sum_{i=1}^{n}(x_{i}/u)\cdot\sigma(y_{i}/v)$ and $\alpha_{0}$ are each greater than [math]. Therefore $\phi^{*}$ is extendable. $\Box$

Lemma 6.6

Let

$\bar{\nu}<\nu\leq\omega_{1}$** 2. 2.

$D$* be a subring of a standard ultrapower of $\mathbb{R}$ over $\omega$* 3. 3.

$\phi:D\to\mathcal{E}\in M[G_{\bar{\nu}}]$* be an extendable $\mathbb{R}$-monomorphism on $D$* 4. 4.

$D^{*}=\bigcup_{\sigma\in\mathcal{F}_{\bar{\nu},\nu}}\sigma[D]$.

Then there is a unique extendable $\mathbb{R}$-monomorphism $\phi^{*}$ on the ring generated by $D^{*}$ which, for any $\sigma\in\mathcal{F}_{\bar{\nu}\nu}$, extends $\sigma[\phi]$.

Proof: If $\nu=\bar{\nu}+1$, then the result follows from Lemma 6.5.

If there is no limit ordinal ordinal $\lambda$, $\bar{\nu}<\lambda\leq\nu$. Then there is $n\in\omega$ such that

[TABLE]

By Lemma 6.5, for any extendable $\phi:D\to\mathcal{E}$ and splitting function $\sigma$, the ring monomorphism on the ring generated by $D\cup\sigma[D]$ extending $\phi\cup\sigma[\phi]$ is extendable. By $n$ iterated applications of Lemma 6.5, there is a unique extendable $\mathbb{R}$-monomorphic extension of $\phi$, $\phi^{*}\supset\bigcup_{\sigma\in\mathcal{F}_{\bar{\nu}\nu}}\sigma[\phi]$, to the ring generated by $\bigcup_{\sigma\in\mathcal{F}_{\bar{\nu}\nu}}\sigma[D]$.

So assume there is a limit ordinal $\lambda$, $\nu_{\bar{\alpha}}<\lambda\leq\nu_{\alpha}$. Let $\lambda$ be the least limit ordinal greater than $\nu_{\bar{\alpha}}$. Let

[TABLE]

Let $D^{*}_{\lambda}$ be the ring generated by $D_{\lambda}$, We show that there is an extendable $\mathbb{R}$-monomorphism of $D^{*}_{\lambda}$ which, for any $\sigma\in\mathcal{F}_{\bar{\nu}\lambda}$, extends $\sigma[\phi]$.

Let $F$ be a finitely generated subring of $D^{*}_{\lambda}$. Let $\{d_{1},\ldots,d_{n}\}$ generate $F$, $\sigma_{1},\ldots,\sigma_{n}\in\mathcal{F}_{\bar{\nu},\lambda}$ and for all $i\leq n$, $c_{i}\in D$ be such that

[TABLE]

By condition P4 in the definition of the simplified morass, there is $n\in\omega$, and $g\in\mathcal{F}_{\nu_{\bar{\alpha}}+n,\lambda}$ and $f_{1},\ldots,f_{n}\in\mathcal{F}_{\nu_{\bar{\alpha}},\nu_{\bar{\alpha}}+n}$ such that, for $i\leq n$,

[TABLE]

For each $m<n$, let $h_{m}$ be the splitting function of $\mathcal{F}_{\bar{\nu}+m,\bar{\nu}+m+1}$. By Lemma 6.5, $\phi\cup h_{1}[\phi]$ may be extended to an extendable $\mathbb{R}$-monomorphism. Furthermore this ring monomorphism may be extended by the splitting functions $h_{2}$ through $h_{m}$. Let $\psi$ be the function on $\bigcup_{\sigma\in\mathcal{F}_{\bar{\nu}\lambda}}\sigma[D]$ resulting after the $n$ splits. Then $\psi$ is extendable and $f_{i}(c_{i})$ is in the domain of $\psi$ for all $i\leq n$. Therefore $g\circ\psi$ is an $\mathbb{R}$-linear order monomorphism and is the restriction of $\phi^{*}$ to a ring containing $F$. Thus

[TABLE]

is a well-defined extendable $\mathbb{R}$-monomorphism of $D^{*}_{\lambda}$.

By an inductive argument on $\nu$, invoking condition P2, and the results above at limits and Lemma 6.5 at successor ordinals, it is straightforward to show that $\bigcup_{\sigma\in\mathcal{F}_{\lambda,\nu}}\sigma[\phi^{*}_{\lambda}]$ has a unique extension to an extendable $\mathbb{R}$-monomorphism of the ring generated by $\bigcup_{\sigma\in\mathcal{F}_{\lambda,\nu}}\sigma[D^{*}_{\lambda}]$. $\Box$

6.2 Extensions by a specified

element

Because ordered subrings of real-closed fields have unique extensions to real-closed subfields, we will be able to restrict our attention to extending subrings by algebraically independent elements. If $X$ is a subring of a real-closed field and $Z$ is a subset of that real-closed field, we let $X[Z]$ be the subring of the real-closed field generated by $X\cup Z$. We continue with conventions similar to those outlined earlier in the section. Specifically we assume:

$M$ is a c.t.m. of ZFC 2. 2.

$P$ is the poset adding generic reals indexed by $\theta$ 3. 3.

$G$ is $P$-generic over $M$ 4. 4.

$F$ is the ring of finite elements of a standard ultrapower of $\mathbb{R}$ over $\omega$ in M[G] 5. 5.

$X\subset F$ 6. 6.

$Y\subset\mathcal{E}$ 7. 7.

$\phi:X\to Y$ is extendable.

We wish to prove analogues of Johnson’s theorems that extendable functions may be extended by a specified element. Throughout the arguments of this section, we will commonly use $x$ to represent an element of $F$, and also to represent the variable in the power series representations of member of $\mathcal{E}$. Presumably the context will make clear which use is intended.

Lemma 6.7

Suppose $\phi:X\to Y$ is an extendable function, $r\in\mathbb{R}$ and $r$ is transcendental over $X$. Then there is an extendable function that extends $\phi$, $\psi:X[r]\to Y[r]$.

Proof: Let $\phi^{*}$ be the $\mathbb{R}$-monomorphism extending $\phi$ to the real closure of $X$. Then $\phi^{*}\upharpoonright_{\mathbb{R}\cap X}$ is $\mathbb{R}$-linear. Suppose that $r$ is transcendental over $X$. The real closure of $X$ and the real closure of $Y$ contain precisely the same real numbers and are full. By Lemma 3.2, there is an $\mathbb{R}$-monomorphic extension of $\phi^{*}$, $\psi^{*}$, to the real closure of the field generated by $X[r]$. Let $\psi=\psi^{*}\upharpoonright_{X[r]}$. Then $\psi$ is extendable. $\Box$

Lemma 6.8

Suppose $\phi:X\to Y$ is an extendable function, $x\in F$ has strict level, and is transcendental over $X$. Then there is $\bar{X}\supseteq X[x]$ and extendable $\psi:\bar{X}\to\mathcal{E}$ that extends $\phi$.

Proof: If $x\in\mathbb{R}$, then apply Lemma 6.7. Assume $x\notin\mathbb{R}$. We may assume that $x$ is infinitesimal. Let $(l,u)$ be the gap formed by $x$ in $X$, $L=\phi[l]$ and $U=\phi[u]$. The Esterle algebra is an $\eta_{1}$-ordering, so there is $y\in\mathcal{E}$ that witnesses the gap $(L,U)$. By application of Johnson’s Lemma 3.3, there is $y\in\mathcal{E}$ that witnesses the gap $(L,U)$ and such that the real closure of the field extending $Y[y]$ is full. Although the existence of such an element can be used to advantage, the element $y$ may fail to have some of the properties we require for a morass construction.

Let $\mu\subseteq\omega_{1}$ be the strict level of $x$. We seek a candidate in $\mathcal{E}$ for the image of $x$ with strict level $\mu$ that is transcendental over $Y$. The Esterle algebra is level dense and upward level dense, so by Lemma 4.5 of [References] there is $y\in\mathcal{E}$, with the strict level $\mu$, such that for all $z\in X$,

[TABLE]

It may happen that $Y[y]$ is not closed under partial sums.

Case 1: There is a largest partial sum of $y$ that is a member of $Y$.

We include in this case that the first term of $y$ is a monomial not in $Y$. Let $t$ be the largest partial sum of $y$ that is also a member of $Y$ and $s=\phi^{-1}(t)\in X$. We shift our attention to the gap formed by $x-s$ in $X$. Then for all $z\in X$

[TABLE]

We note that $x-s$ is transcendental over $X$. Either $x-s$ has the same Archimedean valuation as a member of $X$, or it has a distinct valuation.

Assume $x-s$ has the same Archimedean valuation as a member of $X$. Let $\alpha x^{a}$ be the leading term of $y-t$. Then $x^{a}\in Y$ and $\alpha\notin Y$. The coefficient $\alpha$ has strict level contained in $\mu$. Let $(L^{*},U^{*})$ be the cut formed in $\mathbb{R}\cap X$ by $\alpha$. Then there is a real number with strict level $\mu$ that witnesses the gap $(L^{*},U^{*})$. So we assume that $\alpha$ has strict level $\mu$. By Lemma 6.7 there is an extendable term function, $\psi\supset\phi$, $\psi:X[\alpha]\to Y[\alpha]$, with $\psi(\alpha\phi^{-1}(x^{a}))=\alpha x^{a}$.

We assume that $x-s$ has a valuation distinct from the valuations of members of $X$. We also note that the strict level of $s$ and $t$ are equal and contained in $\mu$. If the strict level of $s$ is a proper subset of $\mu$, then $x-s$ and $y-t$ have level $\mu$. If the strict level of $x-s$ is contained in $\mu$, then there is an element of $y^{*}\in\mathcal{E}$ with a valuation distinct from the valuations of $Y$ having strict level equal to the strict level of $x-s$.

We need consider only the case in which $x$ has valuation distinct from the valuations of $X$. Let $y\in\mathcal{E}$ witness the gap $(L,U)$, and let $y_{0}=\alpha x^{a}$ be the leading term of $y$. If $\alpha>0$, then $x^{a}$ witnesses the gap $(L,U)$.

We assume without loss of generality that $y_{0}=x^{a}$. The strict level of $y_{0}$ equals the strict level of $a\in G_{\omega_{1}}$. It is straightforward to see that there is $b\in G_{\omega_{1}}$ such that

$b$ is positive. 2. 2.

Any element of the support of $b$ is greater than any element of the support of any exponent occurring in any power series of $Y$. 3. 3.

$b$ has strict level $\mu$.

Let $S$ be the union of supports of all exponents among the elements in $Y$. The exponent $b\in G_{\omega_{1}}$ is greater than [math] but less than any positive exponent in $Y$. Furthermore, $b$ is greater than the constant [math] function, the additive identity of $G_{\omega_{1}}$ (and the valuation of standard reals $\mathcal{E}$). However $b$ is greater than any positive valuation occurring in $Y$. Consequently $x^{b}$ is less than any positive infinitesimal of $Y$. Additionally,

$x^{a+b}$ witnesses the gap $(L,U)$. 2. 2.

$x^{a+b}$ is transcendental over $X$. 3. 3.

$x^{a+b}$ has strict level $\mu$. 4. 4.

The ring generated by $Y\cup\{x^{a+b}\}$ in $\mathcal{E}$ is closed under partial sums and coefficients.

Let $\psi:X[x]\to Y[x^{a+b}]$ be the unique $\mathbb{R}$-monomorphism extending $\phi$ such that

[TABLE]

Then $\psi$ is extendable.

Case 2: There is no largest partial sum of $y$ that is a member of $Y$.

Let $D$ be the countable well-ordered sequence of exponents of $y$. Let $D^{\prime}$ be the smallest initial segment of $D$ such that $y\upharpoonright_{D^{\prime}}$ is not a member of $Y$. Let $y^{\prime}=y\upharpoonright_{D^{\prime}}$. Then $y^{\prime}$ witnesses the gap $(L,U)$. The strict level of $y^{\prime}$ is a subset of $\mu$, and is possibly proper subset of $\mu$. If the strict level of $y^{\prime}$ equals $\mu$, let $\psi:X[x]\to Y[y^{\prime}]$ be the unique $\mathbb{R}$-monomorphism extending $\phi$ such that

[TABLE]

Otherwise, let $a\in G_{\omega_{1}}$ be an exponent of $\mathcal{E}$ that has strict level $\mu$ and is greater than all exponents occurring in $Y$. Let

[TABLE]

Then $y^{*}$ witnesses the gap $(L,U)$ and has strict level $\mu$. Let $\psi^{*}:X[x]\to Y[y^{*}]$ be the unique $\mathbb{R}$-monomorphism extending extending $\phi$ such that

[TABLE]

Then $\psi$ satisfies the conditions for an extendable function, except for closure under partial sums. In particular $y^{\prime}$ and $x^{a}$ are not in the range of $\psi^{*}$. Let $(L^{*},U^{*})$ be the gap formed by $x^{a}$ in $Y[y^{*}]$. We observe that $x^{a}$ is infinitesimal with respect to every member of $Y[y^{*}]$. Since $F$ is level dense and upward level dense, there is a positive element of $F$, $\epsilon$, with strict level $\mu$ that is infinitesimal with respect to all elements of $X[x]$. Therefore there is an $\mathbb{R}$-monomorphism $\psi:X[x,\epsilon]\to Y[y^{*},x^{a}]$ extending $\psi^{*}$ and such that

[TABLE]

We note that $y\in Y[y^{*},x^{a}]$ and

[TABLE]

Then $Y[y,x^{a}]$ is closed under coefficients and partial sums, so $\psi$ is extendable. $\Box$

These results permit a simplification of the construction. Given an extendable $\mathbb{R}$-monomorphism, $\phi:D\to\mathcal{E}$, we may extend $\phi$ to an $\mathbb{R}$-monomorphism of $D[\mathbb{R}]$, and then extend by algebraically independent infinitesimals.

7 An $\mathbb{R}$-monomorphism from the finite elements of ${\mathbb{R}}^{\omega}/U$

into the Esterle algebra.

In the Cohen extension adding $\aleph_{2}$ generic reals, we construct a level $\mathbb{R}$-monomorphism, $\phi$, from the finite elements of a morass-definable ultrapower of $\mathbb{R}$ into $\mathcal{E}$. This construction differs significantly from the construction of Woodin [References]. The construction of Woodin relies on the fact that in the Cohen extension of $L$ by $\aleph_{2}$-generic reals, any cut of $\mathcal{E}$ has a countable, cofinal subcut. As Woodin observes, this argument is not generalizable to models of ZFC with higher powers of the continuum. Our construction yields a monomorphism that is level, and therefore respects the “complexity” (with respect to the index of Cohen reals) of the elements of the ultrapower and the Esterle algebra. Consequently, for any $S\subseteq\omega_{2}$, $\phi\cap M[G(S)]$ is an $\mathbb{R}$-monomorphism of ring of finite elements of ${\mathbb{R}}^{\omega}/U\cap M[G(S)]$ to $\mathcal{E}\cap M[G(S)]$.

Theorem 7.1

Suppose $M$ is a c.t.m. of ZFC+CH containing a simplified $(\omega_{1},1)$-morass and $P$ is the poset adding generic reals indexed by ordinals less than $\omega_{2}$. Let $G$ be $P$-generic over $M$, $F\in M[G]$ be the ring of finite elements of an ultrapower of $\mathbb{R}$ over a standard ultrafilter on $\omega$, and $\mathcal{E}\in M[G]$ be the Esterle Algebra computed in $M[G]$. Then there is a level $\mathbb{R}$-monomorphism, $\phi:F\to\mathcal{E}$.

Proof: For each $\nu$, $\alpha<\omega_{1}$, let $G_{\nu}$ be the factor of $G$ adding generic reals indexed by $\theta_{\nu}$ (the ordinal associated with the vertex $\nu$ in the morass), and, for $S\subseteq\omega_{1}$, $G(S)$ be the factor of $G$ adding generic reals indexed by $S$. Let

[TABLE]

[TABLE]

and $X$ be a maximal algebraically independent set of positive morass generators of $F$. Then $F$ is composed of the finite elements of the real closure of the field generated by the morass-closure of $X$.

Let $\langle x_{\alpha}\mid\alpha<\omega_{1}\rangle$ be a well-ordering of $X$. Let $\langle\nu_{\alpha}\mid\alpha<\omega_{1}\rangle$ be a weakly ascending transfinite sequence of countable ordinals such that $x_{\alpha}\in M[G_{\nu_{\alpha}}]$. We will construct an ascending sequence of functions (ordered by inclusion), $\langle\phi_{\alpha}:D_{\alpha}\to\mathcal{E}\mid\alpha<\omega_{1}\rangle$, such that for all $\alpha<\omega_{1}$

$x_{\alpha}\in D_{\alpha}$ 2. 2.

$\phi_{\alpha}:D_{\alpha}\to E_{\alpha}$ is extendable 3. 3.

If $x\in D_{\alpha}$, $\nu\leq\nu_{\alpha}$, the morass generator of $x$ is $x^{*}\in M[G_{\nu}]$ and $\sigma\in\mathcal{F}_{\nu\nu_{\alpha}}$, then $\sigma(x^{*})\in D_{\alpha}$.

At each stage of the construction, $\alpha<\omega_{1}$, the domain of $\phi_{\alpha}$ extends the domain of $\bigcup_{\beta<\alpha,f\in\mathcal{F}_{\beta\alpha}}f(\phi_{\beta})$, so that it includes all morass descendants in $M[G_{\nu_{\alpha}}]$, and $\phi_{\alpha}$ is extendable.

Case: $\alpha=0$.

Let $\langle x_{0,n}\mid n\in\omega\rangle\subseteq M[G_{\nu_{0}}]$ be a well-ordering of the morass-descendants of $x_{0}$ in $M[G_{\nu_{0}}]$. We construct a sequence of $\mathbb{R}$-monomorphisms, $\langle\phi_{0,n}:D_{0,n}\to\mathcal{E}\mid n\in\omega\rangle$ such that for all $n\in\omega$,

$D_{0,0}=\mathbb{Q}[x_{0,0}]$ 2. 2.

$D_{0,n}[x_{0,n+1}]\subseteq D_{0,n+1}$ 3. 3.

$\phi_{0,n}\in M[G_{\nu_{0}}]$ 4. 4.

For all $m<n$, $\phi_{0,m}\subseteq\phi_{0,n}$. 5. 5.

$\phi_{0,n}$ is extendable.

Let $z=x_{0,0}$, and $\mu$ be the strict level of $z$. We may assume that $z$ is positive. If $z\in\mathbb{R}$, let $D_{0,0}=\mathbb{Q}[z]$ and $\phi_{0,0}:D_{0,0}\to\mathcal{E}$ be the identity restricted to $D_{0,0}$.

If $z$ is infinitesimal, let $a\in G_{\omega_{1}}$ be positive and have strict level $\mu$, and

[TABLE]

Let $D_{0,0}=\mathbb{Q}[z]$ and $\phi_{0,0}:D_{0,0}\to\mathcal{E}$ be the $\mathbb{R}$-linear ring monomorphism such that

[TABLE]

If $z=r+\epsilon$, where $\epsilon$ is infinitesimal and positive, let $a\in G_{\omega_{1}}$ have the same strict level as $\epsilon$. Then let

[TABLE]

and $D_{0,0}=\mathbb{Q}[r,\epsilon]$. Let $\phi_{0,0}:D_{0,0}\to\mathcal{E}$ be the ring monomorphism extending the identity on $\mathbb{Q}[r]$ such that

[TABLE]

If $\epsilon<0$, let $y=-x^{a}$ and define $D_{0,0}$ and $\phi_{0,0}$ analogously.

Let $N\in\omega$ and assume that $\langle\phi_{0,n}\mid n\leq N\rangle$ satisfies conditions 1 - 5 above (below $N+1$). Then $\phi_{0,N}$ is extendable and Lemmas 6.7 and 6.8 apply. Let $z=x_{0,N+1}$. Since $z$ is the image of a morass-generator with strict level, $z$ is transcendental over $D_{0,N}$. If $z\in\mathbb{R}$, let

[TABLE]

and

[TABLE]

By Lemma 6.7, there is an $\mathbb{R}$-monomorphic extension of $\phi^{*}$, $\phi_{0,N+1}:D_{0,N+1}\to\mathcal{E}$. The sub-ring of $\mathcal{E}$ generated by a set closed under partial sums and coefficients is also closed under partial sums and coefficients, so $\phi_{0,N+1}$ is extendable.

If $z$ is non-standard, let $R^{*}$ be the set of reals contained in the smallest full real closure of $D_{0,N}\cup\{z\}$. By Lemma 6.7 there is an extendable level $\mathbb{R}$-monomorphism extending $\phi_{0,N}$, $\phi^{*}:D_{0,N}[R^{*}]\to\mathcal{E}$. Let

[TABLE]

Then $D_{0,N+1}\in M[G_{\nu_{0}}]$. By Lemma 6.8 there is an extendable extension of $\phi^{*}$,

[TABLE]

Let

[TABLE]

and

[TABLE]

Then the morass descendants of $x_{0}$ (in $M[G_{\nu_{0}}]$) are elements of the domain of $\phi_{0}$, and $\phi_{0}$ is extendable.

Assume $\alpha$ a successor.

Let $\alpha=\bar{\alpha}+1$. Assume that $\langle D_{\beta}\mid\beta<\alpha\rangle$ and $\langle\phi_{\beta}\mid\beta<\alpha\rangle$ have been defined so that for all $\gamma<\beta<\alpha$

$x_{\gamma}\in D_{\gamma}$ 2. 2.

$D_{\gamma}\subseteq D_{\beta}$ is closed under morass-maps below $\beta$ 3. 3.

$\phi_{\gamma}\subseteq\phi_{\beta}$ 4. 4.

$\phi_{\beta}:D_{\beta}\to\mathcal{E}$ is an extendable $\mathbb{R}$-monomorphism.

If $\nu_{\alpha}=\nu_{\bar{\alpha}}$, then we may argue as in the previous case. Let $\langle x_{\alpha,n}\mid n<\omega\rangle$ be an enumeration of the morass descendants of $x_{\alpha}$ in $M[G_{\nu_{\alpha}}]$. We may extend $D_{\bar{\alpha}}$ to $D_{\alpha}$ containing the morass descendants of $x_{\alpha}$, and $\phi_{\bar{\alpha}}$ to $\phi_{\alpha}:D_{\alpha}\to\mathcal{E}$ so that for all $\gamma<\beta\leq\alpha$

$D_{\gamma}\subseteq D_{\beta}$. 2. 2.

$\phi_{\gamma}\subseteq\phi_{\beta}$. 3. 3.

$\phi_{\alpha}$ is extendable.

If $\nu_{\bar{\alpha}}<\nu_{\alpha}$, then by Lemma 6.6 there is an extendable $\mathbb{R}$-monomorphism, $\phi^{*}$, extending $\bigcup_{\sigma\in\mathcal{F}_{\nu_{\bar{\alpha}}\nu_{\alpha}}}\sigma[\phi_{\bar{\alpha}}]$ to the ring generated by $\bigcup_{\sigma\in\mathcal{F}_{\nu_{\bar{\alpha}}\nu_{\alpha}}}\sigma[D_{\bar{\alpha}}]$, $D^{*}$. Let $D^{\prime}$ be ring generated by $D^{*}$ and the morass descendants of $x_{\alpha}$ in $M[G_{\nu_{\alpha}}]$. Let $\mathbb{R}^{*}$ be the real numbers of the smallest full extension of the real closure of $D^{\prime}$. Then the real closure of $D^{\prime}[\mathbb{R}^{*}]$ is full. Let

[TABLE]

By the preceding case, there is an extendable $\mathbb{R}$-monomorphism $\phi_{\alpha}:D_{\alpha}\to\mathcal{E}$ with $\phi_{\alpha}\supseteq\phi^{*}$.

Finally, assume $\alpha<\omega_{1}$ is a limit ordinal.

If there is $\beta<\alpha$ such that $\nu_{\alpha}=\nu_{\beta}$ then we may proceed as in the case $\alpha=0$ to define $D_{\alpha}$ and $\phi_{\alpha}$.

So we assume that $\nu_{\beta}<\nu_{\alpha}$ for all $\beta<\alpha$. Let

[TABLE]

If $\lambda=\nu_{\alpha}$, then let

[TABLE]

Let $D$ be a finitely generated subring of $D^{*}$ with generators $\{d_{1},\ldots,d_{n}\}$. Then there is $\beta<\alpha$,

[TABLE]

and for, $i\leq n$, morass functions $\sigma_{i}\in\mathcal{F}_{\nu_{\beta}\nu_{\alpha}}$ such that

[TABLE]

By condition P4 of Definition 2.1, there is $\nu_{\beta}\leq\gamma<\lambda$, $f_{1},\ldots,f_{n}\in\mathcal{F}_{\nu_{\beta}\gamma}$ and $g\in\mathcal{F}_{\gamma\lambda}$ such that for all $i\leq n$,

[TABLE]

By Lemma 6.6 there is an extendable $\mathbb{R}$-monomorphism $\phi^{*}$ on the ring generated by $\bigcup_{\sigma\in\mathcal{F}_{\nu_{\beta}\lambda}}\sigma[\phi_{\beta}]$. It follows that there is an extendable $\mathbb{R}$-monomorphism on the ring generated by $\bigcup_{\beta<\alpha}(\bigcup_{\sigma\in\mathcal{F}_{\nu_{\beta}\nu_{\alpha}}}\sigma[D_{\beta}])$ extending $\bigcup_{\beta<\alpha}(\bigcup_{\sigma\in\mathcal{F}_{\nu_{\beta}\nu_{\alpha}}}\sigma[\phi_{\beta}])$. We may then proceed as in earlier cases to define an extendable $\mathbb{R}$-monomorphism $\phi_{\alpha}:D_{\alpha}\to\mathcal{E}$, with $D_{\alpha}$ containing the morass descendants of $x_{\alpha}$ in $M[G_{\nu_{\alpha}}]$.

Finally, assume that

[TABLE]

By the previous argument, there is extendable $\phi^{*}:\bigcup_{\beta<\alpha}(\bigcup_{\sigma\in\mathcal{F}_{\nu_{\beta}\lambda}}\sigma[D_{\beta}])\to\mathcal{E}$ extending the morass images in $M[G_{\lambda}]$ of the $\phi_{\beta}$. By Lemma 6.6 there is an extendable $\mathbb{R}$-monomorphism $\phi^{\prime}_{\alpha}:D^{\prime}_{\alpha}\to\mathcal{E}$ such that

[TABLE]

and

[TABLE]

We proceed as in earlier cases to extend $\phi^{\prime}_{\alpha}$ to an extendable $\mathbb{R}$-monomorphism, $\phi_{\alpha}:D_{\alpha}\to\mathcal{E}$, where $D_{\alpha}$ extends $D^{\prime}$ and contains the morass descendants of $x_{\alpha}$ in $M[G_{\nu_{\alpha}}]$.

Let $D^{*}$ be the ring in $M[G]$ generated by $\bigcup_{\alpha<\omega_{1}}(\bigcup_{\sigma\in\mathcal{F}_{\nu_{\alpha}\omega_{1}}}\sigma[D_{\alpha}])$. Then by earlier argument there is an $\mathbb{R}$-monomorphism, $\phi^{*}:D^{*}\to\mathcal{E}$, such that

[TABLE]

Then $\phi^{*}$ is an $\mathbb{R}$-monomorphism on a domain that contains a transcendental basis for $F$, and extends uniquely to an $\mathbb{R}$-monomorphism, $\phi:F\to\mathcal{E}$. $\Box$

Theorem 7.2

Suppose $M$ is a c.t.m. of $ZFC+CH$ containing a simplified $(\omega_{1},1)$-morass and $M[G]$ is the Cohen extensions adding $\aleph_{2}$ generic reals. Then if $X$ is an infinite compact Hausdorff space in $M[G]$, there is a discontinuous homomorphism of $C(X)$ in $M[G]$.

Proof: By Corollary 6.9 of [References], any non-principal ultrafilter on $\omega$ in $M$ may be extended to a standard ultrafilter in the Cohen extension adding $\aleph_{2}$-generic reals. If $U$ is a standard ultrafilter, then there is a level $\mathbb{R}$-monomorphism from the finite elements of ${\mathbb{R}}^{\omega}/U$ into $\mathcal{E}$. Hence the finite elements of ${\mathbb{R}}^{\omega}/U$ bear a non-trivial submultiplicative norm. The theorem follows from results of B. Johnson [References]. $\Box$

8 Discontinuous homomorphisms of $C(X)$ in

a Cohen extension adding $\aleph_{3}$-generic reals.

The existence of a discontinuous homomorphism of $C(X)$ when the continuum has power $\aleph_{2}$ has been known for a long while (Woodin [References]). Unfortunately the methods of the existing proof do not extend to higher powers of the continuum. It has been suggested that the consistency of a discontinuous homomorphism of $C(X)$ with higher powers of the continuum might be proved with higher-gap morasses. Theorem 7.1 is a modest strengthening of Woodin’s result, as the constructed $\mathbb{R}$-monomorphism is an extendable function (except for the cardinality of the function). We require a simplified $(\omega_{1},2)$-morass for the next construction (Velleman [References]).

Definition 8.1

$(Simplified\>\>(\kappa,2)-morass)$* 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.* 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)

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

Theorem 8.2

Let $M$ be a c.t.m. of $ZFC+CH$ containing a simplified $(\omega_{1},2)$-morass, and $M[G]$ be a generic extension of $M$ adding $\aleph_{3}$ generic reals. Let $X$ be an infinite compact Haussdorf space in $M[G]$, and $C(X)$ be the algebra of continuous real-valued functions of $X$ in $M[G]$. Then there is a discontinuous homomorphism of $C(X)$ in $M[G]$.

Proof: Let $M$ be a c.t.m. 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 adding generic reals indexed by $\omega_{3}$, and $G$ be $P$-generic over $M$. Then $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}}\rangle$ is a simplified $(\omega_{2},1)$-morass, and below $\omega_{1}$, $\langle\overrightarrow{\varphi},\overrightarrow{\mathcal{G}}\rangle$ satisfies the axioms of a simplified $(\omega_{1},1)$-morass. Hence the construction of Theorem 7.1 below $\omega_{1}$ can be completed in $M$. In particular, if $U_{0}\in M$ is a non-principal ultrafilter in $M$, then by Corollary 6.9 of [References], there is $\overline{U}\subseteq M^{P(\omega_{1})}$, a standard term for an ultrafilter that is morass-commutative below $\omega_{1}$, that is forced to extend $U_{0}$. Furthermore, the morass-continuation of $\overline{U}$, $U$, is a standard ultrafilter that is gap-2 morass-commutative and morass-commutative. By Theorem 6.4 of [References], ${\mathbb{R}}^{\omega}/U$ is a gap-2 morass-definable $\eta_{1}$ real-closed field.

We adapt the argument of Theorem 7.1 to the gap-2 morass construction of [References]. Hence we construct a level term function from the finite elements of a standard ultrapower to the Esterle algebra that is closed under morass-embeddings and is forced to be an $\mathbb{R}$-monomorphism. For $\alpha<\omega_{1}$, let $X_{\alpha}=({\mathbb{R}}^{\omega}/U)\cap M^{P_{\alpha}}$ and $Y_{\alpha}=\mathcal{E}\cap M^{P_{\alpha}}$. We consider $X_{\alpha}$ and $Y_{\alpha}$ as the restrictions of ${\mathbb{R}}^{\omega}/U$ and $\mathcal{E}$, 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 construct a morass-commutative level term injection from $X_{\omega_{1}}$ to $Y_{\omega_{1}}$ that is forced to be an $\mathbb{R}$-linear order-monomorphism. 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 $\{x_{\beta}\mid\beta<\omega_{1}\}\subseteq X_{\omega_{1}}$ be a transfinite sequence of terms of strict level for a maximal algebraically independent set of morass generators of the infinitesimal elements of $X_{\omega_{1}}$, such that $x_{\alpha}\in X_{\alpha}$ for all $\alpha<\omega_{1}$. 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 $\alpha\leq\beta<\omega_{1}$,

$D_{\beta}\subseteq X_{\theta_{\beta}}$ is a morass-commutative subring 2. 2.

$E_{\beta}\subseteq Y_{\theta_{\beta}}$ is closed under partial sums and contains all coefficients 3. 3.

$D_{\alpha}\subseteq D_{\beta}$ and $E_{\alpha}\subseteq E_{\beta}$ 4. 4.

$x_{\beta}\in D_{\beta}$ 5. 5.

$F_{\beta}$ is a level term function that is forced to be an order-preserving, $\mathbb{R}$-monomorphism 6. 6.

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

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

Base Case: $\gamma=0$.

Let $y_{0}$ be a positive infinitesimal monomial of $Y_{0}$ having the same strict level as $x_{0}$, and $\mathbb{R}_{0}$ be the reals of the ground model. Let $D_{0}$ be the ring generated by $\mathbb{R}_{0}\cup\{x_{0}\}$, $\mathbb{R}_{0}[x_{0}]$, and $E_{0}=\mathbb{R}_{0}[y_{0}]$. We observe that $E_{0}$ is closed under partial sums. There is an $R$-linear order-monomorphism, $F_{0}:D_{0}\to E_{0}$, with $F_{0}(x_{0})=y_{0}$.

Successor Case: $\gamma=\beta+1$. Let $\langle F_{\alpha}:D_{\alpha}\to E_{\alpha}\mid\alpha\leq\beta\rangle$ be an extendable sequence satisfying conditions 1-6 above. Let $D^{*}$ be the ring generated by $\{g_{\theta_{\beta}}[D_{\beta}]\mid g\in\mathcal{G}_{\theta_{\beta}\theta_{\gamma}}\}$. Then $D^{*}$ is generated by the union of the images of $D_{\beta}$ under the second components of left-branching embeddings of $\mathcal{F}_{\beta\gamma}$. Let $h$ be the right-branching embedding of $\mathcal{F}_{\beta\gamma}$ and $D^{\prime}$ be the ring generated by $D^{*}$ and $h_{\theta_{\beta}}[D_{\beta}]$.

By Lemma 5.2 of [References], $\bigcup\{f_{\theta_{\beta}}[F_{\beta}]\mid f\in\mathcal{F}_{\beta\gamma}\}$ is a level term injection that is forced to be an $\mathbb{R}$-linear order-preserving injection. The Lemma does not guarantee that $\bigcup\{f_{\theta_{\beta}}[F_{\beta}]\mid f\in\mathcal{F}_{\beta\gamma}\}$ extends to a homomorphism of $D^{\prime}$. However, by Lemma 6.6, there is a unique extendable $\mathbb{R}$-monomorphism, $F^{*}:D^{*}\to\mathcal{E}$.

Let $B$ be a transcendental basis of $D_{\beta}$ over ${\mathbb{R}}^{\omega}/U\cap M$ containing terms of strict level. Let $D$ be the ring generated by $B$ and $D_{\beta}\cap M$. Let $V$ be the semigroup generated by $B$. Because $\mathcal{F}_{\beta\gamma}$ is a set of compatible embeddings, Lemma 6.1 applies and, treating $D$ as a vector space (over the field generated by $D_{\beta}\cap M$), $V$ is a basis of elements of strict level for $D$. Therefore, if $F:D\to\mathcal{E}$ is the naturally induced $\mathbb{R}$-linear extension of $\bigcup_{f\in\mathcal{F}_{\beta\gamma}}f_{\theta_{\beta}}[F_{\beta}]$, the image of $V$ under $F$ is a linearly independent subset of $\mathcal{E}$ over $E_{\beta}\cap M$. Therefore $F$ is a linear transformation and an $\mathbb{R}$-monomorphism of $D$. We argue along the lines of Lemma 6.5 to show that $F^{\prime}$ is forced to be order-preserving. By Lemma 6.5, $F^{\prime}\upharpoonright_{D^{*}}$ is forced to be order-preserving. If $z\in D^{\prime}$, there is $n\in\mathbb{N}$, $x_{1},\ldots,x_{n}\in D^{*}$ and $y_{1},\ldots,y_{n}\in f_{\theta_{\beta}}[D_{\beta}]$ such that $z=\sum_{i=1}^{n}x_{i}\cdot y_{i}$. We show that $\sum_{i=1}^{n}x_{i}\cdot y_{i}>0$ iff $\sum_{i=1}^{n}F^{\prime}(x_{i})\cdot F^{\prime}(y_{i})>0$.

If $n=1$, then the sign of $x_{1}\cdot y_{1}$ is the sign of the product of the leading coefficients of $F^{\prime}(x_{1})$ and $F^{\prime}(y_{1})$. However, $F^{\prime}$ is $\mathbb{R}$-linear, so

[TABLE]

Assume that $n\geq 2$, and

[TABLE]

Let

[TABLE]

By assumption, the range of $F^{\prime}$ is closed under partial sums, so there are elements $u_{1},\ldots,u_{j},u_{j+1},\ldots u_{k},v_{1},\ldots,v_{k}\in Y$ such that

$z=\sum_{i=1}^{k}u_{i}\cdot v_{i}$ 2. 2.

For $i\leq j$, every term in the power series expansion of $u_{i}\cdot v_{i}$ has power less than $a_{0}$ 3. 3.

For $j<i\leq k$, every term of the power series expansion of $u_{i}\cdot v_{i}$ has power at least $a_{0}$.

Every term of $z$, expressed as a power series of $\mathcal{E}$, has valuation no less than $a_{0}$, therefore

[TABLE]

and

[TABLE]

If $j<i\leq k$, then $s_{i}=u_{i}\cdot v_{i}$ has valuation no less than $a_{0}$ in $G_{\omega_{1}}$. If the leading terms of $s_{i}$ has exponent greater than $a_{0}$, then $s_{i}$ has Archimedean valuation less than $z$. Let $S$ be the set of all terms of all $s_{i}$, for all $1\leq j\leq i$, having exponent $a_{0}$ and $T$ be the set of all terms of all $s_{i}$, $1\leq i\leq j$, with exponent greater than $a_{0}$. $S$ is nonempty and every element of $T$ has Archimedean valuation less than every element of $S$.

There are $b_{0},c_{0}\in G_{\omega_{1}}$ such that $x^{b_{0}}$ is in the range of $F^{*}$ and $x^{c_{0}}$ is in the range of $f_{\theta_{\beta}}[F_{\beta}]$ and

[TABLE]

Let $u\in D^{*}$ and $v\in f_{\theta_{\beta}}[D_{\beta}]$ be such that

[TABLE]

and

[TABLE]

We observe that the ordered ring $D^{\prime}$ has a unique extension to the field closure of $D^{\prime}$, $\bar{D}$.Treated as a field map, the unique monomorphic extension of $F^{\prime}$, $\bar{F}:\bar{D}\to\mathcal{E}$, is an $\mathbb{R}$-linear field monomorphism of $D^{\prime}$. We have previously observed that $F^{\prime}$ is an $\mathbb{R}$-linear monomorphism, and $F^{*}\cup f_{\theta_{\beta}}[F_{\beta}]$ is an order-preserving injection. So

[TABLE]

and

[TABLE]

The real number, $\alpha_{0}$, is in the domain $\bar{F}$, so $\bar{F}(\alpha_{0})=F^{\prime}(\alpha_{0})=\alpha_{0}$. The range of $F^{\prime}$ is closed under partial sums, so both $\sum_{0<\lambda<\gamma}\alpha_{\lambda}x^{a}_{\lambda}$ and $x^{a_{0}}$ are in the range of $F^{\prime}$. Thus $\sum_{0<\lambda<\gamma}\alpha_{\lambda}x^{a_{\lambda}-a_{0}}$ is in the range of $\bar{F}$. Let

[TABLE]

Every term of $\delta$ is infinitesimal (has Archimedean valuation greater than elements of $\mathbb{R}$). $\bar{F}$ is order preserving on $D^{\prime}$, so $\bar{F}^{-1}(\delta)$ is infinitesimal. Hence

[TABLE]

and the $\sum_{i=1}^{n}x_{i}\cdot y_{i}>0$ iff $\alpha_{0}>0$ iff $F^{\prime}(\sum_{i=1}^{n}x_{i}\cdot y_{i})>0$.

Therefore $F^{\prime}:D^{\prime}\to\mathcal{E}$ is an $\mathbb{R}$-linear order-monomorphism, in which the range of $F^{\prime}$ contains all coefficients and all partial sums. Let $\hat{D}$ be the ring generated by $D^{\prime}$ and $\mathbb{R}\cap M[G_{\theta_{\gamma}}]$. By application of Lemma 6.7, we may extend $F^{\prime}$ to a level order-preserving $\mathbb{R}$-monomorphism, $\hat{F}:\hat{D}\to\mathcal{E}$. Assume that $x_{\gamma}\notin D^{\prime}[\mathbb{R}]$. By Lemma 6.8, there is $D_{\gamma}\supset\hat{D}$, with $x_{\gamma}\in D_{\gamma}$, and an extension of $\hat{F}$, $F_{\gamma}:D_{\gamma}\to\mathcal{E}$, such that $F_{\gamma}$ is an extendable function.

Limit Case: Suppose $\gamma$ is a limit ordinal. Let $D^{\prime}=\bigcup_{\beta<\gamma}(\bigcup_{f\in\mathcal{F}_{\beta\gamma}}{f_{\theta_{\beta}}[D_{\beta}]})$. Then $D^{\prime}$ is a subring of $D_{\omega_{1}}$ and has countable transcendence degree over $\mathbb{R}$. By condition 6 of Definition 8.1 it is straightforward to verify that $F^{\prime}:D^{\prime}\to\mathcal{E}$ defined by $F^{\prime}=\bigcup_{\beta<\gamma}(\bigcup_{f\in\mathcal{F}_{\beta\gamma}}\{f_{\theta_{\beta}}[F_{\beta}]\})$ is an extendable function. Since $D^{\prime}$ is full, we may apply Lemma 6.7 to extend $F^{\prime}$ to an $\mathbb{R}$-linear order-monomorphism of $\hat{D}$. Let $D_{\gamma}=\hat{D}$. Then by Lemma 6.8 there is an extension of $F^{\prime}$, $F_{\gamma}:D_{\gamma}\to\mathcal{E}$ that is an extendable function. Let $F=\{f_{\omega_{1}}[F_{\omega_{1}}]\mid f\in\mathcal{F}_{\omega_{1}\omega_{2}}\}$. By Lemma 3.3 of [References], the domain of $F$ is forced to be the finite elements of ${\mathbb{R}}^{\omega}/U$. $\Box$

Corollary 8.3

Let $M$ be a c.t.m. of $ZFC+CH$ containing a simplified $(\omega_{2},2)$-morass, and $M[G]$ be a generic extension of $M$ adding $\aleph_{4}$ generic reals. Let $X$ be an infinite compact Haussdorf space in $M[G]$. Then there is a discontinuous homomorphism of $C(X)$, the algebra of continuous real-valued functions on $X$ in $M[G]$.

Proof. By Woodin’s argument [References], in the generic extension adding $\aleph_{2}$ generic reals to $L$, all cuts of $\mathcal{E}$ have countable cofinal subcuts. Let $M$ be a model of $ZFC+CH$ containing a simplified $(\omega_{2},2)$-morass and $G$ be the generic extension adding $\aleph_{4}$ generic reals. If the cuts of $\mathcal{E}$ in $M[G]$ admit countable cofinal subcuts, then the argument for Theorem 8.2 yields the existence of a discontinuous homomorphism of $C(X)$ in a model with $2^{\aleph_{0}}=\aleph_{4}$. $\Box$

9 Pressing the continuum

The techniques of this paper, and those of [References] and [References] depend on the construction of functions between sets of terms in the forcing language of Cohen extensions, utilizing commutativity with order-preserving injections indexing ordinals of the Cohen poset. Having presented the details of the construction of level term functions using simplified gap-1 and gap-2 morasses, it is relatively straightforward to see how these constructions extend to simplified higher (finite) gap morasses.

A simplified $(\kappa,n+1)$-morass, for $\kappa$ a regular cardinal and integer $n$, is a family of embeddings between fake simplified $(\kappa,n)$-morass segments that satisfies properties analogous to those relating a simplified $(\omega_{1},2)$-morass to embeddings between fake simplified $(\omega_{1},1)$-morass segments. Central to the utility of these constructions is that the second component of the embeddings are order-preserving injections between ordinals. For a thorough treatment of simplified finite gap morasses, including an inductive definition, see Szalkai [References].

Higher gap morasses will allow the extension of results of this paper and [References] to Cohen extensions adding more than $\aleph_{4}$ generic reals. The definition of gap-2 morass-definable $\eta_{1}$-orderings and $\eta_{1}$-ordered real-closed fields (resp.) are easily generalized to gap-n morass-definable $\eta_{1}$-orderings and $\eta_{1}$-ordered real-closed fields (resp.). We state the following without proof.

Claim 9.1

Let $M$ be a model of $ZFC+CH$ containing a simplified $(\omega_{1},n)$-morass, $P$ be the poset adding $\aleph_{n}$ generic reals and $G$ be $P$-generic over $M$. Then in $M[G]$

Gap-n morass-definable $\eta_{1}$-orderings without endpoints are order-isomorphic. 2. 2.

There exists a gap-n morass-definable, non-principal ultrafilter over $\omega$. 3. 3.

If $U$ is a gap-n morass-definable non-principal ultrafilter over $\omega$, then ${\mathbb{R}}^{\omega}/U$ is a gap-n morass-definable $\eta_{1}$-ordered real-closed field. 4. 4.

There exists an $\mathbb{R}$-isomorphism between gap-n morass-definable $\eta_{1}$-ordered real-closed fields. 5. 5.

There exists an $\mathbb{R}$-monomorphism from the finite elements of a standard ultrapower of $\mathbb{R}$ (over $\omega$) into the Esterle algebra. 6. 6.

If $X$ is an infinite compact Hausdorff space, then there exists a discontinuous homomorphism of $C(X)$, the algebra of continuous real-valued functions on $X$.