# Using Ultrapowers to Compare Continuous Structures

H. Jerome Keisler  University of Wisconsin-Madison, Department of Mathematics,
Madison, WI 53706-1388<EMAIL_ADDRESS>
###### Abstract.

In 1967 the author introduced a pre-ordering of all first order complete
theories where T is lower than U if it is easier for an ultrapower of a model
of $T$ than an ultrapower of a model of $U$ to be saturated. In a long series
of recent papers, Malliaris and Shelah showed that this pre-ordering is very
rich and gives a useful way of classifying simple theories. In this paper we
investigate the analogous pre-ordering in continuous model theory.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on continuous model theory
  3. 3 Comparing two continuous theories
  4. 4 Elementary equivalent structures are equally saturatable
  5. 5 Every continuous theory is $\trianglelefteq$-equivalent to a metric theory
  6. 6 Stable theories
  7. 7 $\trianglelefteq$-minimal theories
  8. 8 $\trianglelefteq$-maximal theories and SOP2
  9. 9 $\trianglelefteq$-minimal unstable theories and the independence property
  10. 10 $\trianglelefteq$-minimal TP2 theories

##  1\. Introduction

The results in this paper bring together two research projects that began in
the 1960’s, the so-called “Keisler order” $\trianglelefteq$, and continuous
model theory. Both of these projects were mostly dormant for four decades, and
then suddenly blossomed into active and highly successful programs during the
last decade.

In the paper [Ke67], the relation $\trianglelefteq$ was introduced and
proposed as a way of classifying first order theories111In this paper,
“theory” always means “complete theory”. with possibly different countable
vocabularies. Informally, $T\trianglelefteq U$ if is it is easier for an
ultrapower of a model of $T$ than an ultrapower of a model of $U$ to be
saturated. Formally, $T\trianglelefteq U$ means that for every cardinal
$\lambda$, every regular ultrafilter $\mathcal{D}$ over a set of cardinality
$\lambda$, and every model $\mathcal{M}$ of $T$ and $\mathcal{N}$ of $U$, if
the ultrapower $\mathcal{N}_{\mathcal{D}}$ is $\lambda^{+}$-saturated then so
is the ultrapower $\mathcal{M}_{\mathcal{D}}$. $T$ and $U$ are
$\trianglelefteq$-equivalent if $T\trianglelefteq U$ and $U\trianglelefteq T$.
The relation $\trianglelefteq$ is obviously transitive. It is shown in [Ke67]
that $T\trianglelefteq T$, so $\trianglelefteq$ is a pre-ordering on the set
of all first order theories. In a remarkable series of recent papers,
Malliaris and Shelah clarified the behavior of $\trianglelefteq$ on first
order theories. They showed that $\trianglelefteq$ gives a useful
classification of simple theories, that is so rich that it reveals
$2^{\aleph_{0}}$ kinds of simple theories with countable vocabularies.

The modern treatment of continuous model theory deals with metric structures.
A surprisingly large part of first order model theory can be generalized to
the continuous case. Many notions and results in first order model theory have
analogues for metric structures, including ultrapowers and $\kappa$-saturated
structures. In recent years, the model theory of metric structures has had
exciting mathematical applications in analysis.

In this paper we study the analogue of the relation $\trianglelefteq$ on the
class of metric theories (complete theories of metric structures). Some care
is needed in our terminology, because we will be comparing metric theories
with possibly different vocabularies.

Metric structures are defined as follows. By a vocabulary $V$ we mean a
countable set of constant, function, predicate symbols. A continuous formula
in $V$ is built from atomic formulas using continuous functions on the real
unit interval $[0,1]$ as connectives, and the quantifiers $\inf$ and $\sup$. A
_real-valued structure with vocabulary_ $V$ (briefly, a _$V$ -structure_) is
like a first order structure except that the formulas take truth values in
$[0,1]$, with $0$ representing truth. A _metric signature_ $L$ over $V$
specifies a distinguished binary predicate symbol $d\in V$, called the
distance predicate of $L$, and a uniform continuity bound for each function
and predicate symbol in $V$. An _$L$ -metric structure_ is a $V$-structure
$\mathcal{M}$ in which the distance predicate of $L$ is a complete metric and
all the function and predicate symbols of $V$ respect the bounds of uniform
continuity. First order structures with equality are just $L$-metric
structures where the distance predicate of $L$ is the discrete metric and all
atomic formulas have truth values in $\\{0,1\\}$. We define a _metric
structure_ to be a real-valued structure $\mathcal{M}$ that is an $L$-metric
structure for some metric signature $L$.

We next define what we mean by a metric theory. By a $V$-_sentence_ we mean a
continuous sentence in the vocabulary $V$. Given an $L$-metric structure
$\mathcal{M}$ where $L$ is a metric signature over a vocabulary $V$, the
_theory of_ $\mathcal{M}$ is the set $\operatorname{Th}(\mathcal{M})$ of all
$V$-sentences that have truth value $0$ in $\mathcal{M}$. We say that $T$ is a
_metric theory_ if $T=\operatorname{Th}(\mathcal{M})$ for some metric
structure $\mathcal{M}$. If $\mathcal{M}$ is a metric structure and
$T=\operatorname{Th}(\mathcal{M})$, we call $\mathcal{M}$ a _metric model_ of
$T$.

We say that a regular ultrafilter $\mathcal{D}$ over a set $I$ of cardinality
$\lambda$ _saturates_ a metric structure $\mathcal{M}$ if the ultrapower
$\mathcal{M}_{\mathcal{D}}$ is $\lambda^{+}$-saturated. For two metric
theories $T,U$ with possibly different vocabularies, we write
$T\trianglelefteq U$ if for every metric model $\mathcal{M}$ of $T$ and
$\mathcal{N}$ of $U$, every regular ultrafilter that saturates $\mathcal{N}$
saturates $\mathcal{M}$. We will prove the following results.

  * A.

Theorem 3.5: For any metric theory $T$ (with a countable vocabulary), we have
$T\trianglelefteq T$. This means that for any two metric models
$\mathcal{M},\mathcal{N}$ of $T$, $\mathcal{M}$ and $\mathcal{N}$ are
saturated by the same regular ultrafilters. It follows that $\trianglelefteq$
is a pre-ordering (reflexive and transitive), and that $T\trianglelefteq U$ if
and only if there exist metric models $\mathcal{M}$ of $T$ and $\mathcal{N}$
of $U$ such that every ultrafilter that saturates $\mathcal{N}$ saturates
$\mathcal{M}$.

Note that from the set of sentences $\operatorname{Th}(\mathcal{M})$ one can
recover the vocabulary $V$ but not the metric signature $L$. In fact, it is
possible for a metric theory $T$ in the sense of this paper to have many
metric models with different distance predicates in $V$. By Result A, one can
and should view $\trianglelefteq$ as a relation between metric theories.

  * B.

Theorem 6.22: For metric theories $T$ and $U$, if $T$ is stable and $U$ is
unstable then $T\vartriangleleft U$.

  * C.

Theorem 7.12: There are just two $\trianglelefteq$-equivalence classes of
stable metric theories, the $\trianglelefteq$-minimal theories and the others.

  * D.

Theorem 8.6: If $T$ satisfies the metric analogue of the property SOP2 then
$T$ is $\trianglelefteq$-maximal among all metric theories.

  * E.

Theorem 9.7: The first order theory $T_{rg}$ of the random graph is
$\trianglelefteq$-minimal among all unstable metric theories.

  * F.

Let $T^{R}_{rg}$ be the randomization of $T_{rg}$. (A model of $T^{R}_{rg}$ is
built from a model of $T_{rg}$ by replacing elements by random elements).
Theorem 10.16: $T^{R}_{rg}$ is $\trianglelefteq$-minimal among all metric
theories with TP2, the tree property of the second kind. Moreover,
$T_{rg}\vartriangleleft T^{R}_{rg}$, and $T^{R}_{rg}$ is
$\trianglelefteq$-equivalent to the first order theory called $T^{*}_{feq}$.

These results are summarized in Figure 1. The increasing direction in the pre-
ordering is toward the upper right. The lower left quadrant of Figure 1 has
the stable theories, the right half has the $\trianglelefteq$-maximal
theories, and the top half has the theories with the independence property
(IP). Every unstable theory has either IP or SOP2. It is open whether or not
$T^{R}_{rg}$ is $\trianglelefteq$-maximal, but Figure 1 is drawn as if it is
not. The results A–C were announced in the author’s plenary lecture at the
March, 2020 meeting of the Association for Symbolic Logic, which was cancelled
by the covid-19 pandemic.

[.]2(52,50)(88,50) [.]2(52,88)(88,52)
SOP2?minstablestable$\neg$minTP2unstable$\neg$TP2unstable$2^{\aleph_{0}}$classes$T_{rg}$$T^{R}_{rg}$SOP2$\neg$maxmaxIP$\neg$IP
Figure 1. The $\trianglelefteq$ order on metric theories

Historically, continuous model theory for real-valued structures was
introduced by C. C. Chang and the author in 1966 in the monograph [CK66]. Four
decades later, the 2008 paper [BBHU08] introduced the modern treatment of
continuous model theory for metric structures. In [CK66] the real-valued
structures had an equality predicate. In the current treatment, the real-
valued structures do not have equality, and are the continuous analogue of
first order structures without equality. Metric structures are the continuous
analogue of first order structures with equality, with the distance predicate
playing the role of equality.

The ultraproduct construction for first order structures was introduced by Łoś
[Lo55]. Frayne, Morel and Scott [FMS62] used ultraproducts modulo regular
ultrafilters. The first order analogue of the result A appeared in Keisler
[Ke67], and the first order analogues of B and C appeared in Shelah [Sh78].
For the next three decades, there was no further progress on the relation
$\trianglelefteq$ on first order theories until a breakthrough by Malliaris in
[Ma09]. In that paper, Malliaris introduced notions, such as the distribution
of a set of formulas in an ultraproduct, that were key to subsequent work on
the $\trianglelefteq$-order. The main result in [Ma09] showed that an
ultrapower of a first order structure modulo a regular $\mathcal{D}$ is
$\lambda^{+}$-saturated if, for every first order formula $\varphi$, the
ultrapower is $\lambda^{+}$-saturated for $\varphi$-types.

The first order analogues of the results D, E, and F appeared in the following
papers. D: Malliaris and Shelah [MS16a]; E: Malliaris [Ma12]; and F: Malliaris
and Shelah [MS13]. Those results show that the relation $\trianglelefteq$ on
first order theories also looks like Figure 1 (with $T^{*}_{feq}$ instead of
$T^{R}_{rg}$). The proofs of the first order results required the construction
of ultrafilters with properties like being good. Kunen’s method of
constructing ultrafilters via independent families of sets played a key role.
Good ultrafilters were introduced in [Ke64] and constructed assuming the
generalized continuum hypothesis (GCH). In 1972, Kunen [Ku72] used independent
families of sets to construct good ultrafilters without the GCH.

The work of Malliaris and Shelah involved two parallel lines of research: the
construction of ultrafilters with special set-theoretic properties, and the
interplay between the $\trianglelefteq$ relation and model-theoretic
properties of first order theories.

The results in this paper are obtained by extending the second of the above
lines of research to metric theories. Along the way we obtain other results
about metric structures that may be of interest in their own right. Stability
and indiscernible sequences in metric theories have been studied extensively
in many papers (see Section 6 below). However, as by-products of his proof of
the results B and C above for first order theories, Shelah in [Sh78] proved
many additional results about indiscernible sequences and the finite cover
property, which are generalized to the continuous case here. Metric theories
with the property TP2 and the independence property have been studied in the
paper [BY13]. The notion of an SOP2 metric theory here may be new.

In the continuous model theory literature, a metric signature is usually held
fixed and the ultraproduct is introduced for families of metric structures
with the same metric signature. To prove the results in this paper we will
need a more general ultraproduct construction, for families of real-valued
structures with the same vocabulary. That leads us to consider the
$\trianglelefteq$ relation between theories of arbitrary real-valued
structures, rather than just between metric theories.

It turns out that the pre-ordering $\trianglelefteq$ is essentially the same
in that case. For a real-valued structure $\mathcal{M}$ with vocabulary $V$,
$\operatorname{Th}(\mathcal{M})$ is again defined as the set of all
$V$-sentences with truth value $0$ in $\mathcal{M}$, and we call
$\operatorname{Th}(\mathcal{M})$ a continuous theory with vocabulary $V$. We
have

$\\{\mbox{first order theories}\\}\subsetneqq\\{\mbox{metric
theories}\\}\subsetneqq\\{\mbox{continuous theories}\\}.$

We define the relation $T\trianglelefteq U$ for continuous theories in the
same way as for metric theories. Theorem 3.6 below is a stronger version of
result A above, showing that $T\trianglelefteq T$ for every continuous theory
$T$, so any two real-valued models $\mathcal{M},\mathcal{N}$ of $T$ (metric or
not) are saturated by the same regular ultrafilters. Theorem 5.1 below will
show that every continuous theory is $\trianglelefteq$-equivalent to a metric
theory. We may therefore identify the $\trianglelefteq$-equivalence classes of
continuous theories with the $\trianglelefteq$-equivalence classes of metric
theories in a way that preserves the $\trianglelefteq$ relation.

We thank Isaac Goldbring, James Hanson, and Maryanthe Malliaris for valuable
discussions related to this work.

##  2\. Preliminaries on continuous model theory

We refer to [BBHU08] and [BU09] for background material on metric and
premetric structures. See the Introduction above for the notion of a
vocabulary $V$ and a metric signature $L$ over $V$.

######  Convention 2.1.

Throughout this paper, $V$ will denote a countable vocabulary. To insure that
the collection of all vocabularies is a set rather than a proper class, we
also require that the elements of a vocabulary are hereditarily finite sets.

By a _real-valued structure_ $\mathcal{M}$ with vocabulary $V$ and universe
set $M$ (briefly, a $V$-structure) we mean an object that has a non-empty
universe set $M$, and interpretations:

  * •

$c^{\mathcal{M}}\in M$ for each constant symbol $c\in V$,

  * •

$F^{\mathcal{M}}\colon M^{n}\to M$ for each $n$-ary function symbol $F\in V$,

  * •

$P^{\mathcal{M}}\colon M^{n}\to[0,1]$ for each $n$-ary predicate symbol $P\in
V$.

We say that $\mathcal{M}$ is a real-valued structure if $\mathcal{M}$ is a
$V$-structure for some $V$.

Let $L$ be a metric signature over $V$ with distance predicate $d\in V$. An
_$L$ -premetric structure_ is a $V$-structure $\mathcal{M}$ such that the
interpretation $d^{\mathcal{M}}$ is a pseudo-metric on $M$, and for each
predicate or function symbol $S\in V$, the interpretation $S^{\mathcal{M}}$
respects the modulus of uniform continuity given by $L$. An _$L$ -metric
structure_ is an $L$-premetric structure such that the interpretation
$d^{\mathcal{M}}$ is a complete metric on $M$. We say that $\mathcal{M}$ is a
_premetric structure_ if there exists a metric signature $L$ such that
$\mathcal{M}$ is an $L$-premetric structure. Similarly for metric
structures.222In [BBHU08], premetric structures are called “prestructures”,
and metric structures are called “structures”. In [BU09], vocabularies are
called “non-metric signatures”. In [Ke21], real-valued structures are called
“general structures”.

The notion of an atomic formula is the same as in first order logic. We assume
familiarity with the notions of a continuous formula, a continuous sentence,
and the truth value $\varphi^{\mathcal{M}}(\vec{a})$ of a continuous formula
$\varphi$ in a metric structure $\mathcal{M}$ at a tuple $\vec{a}$ in $M$. By
a $V$-formula we mean a continuous formula built from symbols in $V$. The
notion of truth value of a $V$-formula is exactly the same for arbitrary real-
valued structures as for metric structures. We write $\varphi(\vec{v})$ to
indicate that all the free variables of $\varphi$ are in $\vec{v}$, and use
similar notation for sets of formulas. A set $\Phi(\vec{v})$ of $V$-formulas
is said to be _satisfiable in_ $\mathcal{M}$ if there exist a
$|\vec{v}|$-tuple $\vec{a}$ in M such that $\varphi^{\mathcal{M}}(\vec{a})=0$
for all $\varphi(\vec{v})\in\Phi(\vec{v})$. A mapping
$h\colon\mathcal{M}\to\mathcal{N}$ is an _elementary embedding_ if for every
tuple $\vec{a}$ in $M$ and every $V$-formula $\varphi(\vec{v})$ we have
$\varphi^{\mathcal{M}}(\vec{a})=\varphi^{\mathcal{N}}(h(\vec{a})).$ The
elementary equivalence relation $\mathcal{M}\equiv\mathcal{N}$ and the
elementary extension relation $\mathcal{M}\prec\mathcal{N}$ are defined as in
[BBHU08], and are applied to arbitrary $V$-structures as well as to metric
structures.

When a vocabulary $V$ is clear from the context, $V$-formulas will simply be
called formulas.

Given a $V$-structure $\mathcal{M}$ and a (possibly uncountable) set
$A\subseteq M$, we let $V_{A}=V\cup\\{c_{a}\colon a\in A\\}$ be the set formed
by adding to $V$ a new constant symbol $c_{a}$ for each element $a\in A$.
$\mathcal{M}_{A}$ denotes the structure obtained from $\mathcal{M}$ by
interpreting $c_{a}$ by $a$ for each $a\in A$. The constant symbols
$c_{a},a\in A$, are called _parameters from $A$_, or _parameters from
$\mathcal{M}$_. By a formula with parameters from $A$ we mean a formula
obtained from a formula without parameters by replacing some of the free
variables by parameters in $A$. We say that a set $\Gamma$ of formulas with
parameters from $A$ is _satisfiable in_ $\mathcal{M}$ if $\Gamma$ is
satisfiable in $\mathcal{M}_{A}$, and _finitely satisfiable in_ $\mathcal{M}$
if every finite subset of $\Gamma$ is satisfiable in $\mathcal{M}_{A}$.

Recall from the Introduction that for a $V$-structure $\mathcal{M}$, the
theory of $\mathcal{M}$ is the set $\operatorname{Th}(\mathcal{M})$ of all
$V$-sentences $\varphi$ such that $\varphi^{\mathcal{M}}=0$. Thus
$Th(\mathcal{M})=Th(\mathcal{N})$ if and only if
$\mathcal{M}\equiv\mathcal{N}$. Also, $T$ is a _continuous theory_ if
$T=\operatorname{Th}(\mathcal{M})$ for some real-valued structure
$\mathcal{M}$, and $T$ is a _metric theory_ if
$T=\operatorname{Th}(\mathcal{M})$ for some metric structure $\mathcal{M}$.

######  Remark 2.2.

For each metric signature $L$ with distance predicate $d$, there is a set of
$V$-sentences that expresses the fact that $d$ is a pseudo-metric and each
function and predicate has the required modulus of uniform continuity.

######  Fact 2.3.

333When we state a fact without giving a reference, the result can be found in
[BBHU08].

For every $L$-premetric structure $\mathcal{M}$, there is an $L$-metric
structure $\mathcal{N}$, the _completion_ of $\mathcal{M}$, such that
$\mathcal{N}\equiv\mathcal{M}$, so $\operatorname{Th}(\mathcal{M})$ is an
$L$-metric theory.

######  Corollary 2.4.

A $V$-theory $T$ is an $L$-metric theory if and only if $T$ contains the set
of sentences mentioned in Remark 2.2.

We say that $\mathcal{M}$ is a _model of_ a set of sentences $U$, in symbols
$\mathcal{M}\models U$, if $\varphi^{\mathcal{M}}=0$ for all $\varphi\in U$.
Thus $\mathcal{M}$ is a model of $Th(\mathcal{M})$. If
$T=\operatorname{Th}(\mathcal{M})$ and $\mathcal{M}\models U$, we also write
$T\models U$. We write $\mathcal{M}\models\varphi$, or
$\mathcal{M}\models\varphi(\vec{a})=0$, if $\varphi^{\mathcal{M}}(\vec{a})=0$.
We write $\mathcal{M}\models\varphi(\vec{a})\leq r$ if
$\varphi^{\mathcal{M}}(\vec{a})\leq r$.

######  Corollary 2.5.

If $T$ is an $L$-metric theory, then every model of $T$ is an $L$-premetric
structure.

###### Proof.

By Remark 2.2. ∎

######  Fact 2.6.

Suppose $\mathcal{M}$ is an $L$-premetric structure and $L$ has distance
predicate $d$. For any continuous formula $\varphi(\vec{v})$ in the vocabulary
of $\mathcal{M}$, $\varphi^{\mathcal{M}}$ is a mapping from $M^{|\vec{v}|}$
into $[0,1]$ that is uniformly continuous with respect to
$(M,d^{\mathcal{M}})$.

######  Definition 2.7.

We say that a $V$-structure $\mathcal{M}$ is $\kappa$-saturated if for every
set $A\subseteq M$ of cardinality $|A|<\kappa$, every set of $V$-formulas with
parameters from $A$ that is finitely satisfiable in $\mathcal{M}$ is
satisfiable in $\mathcal{M}$.

By a _strict continuous formula_ we mean a formula built from atomic formulas
in finitely many steps using only the quantifiers $\sup,\inf$, and the
connectives $0,1,\min,\max,\cdot/2,\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}$ (where
$x\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}y=\max(x-y,0)$).

######  Remark 2.8.

  * •

$|x-y|=\max(x\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern
1.94441pt}}y,y\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}x).$

  * •

$x\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}y=0$ if and only if
$x\leq y$.

  * •

$\mathcal{M}\models\theta\leq r$ if and only if
$\mathcal{M}\models\theta\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}r$.

  * •

We sometimes use the alternate notation $\theta\mathbin{\mathchoice{\kern
2.77774pt\hbox to0.0pt{\hss\hbox{$\displaystyle\leq$}\hss}\raise
4.30554pt\hbox to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern
2.77774pt\hbox to0.0pt{\hss\hbox{$\textstyle\leq$}\hss}\raise 4.30554pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle\leq$}\hss}\raise 3.01389pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle\leq$}\hss}\raise 2.15277pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}\psi$ for the formula
$\theta\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}\psi$.

######  Remark 2.9.

The set of strict $V$-formulas is countable.

######  Fact 2.10.

(See Theorem 6.3 in [BBHU08]). For each $V$-formula $\varphi(\vec{x})$ and
$\varepsilon>0$ there is a strict $V$-formula $\theta(\vec{x})$ such that
$|\varphi^{\mathcal{M}}(\vec{x})-\theta^{\mathcal{M}}(\vec{x})|<\varepsilon$
for every $\vec{x}$ in every $V$-structure $\mathcal{M}$.

######  Lemma 2.11.

If $\kappa>\aleph_{0}$, then a $V$-structure $\mathcal{M}$ is
$\kappa$-saturated if and only if every set of fewer than $\kappa$ formulas
with parameters from $M$ that is finitely satisfiable in $\mathcal{M}$ is
satisfiable in $\mathcal{M}$.

###### Proof.

By Remark 2.9, there are countably many strict $V$-formulas without
parameters. Hence there are $|A|+\aleph_{0}$ strict $V$-formulas with
parameters from $A$. So if $|A|<\kappa$ then there are fewer than $\kappa$
strict $V$-formulas with parameters from $A$. The result now follows from Fact
2.10. ∎

In [BBHU08], the ultraproduct of an indexed family of $L$-metric structures
with the same metric signature $L$ modulo an ultrafilter is defined, and it is
proved to again be an $L$-metric structure. In this paper, we will need the
slightly more general notion of an ultraproduct of an indexed family of
$V$-structures with the same vocabulary $V$. We will use the following notion
of reduction of $V$-structures from the paper [Ke21].

######  Definition 2.12.

For $a,b\in M$, we write $a\doteq^{\mathcal{M}}b$ if for every atomic formula
$\varphi(x,\vec{z})$ and tuple $\vec{c}\in M^{|\vec{z}|}$,
$\varphi^{\mathcal{M}}(a,\vec{c})=\varphi^{\mathcal{M}}(b,\vec{c}).$ A
$V$-structure $\mathcal{M}$ is _reduced_ if whenever $a\doteq^{\mathcal{M}}b$
we have $a=b$.

The relation $\doteq^{\mathcal{M}}$ goes back to Leibniz around 1840, The
_reduction map_ for $\mathcal{M}$ is the mapping that sends each element of
$M$ to its equivalence class under $\doteq^{\mathcal{M}}$. The _reduction_ of
the $V$-structure $\mathcal{M}$ is the reduced $V$-structure $\mathcal{N}$
such that $N$ is the set of equivalence classes of elements of $M$ under
$\doteq^{\mathcal{M}}$, and the reduction map for $\mathcal{M}$ is an
elementary embedding of $\mathcal{M}$ onto $\mathcal{N}$. We say that
$\mathcal{M},\mathcal{M}^{\prime}$ are _isomorphic_ , in symbols
$\mathcal{M}\cong\mathcal{M}^{\prime}$, if there is an elementary embedding
from the reduction of $\mathcal{M}$ onto the reduction of
$\mathcal{M}^{\prime}$. The following fact gives a useful characterization of
reduced $V$-structures $\mathcal{M}$ in the case that $\mathcal{M}$ is a
premetric structure.

######  Fact 2.13.

(See Theorem 3.7 in [BBHU08]). Let $\mathcal{M}$ be an $L$-premetric structure
and let $d$ be the distance predicate of $L$. $\mathcal{M}$ is reduced if and
only if for all $x,y\in M$, if $d^{\mathcal{M}}(x,y)=0$ then $x=y$.

######  Remark 2.14.

  * •

$\cong$ is an equivalence relation on $V$-structures.

  * •

Every $V$-structure is isomorphic to its reduction.

  * •

If there is an elementary embedding of $\mathcal{M}$ _onto_ $\mathcal{N}$,
then $\mathcal{M}\cong\mathcal{N}$.

  * •

$\mathcal{M}\cong\mathcal{N}$ implies $\mathcal{M}\equiv\mathcal{N}$.

  * •

A $V$-structure is $\kappa$-saturated if and only if its reduction is
$\kappa$-saturated.

######  Remark 2.15.

Every reduced $\aleph_{1}$-saturated $L$-premetric structure $\mathcal{M}$ is
an $L$-metric structure.

###### Proof.

By Fact 2.13, $(M,d^{\mathcal{M}})$ is a metric space. Completeness of the
metric follows easily from $\aleph_{1}$-saturation. ∎

######  Convention 2.16.

$\lambda$ will always denote an infinite cardinal, and $I$ will always be a
set of cardinality $\lambda$. When working with ultraproducts, we will use the
notation $\langle r[t]\rangle_{t\in I}$ for a function $r$ with domain $I$.

This convention will help keep things straight in the ultraproduct
construction, by using $t$ as a variable over $I$ and leaving the symbol $i$
as a variable ranging over the natural numbers.

######  Definition 2.17.

Let $\mathcal{D}$ be an ultrafilter over $I$. For each function $r\colon
I\to[0,1]$, $\lim_{\mathcal{D}}r$ is the unique element $s\in[0,1]$ such that
for each real $\varepsilon>0$, the set $\\{t\in
I\colon|r[t]-s|<\varepsilon\\}$ belongs to $\mathcal{D}$.

The following definition is taken from [Ke21].

######  Definition 2.18.

Let $\mathcal{D}$ be an ultrafilter over a set $I$ and $\mathcal{M}_{t}$ be a
$V$-structure for each $t\in I$. The _pre-ultraproduct_
$\prod^{\mathcal{D}}\mathcal{M}_{t}$ is the $V$-structure
$\mathcal{M}^{\prime}=\prod^{\mathcal{D}}\mathcal{M}_{t}$ such that:

  * •

$M^{\prime}=\prod_{t\in I}M_{t}$, the cartesian product.

  * •

For each constant symbol $c\in V$, $c^{\mathcal{M}^{\prime}}=\langle
c^{\mathcal{M}_{t}}\rangle_{t\in I}$.

  * •

For each $n$-ary function symbol $G\in V$ and $n$-tuple
$\vec{a}=\langle\vec{a}[t]\rangle_{t\in I}$ in $M^{\prime}$,

$G^{\mathcal{M}^{\prime}}(\vec{a})=\langle
G^{\mathcal{M}_{t}}(\vec{a}[t])\rangle_{t\in I}.$

  * •

For each $n$-ary predicate symbol $P\in V$ and $n$-tuple $\vec{a}$ in
$M^{\prime}$,

$P^{\mathcal{M}^{\prime}}(\vec{a})=\lim_{\mathcal{D}}\langle
P^{\mathcal{M}_{t}}(\vec{a}[t])\rangle_{t\in I}.$

The _ultraproduct_ $\prod_{\mathcal{D}}\mathcal{M}_{t}$ is the reduction of
the pre-ultraproduct $\prod^{\mathcal{D}}\mathcal{M}_{t}$. For each $a\in
M^{\prime}$ we also let $a_{\mathcal{D}}$ denote the equivalence class of $a$
under $\doteq^{\mathcal{M}^{\prime}}$.

######  Remark 2.19.

It follows from Fact 2.13 that if $\mathcal{M}_{i}$ is an $L$-metric structure
with the same $L$ for each $i\in I$, then $\prod_{\mathcal{D}}\mathcal{M}_{t}$
is exactly the ultraproduct of the metric structures $\mathcal{M}_{i}$ as
defined in [BBHU08].

The next fact is the analogue for $V$-structures of the fundamental theorem of
Łoś. It is proved in the same way as the corresponding result for metric
structures, Theorem 5.4 in [BBHU08].

######  Fact 2.20.

(Łoś Theorem). Let $\mathcal{M}_{t}$ be a $V$-structure for each $t\in I$, let
$\mathcal{D}$ be an ultrafilter over $I$, and let
$\mathcal{M}=\prod_{\mathcal{D}}\mathcal{M}_{t}$ be the ultraproduct. Then for
each formula $\varphi$ and tuple $\vec{b}$ in the cartesian product
$\prod_{t\in I}M_{t}$,

$\varphi^{\mathcal{M}}(\vec{b}_{\mathcal{D}})=\lim_{\mathcal{D}}\langle\varphi^{\mathcal{M}_{t}}(\vec{b}[t])\rangle_{t\in
I}.$

######  Corollary 2.21.

Suppose $\mathcal{D}$ is an ultrafilter over $I$, and for each $t\in I$,
$\mathcal{M}_{t},\mathcal{N}_{t}$ are $V$-structures.

(i) If $\mathcal{M}_{t}\cong\mathcal{N}_{t}$ for each $t\in I$, then
$\prod_{\mathcal{D}}\mathcal{M}_{t}\cong\prod_{\mathcal{D}}\mathcal{N}_{t}$.

(ii) If $\mathcal{M}_{t}\equiv\mathcal{N}_{t}$ for each $t\in I$, then
$\prod_{\mathcal{D}}\mathcal{M}_{t}\equiv\prod_{\mathcal{D}}\mathcal{N}_{t}$.

######  Fact 2.22.

Every ultraproduct of $L$-metric structures with the same $L$ is an $L$-metric
structure.

One of the main reasons that metric signatures were introduced in the
definition of a metric structure was to insure that Fact 2.22 holds.

In the case that all the $\mathcal{M}_{t}$ are the same,
$\mathcal{M}_{t}=\mathcal{M}$ for all $t\in I$, the pre-ultraproduct and
ultraproduct are called the _pre-ultrapower_ and _ultrapower_ respectively,
and are denoted by $\mathcal{M}^{\mathcal{D}}$ and
$\mathcal{M}_{\mathcal{D}}$.

######  Corollary 2.23.

For every $V$-structure $\mathcal{M}$ and ultrafilter $\mathcal{D}$ over $I$,
the diagonal embedding $a\mapsto I\times\\{a\\}$ is an elementary embedding of
$\mathcal{M}$ into $\mathcal{M}^{\mathcal{D}}$.

######  Definition 2.24.

An ultrafilter $\mathcal{D}$ over $I$ is said to be _regular_ if there is a
subset $\mathcal{X}\subseteq\mathcal{D}$ of cardinality $\lambda$ such that
each $t\in I$ belongs to only finitely many $X\in\mathcal{X}$. We also say
that $\mathcal{X}$ _regularizes_ $\mathcal{D}$.

######  Fact 2.25.

([FMS62]) For each infinite set $I$ there exist regular ultrafilters over $I$.

Hint: If $I$ is the set of all finite subsets of $J$, let
$\mathcal{X}=\\{X_{j}\colon j\in J\\}$ where $X_{j}=\\{Y\subseteq
I\colon\\{j\\}\subseteq Y\\}$.

######  Remark 2.26.

Let $\mathcal{D}$ is a regular ultrafilter over $I$.

  * (i)

If $\mathcal{M}_{t}$ is a $V$-structure for each $t\in I$, then the
ultraproduct $\prod_{\mathcal{D}}\mathcal{M}_{t}$ is $\aleph_{1}$-saturated.

  * (ii)

If $\mathcal{M}_{i}$ is an $L$-premetric structure for each $i\in I$, then the
ultraproduct $\prod_{\mathcal{D}}\mathcal{M}_{t}$ is an $L$-metric structure.

###### Proof.

(i) Theorem 6.1.1 of [CK12] gives the corresponding result in first order
logic. It is routine to modify the proof in [CK12] to obtain the result stated
here.

(ii) By definition, $\prod_{\mathcal{D}}\mathcal{M}_{t}$ is a reduced
structure. By Remark 2.2 and the Łoś theorem,
$\prod_{\mathcal{D}}\mathcal{M}_{t}$ is an $L$-premetric structure. By Remark
2.15, $\prod_{\mathcal{D}}\mathcal{M}_{t}$ is $\aleph_{1}$-saturated. Then by
(i) above, $\prod_{\mathcal{D}}\mathcal{M}_{t}$ is an $L$-metric structure. ∎

By a _two-valued structure_ , or a _first order structure without equality_ ,
we mean a $V$-structure $\mathcal{M}$ such that for every $n$-ary predicate
symbol $P\in V$, $P^{\mathcal{M}}\colon M^{n}\to\\{0,1\\}$. By a _first order
structure_ we mean a two-valued $L$-metric structure $\mathcal{M}$ such that:

  * •

$L$ has a distance predicate $d$ and the trivial modulus of uniform continuity
for each symbol in $V$.

  * •

$d^{\mathcal{M}}(x,y)$ is the discrete metric; $d^{\mathcal{M}}(x,y)=0$ if
$x=y$, and $d^{\mathcal{M}}(x,y)=1$ otherwise.

By a _first order formula_ we mean a formula that is built from atomic
formulas using only the quantifiers $\sup,\inf$, and the connectives
$0,1,\min,\max,1\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}x$. Note that every
first order formula is a strict continuous formula. A first order formula is
traditionally written with the notation
$\forall,\exists,\top,\bot,\vee,\wedge,\neg$ instead of
$\sup,\inf,0,1,\min,\max,1\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}x$. First order
structures and formulas are sometimes called classical structures and
formulas.

######  Fact 2.27.

([FMS62]. See Proposition 4.3.7 of [CK12].) If $\mathcal{D}$ is a regular
ultrafilter over $I$ and $\mathcal{M}$ is an infinite first order structure,
then $\mathcal{M}_{\mathcal{D}}$ has cardinality $|M|^{|I|}$.

The following lemma is well-known and easily checked, but is stated explicitly
here for completeness.

######  Lemma 2.28.

Let $\mathcal{M}$ be a two-valued structure.

  * (i)

For every first order formula $\varphi(\vec{v})$ and $|\vec{v}|$-tuple
$\vec{a}$ in $\mathcal{M}$, $\varphi^{\mathcal{M}}(\vec{a})\in\\{0,1\\}$.

  * (ii)

Suppose $(\mathcal{M},\vec{a})$ and $(\mathcal{M},\vec{b})$ satisfy the same
first order sentences. Then $(\mathcal{M},\vec{a})$ and
$(\mathcal{M},\vec{b})$ satisfy the same continuous sentences, that is,
$(\mathcal{M},\vec{a})\equiv(\mathcal{M},\vec{b})$ as $V$-structures.

  * (iii)

Suppose $\mathcal{M}$ is a first order structure that is $\kappa$-saturated in
the sense of first order model theory. Then $\mathcal{M}$ is
$\kappa$-saturated as a $V$-structure.

If $\mathcal{M}$ is a first order structure with vocabulary $V$, we abuse
notation by letting $\operatorname{Th}(\mathcal{M})$ denote the set of all
first order $V$-sentences true in $\mathcal{M}$, rather than the set of
continuous $V$-sentences true in $\mathcal{M}$. By a _first order theory_ we
mean the set $\operatorname{Th}(\mathcal{M})$ where $\mathcal{M}$ is a first
order structure.

##  3\. Comparing two continuous theories

######  Convention 3.1.

For the rest of this paper, $\mathcal{D}$ will always denote a regular
ultrafilter over a set $I$ of cardinality $\lambda$.

We now define the relation $\trianglelefteq$ on the set of all continuous
theories, which contains the set of all metric theories. The definitions will
be the same as for in first order model theory, but using the continuous
notions of ultrapower and saturation.

######  Definition 3.2.

Let $\mathcal{M}$ be a $V$-structure. We say that $\mathcal{D}$ _saturates_
$\mathcal{M}$ if the ultrapower $\mathcal{M}_{\mathcal{D}}$ is
$\lambda^{+}$-saturated (or equivalently, the pre-ultrapower
$\mathcal{M}^{\mathcal{D}}$ is $\lambda^{+}$-saturated).

Given a continuous theory $T$, we say that $\mathcal{D}$ _saturates_ $T$, in
symbols $\mathcal{D}\in\operatorname{Sat}(T)$, if $\mathcal{D}$ saturates
every model of $T$.

######  Definition 3.3.

Given continuous theories $T,U$ with possibly different vocabularies, we write
$T\trianglelefteq U$ if $\operatorname{Sat}(U)\subseteq\operatorname{Sat}(T)$.

We write $T\vartriangleleft U$ if $T\trianglelefteq U$ but not
$U\trianglelefteq T$.

We say that $T,U$ are $\trianglelefteq$-_equivalent_ , in symbols
$T\trianglelefteq\trianglerighteq U$, if $T\trianglelefteq U$ and
$U\trianglelefteq T$.

######  Remark 3.4.

$T\trianglelefteq\trianglerighteq U$ if and only if
$\operatorname{Sat}(T)=\operatorname{Sat}(U)$.

By Lemma 2.28, for first order theories, the relations $T\trianglelefteq U$
and $T\trianglelefteq\trianglerighteq U$ are the same as in [Ke67].

Note that every ultraproduct of $V$-structures is reduced. By definition, the
reduction map is an elementary embedding from the pre-ultraproduct onto the
ultraproduct. We will usually work with the pre-ultraproduct rather than with
the ultraproduct. One reason for this choice is that pre-ultrapowers of
$V$-structures commute with expansions. If $\mathcal{M}$ is a $V$-structure
and $W$ is a subset of $V$, the _$W$ -part_444The $W$-part is called the
$W$-reduct in first order model theory (see [CK12]). We say $W$-part here
because the word reduction is used in a different way in continuous model
theory. of $\mathcal{M}$ is the $W$-structure $N$ with the same universe as
$\mathcal{M}$ such that for each symbol $S\in W$,
$S^{\mathcal{N}}=S^{\mathcal{M}}$. If $\mathcal{N}$ is the $W$-part of
$\mathcal{M}$, we say that $\mathcal{M}$ is an _expansion_ of $\mathcal{N}$.
In that case, the pre-ultrapower $\mathcal{N}^{\mathcal{D}}$ is exactly the
$W$-part of the pre-ultrapower $\mathcal{M}^{\mathcal{D}}$, while the
ultrapower $\mathcal{N}_{\mathcal{D}}$ is only the reduction of the $W$-part
of the ultrapower $\mathcal{M}_{\mathcal{D}}$. One reason that ultraproducts
of $V$-structures will be useful in studying the $\trianglelefteq$ relation on
metric theories is that it allows us to take ultraproducts of arbitrary
expansions of metric structures to a larger vocabulary.

In the next section we will prove the following theorem, which generalizes a
result in [Ke67] from first order structures to real-valued structures.

######  Theorem 3.5.

For any continuous theory $T$, and hence for every metric theory $T$, we have
$T\trianglelefteq\trianglerighteq T$.

######  Corollary 3.6.

For any two models $\mathcal{M},\mathcal{N}$ of a continuous theory $T$, a
regular ultrafilter $\mathcal{D}$ saturates $\mathcal{M}$ if and only if
$\mathcal{D}$ saturated $\mathcal{N}$. Thus $\mathcal{D}$ saturates $T$ if and
only if $\mathcal{D}$ saturates some model of $T$.

It is obvious that the relation $\trianglelefteq$ is transitive, so it follows
that $\trianglelefteq$ is a pre-ordering on the set of all continuous
theories.

######  Definition 3.7.

A continuous theory $T$ is $\trianglelefteq$-_minimal_ if $T\trianglelefteq U$
for every continuous theory $U$. A continuous theory $U$ is
$\trianglelefteq$-_maximal_ if $T\trianglelefteq U$ for every continuous
theory $T$. We define $\operatorname{Sat}(T)=\operatorname{Sat}(\mathcal{M})$
where $\mathcal{M}$ is a model of $T$.

Note that any two $\trianglelefteq$-minimal theories are
$\trianglelefteq$-equivalent, and any two $\trianglelefteq$-maximal theories
are $\trianglelefteq$-equivalent.

We recall some definitions from [Ke64].

######  Definition 3.8.

Let $\mathcal{D}$ be an ultrafilter over $I$. For any set $J$, let
$\mathcal{P}_{\aleph_{0}}(I)$ be the set of finite subsets of $J$. A mapping
$\delta\colon\mathcal{P}_{\aleph_{0}}(J)\to\mathcal{D}$ is called _monotone_
if $\delta(u)\supseteq\delta(v)$ whenever $u\subseteq
v\in\mathcal{P}_{\aleph_{0}}(J)$, and is called _multiplicative_ if
$\delta(u\cup v)=\delta(u)\cap\delta(v)$ whenever
$u,v\in\mathcal{P}_{\aleph_{0}}(J)$. We say that a mapping
$\delta^{\prime}\colon\mathcal{P}_{\aleph_{0}}(I)\to\mathcal{D}$ _refines_
$\delta$ if $\delta^{\prime}(u)\subseteq\delta(u)$ for all
$u\in\mathcal{P}_{\aleph_{0}}(J)$. $\mathcal{D}$ is said to be _good_ if
whenever $|J|\leq|I|$, every monotone function
$\delta\colon\mathcal{P}_{\aleph_{o}}(J)\to\mathcal{D}$ has a multiplicative
refinement.

######  Fact 3.9.

(Kunen [Ku72] in ZFC, [Ke64] under the GCH) For every $I$, there exists a good
ultrafilter over $I$.

######  Lemma 3.10.

([Ke67] for the first order case).

  * (i)

There exist $\trianglelefteq$-minimal theories. $T$ is
$\trianglelefteq$-minimal if and only if $\operatorname{Sat}(T)$ is the class
of all regular ultrafilters.

  * (ii)

There exist $\trianglelefteq$-maximal theories. $T$ is
$\trianglelefteq$-maximal if and only if $\operatorname{Sat}(T)$ is the class
of all good ultrafilters.

###### Proof.

(i): By Fact 2.27, for an infinite set $M$,
$|M_{\mathcal{D}}|=|M^{I}|\geq|I|^{+}$. Therefore the first order theory of an
infinite set with only the equality predicate is $\trianglelefteq$-minimal, so
(i) holds.

(ii): In [Ke67], it is proved that every $\mathcal{D}$ that saturates (for
example) $(\mathcal{P}_{\aleph_{0}}(\mathbb{N}),\subseteq)$ or
$(\mathbb{N},+,\times)$ is good. In [Ke64], it is proved (in ZFC) that an
ultrafilter $\mathcal{D}$ is good if and only if it saturates every first
order structure. The same proof works for real-valued structures, and (ii)
follows. ∎

######  Corollary 3.11.

(i) A first order theory is $\trianglelefteq$-minimal if and only it is
$\trianglelefteq$-minimal among first order theories.

(ii) A first order theory is $\trianglelefteq$-maximal if and only it is
$\trianglelefteq$-maximal among first order theories.

######  Corollary 3.12.

([Ke67] for the first order case). If $T$ is $\trianglelefteq$-minimal and $U$
is $\trianglelefteq$-maximal, then $T\vartriangleleft U$.

###### Proof.

If $I$ is uncountable, there is a regular ultrafilter over $I$ that is not
good. ∎

Let us take a brief look at the big picture. Let $\mathbb{F},\mathbb{M},$ and
$\mathbb{C}$ be the sets of $\trianglelefteq$-equivalence classes of first
order theories, metric theories, and continuous theories respectively. Then
each of $(\mathbb{F},\trianglelefteq),(\mathbb{M},\trianglelefteq),$ and
$(\mathbb{C},\trianglelefteq)$ is a partial ordering. By Lemma 3.10,
$(\mathbb{M},\trianglelefteq)$ has a $\trianglelefteq$-minimal element
$\min_{\mathbb{M}}$ and a $\trianglelefteq$-maximal element
$\max_{\mathbb{M}}$, and $\min_{\mathbb{M}}\vartriangleleft\max_{\mathbb{M}}$.
Similarly for $\mathbb{C}$ and $\mathbb{F}$.

Let $\tau\colon\mathbb{F}\to\mathbb{M}$ be the mapping that sends the
equivalence class of $Th(\mathcal{M})$ in $\mathbb{F}$ to the equivalence
class of $Th(\mathcal{M})$ in $\mathbb{M}$ for every first order structure
$\mathcal{M}$.

######  Corollary 3.13.

$\tau$ is an isomorphism from $(\mathbb{F},\trianglelefteq)$ onto a
substructure of $(\mathbb{M},\trianglelefteq)$. Moreover,
$\tau(\min_{\mathbb{F}})=\min_{\mathbb{M}}$, and
$\tau(\max_{\mathbb{F}})=\max_{\mathbb{M}}$.

###### Proof.

By Corollary 3.11. ∎

We have not been able to answer the following important question.

######  Question 3.14.

Is every metric theory $\trianglelefteq$-equivalent to a first order theory?
Equivalently, is the mapping $\tau\colon\mathbb{F}\to\mathbb{M}$ onto?

The answer is “no” if and only if there are “new”
$\trianglelefteq$-equivalence classes of metric theories that do not
correspond to $\trianglelefteq$-equivalence classes of first order theories.

We will see later, in Theorem 5.1, that every $\trianglelefteq$-equivalence
class in $\mathbb{C}$ contains a metric theory. It follows that the mapping
that sends the equivalence class of each metric theory in $\mathbb{M}$ its
equivalence class in $\mathbb{C}$ is an isomorphism from
$(\mathbb{M},\trianglelefteq)$ onto $(\mathbb{C},\trianglelefteq)$. So we
regard the partial ordering $(\mathbb{C},\trianglelefteq)$ as the same as
$(\mathbb{M},\trianglelefteq)$. Beginning in Section 6, we will investigate
the structure of the partial ordering $(\mathbb{M},\trianglelefteq)$.

One can also turn things around and use theories to compare ultrafilters
instead of using ultraproducts to compare theories.

######  Definition 3.15.

$\operatorname{Sat}_{\mathbb{M}}(\mathcal{D})$ is the set of $H\in\mathbb{M}$
such that $\mathcal{D}$ saturates some, or equivalently every, $T\in H$.
$\operatorname{Sat}_{\mathbb{F}}(\mathcal{D})$ is defined similarly with
$\mathbb{F}$ in place of $\mathbb{M}$.

######  Corollary 3.16.

(i) For all $H\in\mathbb{F}$,
$H\in\operatorname{Sat}_{\mathbb{F}}(\mathcal{D})\Leftrightarrow\tau(H)\in\operatorname{Sat}_{\mathbb{M}}(\mathcal{D})$.

(ii)
$\\{\mathcal{D}\colon\operatorname{Sat}_{\mathbb{F}}(\mathcal{D})=\mathbb{F}\\}=\\{\mathcal{D}\colon\operatorname{Sat}_{\mathbb{M}}(\mathcal{D})=\mathbb{M}\\}=\\{\mathcal{D}\colon\mathcal{D}\mbox{
is good}\\}.$

###### Proof.

(i) is trivial. (ii) follows from Lemma 3.10 and Corollary 3.11. ∎

We now review some results of Malliaris and Shelah about first order theories.
In $(\mathbb{F},\trianglelefteq)$, the lowest three equivalence classes are,
in increasing order, the minimal class, the class of stable non-minimal first
order theories, and the equivalence class $rg$ of the theory $T_{rg}$ of the
random graph. $rg$ is the minimal class of unstable first order theories. The
maximal class contains the class of SOP2 first order theories, and Malliaris
and Shelah conjecture that the maximal class is exactly the class of SOP2
first order theories. An first order theory is simple if and only if it has
neither SOP2 nor TP2 (and $T_{rg}$ is simple). There are continuum many
distinct equivalence classes of simple first order theories. There is an first
order theory $T^{*}_{feq}$ that is minimal among TP2 theories, and hence
minimal among non-simple first order theories. Also, $T_{rg}\vartriangleleft
T^{*}_{feq}$. It is open whether or not the $T^{*}_{feq}$ is maximal.

As mentioned in the Introduction, these first order results show that the
partial ordering $(\mathbb{F},\trianglelefteq)$ looks like Figure 1. In this
paper we will show that the same picture applies to the partial ordering
$(\mathbb{M},\trianglelefteq)$. To do so, we will generalize several notions
and results from first order model theory to continuous model theory.

##  4\. Elementary equivalent structures are equally saturatable

In this section we will prove Theorem 3.5 above. The proof will be longer than
the proof of the corresponding first order result in [Ke67], because
continuous logic does not have negation, and the $\inf$ quantifier may only
have approximate witnesses in a real-valued structure. We will first prove two
lemmas, and use those lemmas to prove a stronger version of Theorem 3.5 that
concerns ultraproducts, rather than ultrapowers.

In what follows, $\mathcal{M},\mathcal{N}$ will be $V$-structures.

######  Lemma 4.1.

Suppose $\mathcal{M}\equiv\mathcal{N}$. Let $\Psi(\vec{v})$ be a finite set of
formulas with the free variables $\vec{v}$. For every $n\in\mathbb{N}$ and
every $|\vec{v}|$-tuple $\vec{a}$ in $\mathcal{M}$ there is a
$|\vec{v}|$-tuple $\vec{b}$ in $\mathcal{N}$ such that for every
$\psi\in\Psi$, $\psi^{\mathcal{M}}(\vec{a})$ is within $2^{-n}$ of
$\psi^{\mathcal{N}}(\vec{b}).$

###### Proof.

Let $\vec{a}\in\mathcal{M}^{|\vec{v}|}$. For each $\psi\in\Psi$, let
$r_{\psi}$ be a dyadic rational that is within $2^{-(n+1)}$ of
$\psi^{\mathcal{M}}(\vec{a})$. Since $\Psi$ is finite,

$\max_{\psi\in\Psi}|\psi(\vec{v})-r_{\psi}|\mathbin{\mathchoice{\kern
2.77774pt\hbox to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}2^{-(n+2)}$

is a formula that is satisfied by $\vec{a}$ in $\mathcal{M}$. Then by Remark
2.8,

$\mathcal{M}\models(\inf_{\vec{v}})\max_{\psi\in\Psi}|\psi(\vec{v})-r_{\psi}|\leq
2^{-(n+2)},$

and since $\mathcal{M}\equiv\mathcal{N}$,

$\mathcal{N}\models(\inf_{\vec{v}})\max_{\psi\in\Psi}|\psi(\vec{v})-r_{\psi}|\leq
2^{-(n+2)}.$

Therefore there is a $|\vec{v}|$-tuple $\vec{b}$ in $\mathcal{N}$ such that

$\mathcal{N}\models\max_{\psi\in\Psi}|\psi(\vec{b})-r_{\psi}|\leq 2^{-(n+1)},$

and hence

$|\psi^{\mathcal{M}}(\vec{a})-\psi^{\mathcal{N}}(\vec{b})|\leq 2^{-n}.$

for every $\psi\in\Psi$. ∎

######  Corollary 4.2.

Suppose $\mathcal{M}\equiv\mathcal{N}$, and $\mathcal{N}$ is
$\aleph_{1}$-saturated. Then for every $\vec{a}\in\mathcal{M}^{|\vec{v}|}$
there exists $\vec{b}\in\mathcal{N}^{|\vec{v}|}$ such that
$(\mathcal{N},\vec{b})\equiv(\mathcal{M},\vec{a})$.

######  Lemma 4.3.

Suppose $\mathcal{M}\equiv\mathcal{N}$. Let $\Psi(\vec{u},\vec{v})$ be a
finite set of formulas with the free variables $\vec{u},\vec{v}$. For every
$n\in\mathbb{N}$ and every $|\vec{v}|$-tuple $\vec{a}$ in $\mathcal{M}$ there
is a $|\vec{v}|$-tuple $\vec{b}$ in $\mathcal{N}$ such that

  * (i)

For every $\vec{c}\in\mathcal{M}^{|\vec{u}|}$ there exists
$\vec{e}\in\mathcal{N}^{|\vec{u}|}$ such that for every $\psi\in\Psi$,
$\psi^{\mathcal{M}}(\vec{c},\vec{a})$ is within $2^{-n}$ of
$\psi^{\mathcal{N}}(\vec{e},\vec{b})$.

  * (ii)

For every $\vec{e}\in\mathcal{N}^{|\vec{u}|}$ there exists
$\vec{c}\in\mathcal{M}^{|\vec{u}|}$ such that for every $\psi\in\Psi$,
$\psi^{\mathcal{M}}(\vec{c},\vec{a})$ is within $2^{-n}$ of
$\psi^{\mathcal{N}}(\vec{e},\vec{b})$.

###### Proof.

Let $(z_{1},\ldots,z_{k})$ be a tuple of disjoint $|\vec{u}|$-tuples of
variables that are also disjoint from $\vec{u}$ and $\vec{v}$. In
$\mathcal{M}$, we say that an assignment of $(z_{1},\ldots,z_{k})$ is
$2^{-n}$-_dense over_ $\vec{a}$ if for every $|\vec{u}|$-tuple $\vec{c}$ there
is an $\ell\leq k$ such that for every $\psi\in\Psi$, $\psi(\vec{c},\vec{a})$
is within $2^{-n}$ of $\psi(z_{\ell},\vec{a})$. It is clear that in
$\mathcal{M}$, for any $\vec{a}$ and any $n\in\mathbb{N}$, there exists an
assignment $(z_{1},\ldots,z_{k})$ that is $2^{-n}$-dense over $\vec{a}$.

Let $\theta(z_{1},\ldots,z_{k},\vec{v})$ be the formula

$(\sup_{\vec{u}})\min_{\ell\leq
k}\max_{\psi\in\Psi}(|\psi(\vec{u},\vec{v})-\psi(z_{\ell},\vec{v})|).$

Note that in $\mathcal{M}$, $(z_{1},\ldots,z_{k})$ is $2^{-n}$-dense over
$\vec{a}$ if and only if

$\mathcal{M}\models\theta(z_{1},\ldots,z_{k},\vec{a})\leq 2^{-n}.$

Now fix a $|\vec{v}|$-tuple $\vec{a}$ in $\mathcal{M}$. Suppose
$z=(z_{1},\ldots,z_{k})$ is $2^{-(n+2)}$-dense over $\vec{a}$ in
$\mathcal{M}$. Since $\mathcal{N}\equiv\mathcal{M}$ and $\theta$ is a formula,
it follows from Lemma 4.1 that there exist tuples
$(w_{1},\ldots,w_{k},\vec{b})$ in $\mathcal{N}$ such that

  * (1)

$(w_{1},\ldots,w_{k})$ is $2^{-(n+1)}$-dense over $\vec{b}$ in $\mathcal{N}$.

  * (2)

For each $\psi\in\Psi$ and $1\leq\ell\leq k$,
$\psi^{\mathcal{N}}(w_{\ell},\vec{b})$ is within $2^{-(n+1)}$ of
$\psi^{\mathcal{M}}(z_{\ell},\vec{a})$.

Proof of (i): Let $\vec{c}\in\mathcal{M}^{|\vec{u}|}$. By (1), there is an
$\ell\in\\{1,\ldots,k\\}$ such that for every $\psi\in\Psi$,
$\psi^{\mathcal{M}}(z_{\ell},\vec{a})$ is within $2^{-(n+2)}$ of
$\psi^{\mathcal{M}}(\vec{c},\vec{a})$. Take $\vec{e}=z_{\ell}$. By (2), for
each $\psi\in\Psi$, $\psi^{\mathcal{N}}(\vec{e},\vec{b})$ is within
$2^{-(n+1)}$ of $\psi^{\mathcal{M}}(z_{\ell},\vec{a})$. Therefore
$\psi^{\mathcal{N}}(\vec{e},\vec{b})$ is within $2^{-n}$ of
$\psi^{\mathcal{M}}(\vec{c},\vec{a})$, as required.

Proof of (ii): Let $\vec{e}\in\mathcal{N}^{|\vec{u}|}$. By (1), there is an
$\ell\in\\{1,\ldots,k\\}$ such that for every $\psi\in\Psi$,
$\psi^{\mathcal{N}}(w_{\ell},\vec{b})$ is within $2^{-(n+1)}$ of
$\psi^{\mathcal{N}}(\vec{e},\vec{b})$. Take $\vec{c}=z_{\ell}$. By (2), for
each $\psi\in\Psi$, $\psi^{\mathcal{M}}(\vec{c},\vec{a})$ is within
$2^{-(n+1)}$ of $\psi^{\mathcal{N}}(w_{\ell},\vec{b})$. Therefore
$\psi^{\mathcal{M}}(\vec{c},\vec{a})$ is within $2^{-n}$ of
$\psi^{\mathcal{N}}(\vec{e},\vec{b})$. ∎

The next result, Theorem 4.4, is the stronger form of Theorem 3.5 that we
promised.

######  Theorem 4.4.

Suppose $\mathcal{D}$ is a regular ultrafilter over a set $I$ of infinite
cardinality $\lambda$, $\\{\mathcal{M}_{t}\colon t\in I\\}$ and
$\\{\mathcal{N}_{t}\colon t\in I\\}$ are families of real-valued structures
with the same vocabulary, and $\mathcal{M}_{t}\equiv\mathcal{N}_{t}$ for each
$t\in I$. Then $\prod^{\mathcal{D}}\mathcal{M}_{t}$ is $\lambda^{+}$-saturated
if and only if $\prod^{\mathcal{D}}\mathcal{N}_{t}$ is
$\lambda^{+}$-saturated.

Our proof of Theorem 4.4 will use a continuous analogue of the notion of a
distribution from Malliaris [Ma12]. Distributions give a useful criterion for
a set of formulas with parameters to be satisfiable in a regular ultraproduct.
In what follows, we let
$\mathcal{M}^{\prime}=\prod^{\mathcal{D}}\mathcal{M}_{t}$ be a pre-
ultraproduct, $A$ be a set of parameters in $\mathcal{M}^{\prime}$ of
cardinality $|A|\leq\lambda$, and $\Gamma=\Gamma(\vec{x},A)$ be a set of at
most $\lambda$ continuous formulas in the parameters $A$. For each $t\in I$
and $a\in A$, $a[t]$ will be the corresponding element of $\mathcal{M}_{t}$,
and $A[t]=\\{a[t]\colon a\in A\\}$.

######  Definition 4.5.

We define the approximation $\Gamma^{ap}$ of $\Gamma$ to be the set of
formulas

$\Gamma^{ap}=\Gamma^{ap}(\vec{x},A)=\\{\psi(\vec{x},A)\mathbin{\mathchoice{\kern
2.77774pt\hbox to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern
1.94441pt}}2^{-n}\colon\psi(\vec{x},A)\in\Gamma,n\in\mathbb{N}\\}.$

######  Remark 4.6.

A tuple $\vec{c}$ satisfies $\Gamma$ in $\mathcal{M}^{\prime}$ if and only if
$\vec{c}$ satisfies $\Gamma^{ap}$ in $\mathcal{M}^{\prime}$. $\Gamma$ is
finitely satisfiable in $\mathcal{M}^{\prime}$ if and only if $\Gamma^{ap}$ is
finitely satisfiable in $\mathcal{M}^{\prime}$.

###### Proof.

By Fact 2.26, $\mathcal{M}^{\prime}$ is $\aleph_{1}$-saturated. ∎

######  Definition 4.7.

A _distribution_ of $\Gamma$ in $\mathcal{M}^{\prime}$ is a monotone mapping
$\delta\colon\mathcal{P}_{\aleph_{0}}(\Gamma^{ap})\to\mathcal{D}$ such that:

  * (a)

$\delta\colon\mathcal{P}_{\aleph_{0}}(\Gamma^{ap})\to\mathcal{X}$ for some
$\mathcal{X}$ that regularizes $\mathcal{D}$.

  * (b)

For each $\Psi\in\mathcal{P}_{\aleph_{0}}(\Gamma^{ap})$ and
$t\in\delta(\Psi)$,

$\mathcal{M}_{t}\models(\inf_{\vec{x}})\max_{\psi\in\Psi}\psi(\vec{x},A[t])=0.$

For each $t\in I$, we define

$\Gamma(\delta,t)=\\{\psi\in\Gamma^{ap}\colon t\in\delta(\\{\psi\\})\\}.$

A distribution $\delta$ of $\Gamma$ in $\mathcal{M}^{\prime}$ is _accurate_ if

  * (c)

For all $\Psi\in\mathcal{P}_{\aleph_{0}}(\Gamma^{ap})$,

$\delta(\Psi)=\left\\{t\in\left(\bigcap_{\psi\in\Psi}\delta(\\{\psi\\})\right)\colon\mathcal{M}_{t}\models(\inf_{\vec{x}})\max_{\psi\in\Psi}\psi(\vec{x},A[t])=0\right\\}.$

If $\mathcal{M}$ is a first order structure and $\Gamma$ is a set of first
order formulas, the notion of a distribution is the same as above except that
$\Gamma^{ap}$ is replaced by $\Gamma$, and
$(\inf_{\vec{x}})\max_{\psi\in\Psi}\psi(\vec{x},A[t])=0$ is replaced by
$(\exists\vec{x})\bigwedge_{\psi\in\Psi}\psi(\vec{x},A[t])$.

######  Remark 4.8.

Let $\delta$ be a distribution of $\Gamma$ in $\mathcal{M}^{\prime}$, and let
$\Psi\in\mathcal{P}_{\aleph_{0}}(\Gamma^{ap})$.

(i) For each $t\in I$, $\Gamma(\delta,t)$ is finite.

(ii) $\delta(\Psi)\subseteq\\{t\in
I\colon\Psi\subseteq\Gamma(\delta,t)\\}\in\mathcal{D}$.

(iii) If $\delta$ is multiplicative, then $\delta(\Psi)=\\{t\in
I\colon\Psi\subseteq\Gamma(\delta,t)\\}$ and $\delta$ is accurate.

(iv) We think of $\Gamma(\delta,t)$ as the part of $\Gamma^{ap}$ that matters
at the index $t$.

######  Lemma 4.9.

The following are equivalent.

  * (i)

$\Gamma$ is finitely satisfiable in $\mathcal{M}^{\prime}$.

  * (ii)

$\Gamma$ has a distribution in $\mathcal{M}^{\prime}$.

  * (iii)

$\Gamma$ has an accurate distribution in $\mathcal{M}^{\prime}$.

###### Proof.

(ii) $\Rightarrow$ (i): By Remark 2.26, $\mathcal{M}^{\prime}$ is
$\aleph_{1}$-saturated, so by Remark 4.6, $\Gamma$ is finitely satisfiable in
$\mathcal{M}^{\prime}$ if and only if $\Gamma^{ap}$ is finitely satisfiable in
$\mathcal{M}^{\prime}$. If $\delta$ is a distribution of $\Gamma$ in
$\mathcal{M}^{\prime}$ and $\Psi$ is a finite subset of $\Gamma^{ap}$, then
$\delta(\Psi)\in\mathcal{D}$, so by (c), the Łoś Theorem, and the
$\aleph_{1}$-saturation of $\mathcal{M}^{\prime}$, $\Psi$ is satisfiable in
$\mathcal{M}^{\prime}$.

(i) $\Rightarrow$ (iii): Suppose $\Gamma^{ap}$ is finitely satisfiable in
$\mathcal{M}^{\prime}$. Since $\mathcal{D}$ is regular, $\mathcal{D}$ has a
regularizing set $\mathcal{X}$. Thus $\mathcal{X}\subseteq\mathcal{D}$,
$|\mathcal{X}|=\lambda$, and each $t\in I$ belongs to only finitely many
$X\in\mathcal{X}$. So there is an injective function
$h\colon\Gamma^{ap}\to\mathcal{X}$. For $\psi\in\Gamma^{ap}$ let
$\delta(\\{\psi\\})=h(\psi)$. Then the mapping $\delta$ defined by condition
(c) of Definition 4.7 is an accurate distribution of $\Gamma$ in
$\mathcal{M}^{\prime}$. ∎

Since $\mathcal{D}$ is regular and $I$ is infinite, we may choose sets
$J_{1}\supseteq J_{2}\supseteq\cdots$ in $\mathcal{D}$ such that
$\bigcap_{n}J_{n}$ is empty. Put $J_{0}=I$, and for each $t\in I$, let $n(t)$
be the greatest $n\in\mathbb{N}$ such that $t\in J_{n}$.

######  Lemma 4.10.

The following are equivalent.

  * (i)

$\Gamma$ is satisfiable in $\mathcal{M}^{\prime}$.

  * (ii)

$\Gamma$ has a multiplicative distribution in $\mathcal{M}^{\prime}$.

  * (iii)

$\Gamma$ is finitely satisfiable in $\mathcal{M}^{\prime}$ and every accurate
distribution of $\Gamma$ in $\mathcal{M}^{\prime}$ has a multiplicative
refinement.

###### Proof.

(i) $\Rightarrow$ (ii): Suppose $\vec{c}$ satisfies $\Gamma$ in
$\mathcal{M}^{\prime}$. Let $\mathcal{X}$ regularize $\mathcal{D}$ and let
$h\colon\Gamma^{ap}\to\mathcal{X}$ be injective. For each
$\Psi\in\mathcal{P}_{\aleph_{0}}(\Gamma^{ab})$, let

$\delta(\Psi)=\bigcap_{\psi\in\Psi}(h(\psi)\cap\\{t\in
I\colon\mathcal{M}_{t}\models\psi(\vec{c}[t],A[t])=0\\}).$

It is clear that $\delta$ is multiplicative, and each $t\in I$ belongs to
$\delta(\Psi)$ for only finitely many $\Psi$. Consider a formula
$\psi(\vec{x},A)\in\Gamma^{ap}$. For some $\theta\in\Gamma$ and
$n\in\mathbb{N}$, we have
$\psi(\vec{x},A)=(\theta(\vec{a},A)\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}2^{-n})$, and
$\mathcal{M}^{\prime}\models\theta(\vec{c},A)$. By the Łoś Theorem,

$\\{t\in
I\colon\mathcal{M}_{t}\models\psi(\vec{c}[t],A[t])=0\\}\in\mathcal{D}.$

Therefore $\delta$ maps $\mathcal{P}_{\aleph_{0}}(\Gamma^{ap})$ into
$\mathcal{D}$, and hence $\delta$ is a multiplicative distribution of $\Gamma$
in $\mathcal{M}^{\prime}$.

(ii) $\Rightarrow$ (i): Now suppose $\delta$ is a multiplicative distribution
of $\Gamma$ in $\mathcal{M}^{\prime}$. Let $t\in I$. By (a), the set

$\Psi=\\{\psi\in\Gamma^{ap}\colon t\in\delta(\\{\psi\\})\\}$

belongs to $\mathcal{P}_{\aleph_{0}}(\Gamma^{ap})$. By multiplicity,
$\delta(\Psi)=\bigcap_{\psi\in\Psi}\delta(\\{\psi\\})$, so
$t\in\delta(\Psi)\in\mathcal{D}.$ By 4.7 (b), there is a tuple $\vec{c}[t]$ in
$\mathcal{M}_{t}$ such that

$(\forall\psi\in\Psi)\mathcal{M}_{t}\models\psi(\vec{c}[t],A[t])\leq
2^{-n(t)}.$

Then for each $\Psi\in\mathcal{P}_{\aleph_{0}}(\Gamma^{ab})$ and
$n\in\mathbb{N}$, we have

$X:=\delta(\Psi)\cap J_{n}\in\mathcal{D},\quad(\forall t\in
X)(\forall\psi\in\Psi)\mathcal{M}_{t}\models\psi(\vec{c}[t],A[t])\leq 2^{-n}.$

By the Łoś Theorem, for each $\psi\in\Gamma^{ab}$ and $n\in\mathbb{N}$ we have
$\mathcal{M}^{\prime}\models\psi(\vec{c},A)\leq 2^{-n}$, so
$\mathcal{M}^{\prime}\models\psi(\vec{c},A)=0$.

(iii) $\Rightarrow$ (ii): Assume (iii). By Lemma 4.9, $\Gamma$ has an accurate
distribution in $\mathcal{M}^{\prime}$, so by (iii), $\Gamma$ has a
multiplicative distribution in $\mathcal{M}^{\prime}$.

(i) $\Rightarrow$ (iii): Suppose $\vec{c}$ satisfies $\Gamma$ in
$\mathcal{M}^{\prime}$ and $\delta$ is an accurate distribution of $\Gamma$ in
$\mathcal{M}^{\prime}$. For $\Psi\in\mathcal{P}_{\aleph_{0}}(\Gamma^{ap})$,
let

$\delta^{\prime}(\Psi)=\left\\{t\in\delta(\Psi)\colon\mathcal{M}_{t}\models\max_{\psi\in\Psi}\psi(\vec{c}[t],A[t])=0)\right\\}.$

Then $\delta^{\prime}$ is a distribution of $\Gamma$ in $\mathcal{M}^{\prime}$
that refines $\delta$. Suppose
$t\in\delta^{\prime}(\Psi)\cap\delta^{\prime}(\Theta)$. Then

$t\in\delta(\Psi)\cap\delta(\Theta)\mbox{ and
}\mathcal{M}_{t}\models\max_{\psi\in\Psi\cup\theta}\psi(\vec{c}[t],A[t])=0).$

Since $\delta$ is accurate, $t\in\delta(\Psi\cup\Theta)$, so
$t\in\delta^{\prime}(\Psi\cup\Theta)$ and $\delta^{\prime}$ is multiplicative.
∎

###### Proof of Theorem 4.4.

Let $\mathcal{M}^{\prime}=\prod^{\mathcal{D}}\mathcal{M}_{t}$ and
$\mathcal{N}^{\prime}=\prod^{\mathcal{D}}\mathcal{N}_{t}$. We suppose that
$\mathcal{M}^{\prime}$ is $\lambda^{+}$-saturated, and prove that
$\mathcal{N}^{\prime}$ is $\lambda^{+}$-saturated. Let $B$ be a
$\lambda$-sequence of elements of $\mathcal{N}^{\prime}$, and assume that
$\Gamma(u,B)$ is a set of at most $\lambda$ formulas with parameters from $B$
and at most $u$ free that is is finitely satisfiable in
$\mathcal{N}^{\prime}$. By Lemma 2.11, to show that $\mathcal{N}^{\prime}$ is
$\lambda^{+}$-saturated it suffices to prove that $\Gamma(u,B)$ is satisfiable
in $\mathcal{N}^{\prime}$. By Lemma 4.9, $\Gamma(u,B)$ has a distribution
$\delta_{1}$ in $\mathcal{N}^{\prime}$. For each $\psi\in\Gamma^{ap}(u,B)$, we
have $\psi=(\theta(\psi)\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}2^{-m(\psi)})$ for some
$\theta(\psi)\in\Gamma(u,A)$ and $m(\psi)\in\mathbb{N}$, and we define
$\psi^{+}=(\theta(\psi)\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}2^{-(m(\psi)+1)})$.
Then $\psi\mapsto\psi^{+}$ is a bijection from $\Gamma^{ap}(u,B)$ onto itself.
Define $\Psi^{+}=\\{\psi^{+}\colon\psi\in\Psi\\}$.

For each $t\in I$, let
$m(t)=\max\\{m(\psi)\colon\psi\in\Gamma(\delta_{1},t)\\}$. By Lemma 4.3, there
is a $\lambda$-sequence $A$ of elements of $\mathcal{M}^{\prime}$ such that
for every $t\in I$,

  * (1)

For every $e[t]\in\mathcal{N}_{t}$ there exists $c[t]\in\mathcal{M}_{t}$ such
that for every $\psi$ in $\Gamma(\delta_{1},t)\\}$,
$\theta(\psi)^{\mathcal{N}_{t}}(e[t],B[t])$ is within $2^{-(m(t)+1)}$ of
$\theta(\psi)^{\mathcal{M}_{t}}(c[t],A[t])$.

  * (2)

For every $c[t]\in\mathcal{M}_{t}$ there exists $e[t]\in\mathcal{N}_{t}$ such
that for every $\psi$ in $\Gamma(\delta_{1},t)\\}$,
$\theta(\psi)^{\mathcal{N}_{t}}(e[t],B[t])$ is within $2^{-(m(t)+1)}$ of
$\theta(\psi)^{\mathcal{M}_{t}}(c[t],A[t])$.

It follows from (1) that the mapping
$\delta_{2}\colon\mathcal{P}_{\aleph_{0}}(\Gamma^{ap}(u,A))\to\mathcal{D}$
such that

$\delta_{2}(\Psi(u,A))=\delta_{1}(\Psi^{+}(u,B))$

is a distribution of $\Gamma(u,A)$ in $\mathcal{M}^{\prime}$, so by Lemma 4.9,
$\Gamma(u,A)$ is finitely satisfiable in $\mathcal{M}^{\prime}$. Since
$\mathcal{M}^{\prime}$ is $\lambda^{+}$-saturated, $\Gamma(u,A)$ is
satisfiable in $\mathcal{M}^{\prime}$. Then by Lemma 4.10, there is a
multiplicative distribution $\delta_{3}$ of $\Gamma(u,A)$ in
$\mathcal{M}^{\prime}$. It follows from (2) that the mapping
$\delta_{4}\colon\mathcal{P}_{\aleph_{0}}(\Gamma^{ap}(u,B))\to\mathcal{D}$
such that

$\delta_{4}(\Psi(u,B))=\delta_{3}(\Psi^{+}(u,A))$

is a multiplicative distribution of $\Gamma(u,B)$ in $\mathcal{N}^{\prime}$,
so by Lemma 4.10, $\Gamma(u,B)$ is satisfiable in $\mathcal{N}^{\prime}$. ∎

##  5\. Every continuous theory is $\trianglelefteq$-equivalent to a metric
theory

In this short section we will prove the following.

######  Theorem 5.1.

For every continuous theory $T$ there us a metric theory $U$ (which may have a
different vocabulary) such that $T\trianglelefteq\trianglerighteq U$.

We will see that Theorem 5.1 is an easy consequence of Theorem 3.5 above and
results about premetric expansions from the paper [Ke21] (Fact 5.3–5.7 below).

We say that a sequence
$\langle\varphi_{m}(\vec{x},\vec{y})\rangle_{m\in\mathbb{N}}$ of formulas is
_Cauchy_ in a $V$-structure $\mathcal{M}$ if for each $\varepsilon>0$ there
exists $m$ such that for all $k\geq m$,

$\mathcal{M}\models(\sup_{\vec{x}})(\sup_{\vec{y}})|\varphi_{m}(\vec{x},\vec{y})-\varphi_{k}(\vec{x},\vec{y})|\leq\varepsilon.$

If $\langle\varphi_{m}(\vec{x},\vec{y})\rangle_{m\in\mathbb{N}}$ is Cauchy in
$\mathcal{M}$, then there is a unique mapping from $M^{|\vec{x}|}\times
M^{|\vec{y}|}$ into $[0,1]$, denoted by $[\lim\varphi_{m}]^{\mathcal{M}}$,
such that

$(\forall\vec{b}\in M^{|\vec{x}|})(\forall\vec{c}\in
M^{|\vec{y}|})[\lim\varphi_{m}]^{\mathcal{M}}(\vec{b},\vec{c})=\lim_{m\to\infty}\varphi_{m}^{\mathcal{M}}(\vec{b},\vec{c}).$

######  Definition 5.2.

Let $T$ be a $V$-theory, let $D$ be a predicate symbol that may or may not
belong to $V$, and let $V_{D}=V\cup\\{D\\}$. We say that a $V_{D}$-theory
$T_{e}$ is a _premetric expansion_ of $T$ if:

  * (i)

There is a metric signature $L_{e}$ over $V_{D}$ with distance predicate $D$
such that $T_{e}$ is an $L_{e}$-metric theory.

  * (ii)

There is a Cauchy sequence $\langle d\rangle=\langle d_{m}\rangle$ of
$V$-formulas such that the models of $T_{e}$ are exactly the
$V_{D}$-structures of the form $\mathcal{M}_{e}=(\mathcal{M},[\lim
d_{m}]^{\mathcal{M}})$, where $\mathcal{M}$ is a model of $T$.

It follows that $\mathcal{M}_{e}$ is an $L_{e}$-premetric structure, which we
call the _premetric expansion_ of $\mathcal{M}$ to $T_{e}$.

######  Fact 5.3.

(Expansion Theorem, Theorem 3.3.4 of [Ke21]) Every continuous theory $T$ has a
premetric expansion.

######  Fact 5.4.

(By Propositions 3.4.5 and 4.3.4 of [Ke21]) Every reduced $V$-structure
$\mathcal{M}$ has a unique topology that is metrizable by a metric $D$ (not
unique) such that $(\mathcal{M},D)$ is a premetric expansion of $\mathcal{M}$.

The following fact can be used to show that when $T_{e}$ is a premetric
expansion of $T$, various properties hold for $T$ if and only if they hold for
$T_{e}$.

######  Fact 5.5.

(Lemma 4.2.1 in [Ke21]) Suppose $T_{e}$ is a premetric expansion of $T$. Then
for every continuous formula $\varphi(\vec{x})$ in the vocabulary of $T_{e}$,
and every real $\varepsilon>0$, there is a formula $\theta(\vec{x})$ in the
vocabulary of $T$ such that

$T_{e}\models(\sup_{\vec{x}})|\varphi(\vec{x})-\theta(\vec{x})|\leq\varepsilon.$

######  Fact 5.6.

(Proposition 3.1.6 of [Ke21]). Suppose $T_{e}$ is a premetric expansion of
$T$. Then for every pre-ultraproduct $\prod^{\mathcal{D}}\mathcal{M}_{t}$ of
models of $T$,
$(\prod^{\mathcal{D}}\mathcal{M}_{t})_{e}=\prod^{\mathcal{D}}((\mathcal{M}_{t})_{e})$.

######  Fact 5.7.

(Proposition 4.1.6 of [Ke21]) If $\mathcal{M}_{e}$ is a premetric expansion of
$\mathcal{M}$, then for each infinite cardinal $\kappa$, $\mathcal{M}_{e}$ is
$\kappa$-saturated if and only if $\mathcal{M}$ is $\kappa$-saturated.

######  Theorem 5.8.

If $T$ is a continuous theory and $T_{e}$ is a premetric expansion of $T$,
then $T_{e}\trianglelefteq\trianglerighteq T$.

###### Proof of Theorem 5.8.

Let $\mathcal{M}$ be a model of $T$. By Fact 5.6,
$\prod^{\mathcal{D}}(\mathcal{M}_{e})=(\prod^{\mathcal{D}}\mathcal{M})_{e}$,
so $\prod^{\mathcal{D}}(\mathcal{M}_{e})$ is a premetric expansion of
$\prod^{\mathcal{D}}\mathcal{M}$. By Fact 5.7,
$\prod^{\mathcal{D}}(\mathcal{M}_{e})$ is $\lambda^{+}$-saturated if and only
if $\prod^{\mathcal{D}}\mathcal{M}$ is $\lambda^{+}$-saturated. Therefore by
Theorem 3.5, we have $T_{e}\trianglelefteq\trianglerighteq T$. ∎

###### Proof of Theorem 5.1.

By Fact 5.3, $T$ has a pre-metric expansion $T_{e}$. By definition, $T_{e}$ is
a metric theory. By Theorem 5.8, $T_{e}\trianglelefteq\trianglerighteq T$. ∎

##  6\. Stable theories

In view of the Theorem 5.8, from here on we may confine our attention to
metric theories.

######  Convention 6.1.

For the rest of this paper, $L$ will be a metric signature over $V$ with
distance predicate $d\in V$, and $T$ will be an $L$-metric theory.

Thus by Corollary 2.5, every model of $T$ is an $L$-premetric structure.

In this section we study the $\trianglelefteq$ ordering on stable metric
theories. Theorem 6.22 (iii) below will show that if $T,U$ are metric
theories, $T$ is stable, and $U$ is unstable, then $T\vartriangleleft U$.

The papers [BBHU08] and [BU09] gave equivalent definitions of a stable metric
theory. Here we will use the definition in [BU09] (Definition 6.2 below).

For every result proved in this section, the corresponding result for first
order theories was proved by Shelah in Section VI of [Sh78]. The arguments in
this section are similar to the arguments in [Sh78], but generalized to the
continuous case. Many first order results about Morley sequences and
indiscernible sets have been extended to continuous model logic (e.g. in
[BU09], [BY03], [BY09], [BY13], and [EG12]), but we will need to extend some
additional results about indiscernible sets from first order logic to
continuous logic here. It will be convenient to use the properties of a stable
independence relation (Definition 6.5) rather than the notion of a non-forking
extension of a type, and sometimes to use ultraproducts of real-valued
structures rather than just metric structures.

######  Definition 6.2.

(Local stability. See Definition 7.1 of [BU09]).

(i) A continuous formula $\varphi(\vec{x},\vec{y})$ is _unstable with bounds
$(s,r)$ in $T$_ if $0\leq s<r\leq 1$ and in some model $\mathcal{M}\models T$
there is an infinite sequence
$\langle\vec{a}_{h},\vec{b}_{h}\rangle_{h\in\mathbb{N}}$ such that whenever
$h<k$, $\varphi^{\mathcal{M}}(\vec{a}_{h},\vec{b}_{k})\leq s$ and
$\varphi^{\mathcal{M}}(\vec{a}_{k},\vec{b}_{h})\geq r$. $\varphi$ is
_unstable_ in $T$ if it is unstable in $T$ for some $(s,r)$, and is _stable_
in $T$ otherwise.

(ii) $T$ is stable if every continuous formula $\varphi(\vec{x},\vec{y})$ is
stable in $T$.

Note that if $\varphi(\vec{x},\vec{y})$ is unstable in $T$, then there is a
model $\mathcal{M}$ as in (i) that has a countable dense set.

######  Corollary 6.3.

$T$ is unstable if and only if there is a continuous formula
$\theta(\vec{x},\vec{y})$ that is unstable for $(0,1)$ in $T$. That is, in
some model $\mathcal{M}\models T$ there is an infinite sequence
$\langle\vec{a}_{h},\vec{b}_{h}\rangle_{h\in\mathbb{N}}$ such that whenever
$h<k$, $\varphi^{\mathcal{M}}(\vec{a}_{h},\vec{b}_{k})=0$ and
$\varphi^{\mathcal{M}}(\vec{a}_{k},\vec{b}_{h})=1$.

###### Proof.

Suppose $\varphi(\vec{x},\vec{y})$ is unstable for $(s,r)$ in $T$. There is a
continuous function $f\colon[0,1]\to[0,1]$ that maps $[0,s]$ to $0$ and maps
$[r,1]$ to $1$. Then the formula $f(\varphi(\vec{x},\vec{y}))$ is unstable for
$(0,1)$ in $T$. ∎

As is often done in the literature, we assume for convenience that there is an
uncountable inaccessible cardinal $\Upsilon$. That assumption simplifies
things but can be avoided. We call a set _small_ if it has cardinality less
than $\Upsilon$. By a _monster structure_ we mean a $\Upsilon$-saturated
metric structure of cardinality $\leq\Upsilon$.

######  Fact 6.4.

Up to isomorphism, $T$ has a unique monster model $\mathfrak{M}$. Every small
model of $T$ is elementarily embeddable in $\mathfrak{M}$.

###### Proof.

The proof that $T$ a unique reduced $\Upsilon$-saturated model $\mathfrak{M}$
of cardinality $\leq\Upsilon$ is similar to the proof of the analogous result
in first order model theory. Since $\mathfrak{M}$ is reduced and
$\aleph_{1}$-saturated, it is a metric structure by Remark 2.15. ∎

We recall some notation from the literature. We sometimes write $AB$ for
$A\cup B$. We write $A\equiv_{B}A^{\prime}$ if
$\mathfrak{M}_{BA}\equiv\mathfrak{M}_{BA^{\prime}}$. Given an $n$-tuple
$\vec{a}$ in the monster model $\mathfrak{M}$ of $T$ and a subset $B$ of
$\mathfrak{M}$, the _type_ of $\vec{a}$ over $B$ is the set
$\operatorname{tp}(\vec{a}/B)$ of all formulas $\varphi(\vec{x},B)$ with
parameters in $B$ such that $\mathfrak{M}\models\varphi(\vec{a},B)=0.$ The set
of all $n$-types over $B$ is
$S_{n}(B)=\\{\operatorname{tp}(\vec{a}/B)\colon\vec{a}\in\mathfrak{M}^{n}\\}$.
For each $p\in S_{n}(B)$ and formula $\psi(\vec{x},B)$ with $|\vec{x}|=n$,
$p^{\psi}$ is the unique $r\in[0,1]$ such that $|\psi-r|\in p$.

The following definition is equivalent to Definition 14.13 of [BBHU08], in
view of Theorem 8.10 of [BU09].

######  Definition 6.5.

A _stable independence relation_ for $T$ is a ternary relation
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hbox
to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 2.85764pt}}$
on the small subsets of the monster model $\mathfrak{M}$ of $T$ that has the
following properties:

  * •

_Invariance under automorphisms of $\mathfrak{M}$._

  * •

_Symmetry:_ $A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}B$ if and only if $B\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}A$.

  * •

_Transitivity:_ $A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}BD$ if and only if $A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}B$ and $A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{BC}D$.

  * •

_Finite Character:_ $A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}B$ if and only if $\vec{a}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern
4.00069pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
2.71246pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern
2.85764pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hbox
to0.0pt{$$}\hss}\lower 1.93747pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 2.85764pt}}_{C}B$ for all
finite $\vec{a}\subseteq A$.

  * •

_Existence:_ For all $A,B,C$ there exists $A^{\prime}$ such that
$A^{\prime}\equiv_{C}A$ and $A^{\prime}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern
4.00069pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
2.71246pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern
2.85764pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hbox
to0.0pt{$$}\hss}\lower 1.93747pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 2.85764pt}}_{C}B$.

  * •

_Strong Local Character:_ For each finite $\vec{a}$, there exists $C\subseteq
B$ of cardinality $\leq\aleph_{0}$ with $\vec{a}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern
4.00069pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
2.71246pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern
2.85764pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hbox
to0.0pt{$$}\hss}\lower 1.93747pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 2.85764pt}}_{C}B$.

  * •

_Stationarity:_ For all small $B$, algebraically closed $C$, and tuples
$\vec{a},\vec{a}\,^{\prime}$,

$\mbox{If}\quad\vec{a}\equiv_{C}\vec{a}^{\prime},\quad\vec{a}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern
4.00069pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
2.71246pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern
2.85764pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hbox
to0.0pt{$$}\hss}\lower 1.93747pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}B,\quad\vec{a}\,^{\prime}\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern
4.00069pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
2.71246pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern
2.85764pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hbox
to0.0pt{$$}\hss}\lower 1.93747pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}B,\quad\mbox{then}\quad\vec{a}\equiv_{BC}\vec{a}^{\prime}.$

######  Fact 6.6.

(See Theorems 14.6 and 14.14 of [BBHU08].) $T$ is stable if and only if there
is a stable independence relation for $T$, and also if and only if there is a
unique stable independence relation for $T$.

######  Convention 6.7.

Hereafter, when $T$ is a stable theory, $\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern
4.00069pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
2.71246pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern
2.85764pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hbox
to0.0pt{$$}\hss}\lower 1.93747pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 2.85764pt}}$ will denote the
unique stable independence relation for $T$.

By a proper tuple we mean a tuple of distinct elements. Let $n\in\mathbb{N}$.
A set $\mathcal{I}\subseteq\mathfrak{M}^{n}$ is called $B$-indiscernible if
every two proper tuples of elements of $\mathcal{I}$ of the same length have
the same type over $B$. A sequence
$\mathcal{I}=\langle\vec{a}_{h}\rangle_{h\in\mathbb{N}}$ of elements of
$\mathfrak{M}^{n}$ is $B$-indiscernible if every two strictly increasing
finite subsequences of $\mathcal{I}$ of the same length have the same type
over $B$. Indiscernible means $\emptyset$-indiscernible.

######  Lemma 6.8.

Suppose $T$ is stable. Then every $B$-indiscernible sequence is a
$B$-indiscernible set.

###### Proof.

The proof is the same as the proof of the corresponding result in first order
logic (see, for example, Proposition 7.1 in [Pi83]), but using the continuous
notion of a stable formula in Definition 6.2 instead of the first order notion
of a stable formula. ∎

The next lemma is a continuous analogue of Definition III.1.5 and Lemma
III.1.7 in [Sh78].

######  Lemma 6.9.

Suppose $T$ is stable, $n\in\mathbb{N}$,
$\mathcal{I}\subseteq\mathfrak{M}^{n}$ is an infinite indiscernible set, and
$B\subseteq\mathfrak{M}$ is small. There is a unique type
$\operatorname{Av}(\mathcal{I},B)\in S_{k}(B)$ such that for each formula
$\psi(\vec{x},B)$ with $|\vec{x}|=n$ and $\varepsilon>0$, for all but finitely
many $\vec{a}\in\mathcal{I}$ we have

$|\psi^{\mathfrak{M}}(\vec{a},B)-\operatorname{Av}(\mathcal{I},B)^{\psi}|<\varepsilon.$

###### Proof.

Consider a formula $\psi(\vec{x},B)$. We have
$\psi^{\mathfrak{M}}(\vec{x},B)=\psi^{\mathfrak{M}}(\vec{x},\vec{b})$ for some
finite $\vec{b}\subseteq B$. It suffices to show that there is a unique value
$t(\psi)\in[0,1]$ such that for every $\varepsilon>0$, for all but finitely
many $\vec{a}\in\mathcal{I}$ we have

$|\psi^{\mathfrak{M}}(|\vec{a},\vec{b})-t(\psi)|<\varepsilon,$

because then

$\operatorname{Av}(\mathcal{I},B)=\\{\psi(\vec{x},\vec{b})\colon t(\psi)=0\\}$

has the required property. It is clear that there is at most one such value
$t(\psi)$.

Let is call a set $X\subseteq[0,1]$ _broad_ if $X$ is a closed interval, and
$\psi^{\mathfrak{M}}(\vec{a},\vec{b})\in X$ for all but finitely many
$\vec{a}\in\mathcal{I}$. Note that any intersection of finitely many broad
sets is broad.

Claim 6.9.1: Suppose $0\leq s<r\leq 1$. Then at least one of the intervals
$[0,r],[s,1]$ is broad.

Proof of Claim 6.9.1. Assume not. Then there are infinitely many
$\vec{a}\in\mathcal{I}$ such that $\psi^{\mathfrak{M}}(\vec{a},\vec{b})<s$,
and infinitely many $\vec{c}\in\mathcal{I}$ such that
$\psi^{\mathfrak{M}}(\vec{c},\vec{b})>r$. Hence there is an infinite
subsequence $\langle\vec{c}_{h}\rangle_{h\in\mathbb{N}}$ of $\mathcal{I}$ such
that for all even $h\in\mathbb{N}$ we have
$\psi^{\mathfrak{M}}(\vec{c}_{h},\vec{b})<s$, and for all odd $j\in\mathbb{N}$
we have $\psi^{\mathfrak{M}}(\vec{c}_{j},\vec{b})>r$. Taking
$\vec{b}_{h}=\vec{b}$ for all $h$, we see that $\psi(\vec{x},\vec{y})$ is
unstable in $T$. But since $T$ is stable by hypothesis,
$\psi(\vec{x},\vec{y})$ is stable in $T$ by Fact 6.6. This proves Claim 6.9.1.

Claim 6.9.2: For each positive $k\in\mathbb{N}$ there exists a broad set
$X_{k}$ of length $\leq 2/k$.

Proof of Claim 6.9.2. There is a least integer $m_{0}\leq k$ such that
$[0,m_{0}/k]$ is broad, and a greatest integer $m_{1}\leq k$ such that
$[m_{1}/k,1]$ is broad. Then the set

$X_{k}=[0,m_{0}/k]\cap[m_{1}/k,1]=[m_{1}/k,m_{0}/k]$

is broad, but neither of the sets $[0,(m_{0}-1)/k]$, $[(m_{1}+1)/k,1]$ is
broad. By Claim 6.9.1, we cannot have $m_{0}-1>m_{1}+1$. Hence $m_{0}\leq
m_{1}+2$, so the interval $X_{k}$ has length $\leq 2/k$. This proves Claim
6.9.2.

Since $[0,1]$ is compact, $\bigcap_{n=1}^{\infty}X_{k}$ is non-empty, and
contains exactly one point $t\in[0,1]$. The result now follows with
$t(\psi)=t$. ∎

######  Corollary 6.10.

Suppose $T$ is stable, $n\in\mathbb{N}$,
$\mathcal{I}\subseteq\mathfrak{M}^{n}$ is an infinite indiscernible set, and
$B\subseteq\mathfrak{M}$ is small. Then:

  * (i)

If $\mathcal{I}$ is $B$-indiscernible, then each $\vec{a}\in\mathcal{I}$
realizes $\operatorname{Av}(\mathcal{I},B)$.

  * (ii)

For any infinite $\mathcal{H}\subseteq\mathcal{I}$,
$\operatorname{Av}(\mathcal{H},B)=\operatorname{Av}(\mathcal{I},B)$.

###### Proof.

(i) is clear.

(ii) By Lemma 6.9 we have
$\operatorname{Av}(\mathcal{I},B)\subseteq\operatorname{Av}(\mathcal{H},B)$.
Since $\operatorname{Av}(\mathcal{I},B)\in S_{n}(B)$ and
$\operatorname{Av}(\mathcal{H},B)\in S_{n}(B)$,
$\operatorname{Av}(\mathcal{H},B)=\operatorname{Av}(\mathcal{J},B)$. ∎

We will only need the following lemma for subsets of $\mathfrak{M}$, but a
slightly more complicated argument will prove the corresponding result for
subsets of $\mathfrak{M}^{n}$.

######  Lemma 6.11.

Suppose $T$ is stable, and $C$ is small and algebraically closed. For each
element $a\in\mathfrak{M}\setminus C$ there is an infinite $C$-indiscernible
set $\mathcal{I}\subseteq\mathfrak{M}$ such that $a\in\mathcal{I}$, and
$X\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}Y$ for any disjoint $X,Y\subseteq\mathcal{I}$.

###### Proof.

The proof is the same as the corresponding argument in first order logic. By
Existence and Symmetry, for any $X$ and $Y$ we have
$X\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}\emptyset$ and $\emptyset\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern
4.00069pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
2.71246pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern
2.85764pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hbox
to0.0pt{$$}\hss}\lower 1.93747pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 2.85764pt}}_{C}Y$. By
Existence, one can inductively build a sequence $\mathcal{I}=\langle
a_{h}\rangle_{h\in\mathbb{N}}$ in $\mathfrak{M}$ such that $a_{0}=a$, and for
each $n\in\mathbb{N}$ we have

$a_{n}\equiv_{C}a\mbox{ and }a_{n}\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}\\{a_{h}\colon h<n\\}.$

By Anti-reflexivity and Finite Character, $a_{n}\neq a_{h}$ whenever $h<n$, so
$\mathcal{I}$ is infinite. Using Symmetry and Transitivity for
$\mathop{\mathchoice{\kern 5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hbox
to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 2.85764pt}}$,
one can show by induction on $n$ that $X\mathop{\mathchoice{\kern
5.71527pt\hbox to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower
3.87495pt\hbox to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern
4.00069pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
2.71246pt\hbox to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern
2.85764pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hbox
to0.0pt{$$}\hss}\lower 1.93747pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 2.85764pt}}_{C}Y$ for each
pair $X,Y$ of disjoint subsets of $\\{a_{0},\ldots,a_{n}\\}$. By Symmetry and
Finite Character, it follows that $X\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}Y$ for any disjoint $X,Y\subseteq\mathcal{I}$.

By Lemma 6.8, to show that $\mathcal{I}$ is a $C$-indiscernible set, it
suffices to show that $\mathcal{I}$ is a $C$-indiscernible sequence. This
means that for all $n\in\mathbb{N}$, $X\equiv_{C}Y$ for all strictly
increasing subsequences

$X=\\{x_{1},\ldots,x_{n}\\},Y=\\{y_{1},\ldots,y_{n}\\}$

of $\mathcal{I}$ of length $n$. We argue by induction on $n$. The result holds
trivially for $n=1$. Suppose $(X,u)$, $(Y,v)$ are strictly increasing
subsequences of $\mathcal{I}$ of length $n+1$, and $X\equiv_{C}Y$. Without
loss of generality we may assume that $u$ is equal to or before $v$ in
$\mathcal{I}$. Then $u\equiv_{C}v$, $X\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}v$, and $Y\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}v$, so by Stationarity we have $X\equiv_{Cv}Y$ and hence
$(X,v)\equiv_{C}(Y,v)$. By a similar argument but using Symmetry, we have
$u\equiv_{CX}v$, and hence

$(X,u)\equiv_{C}(X,v)\equiv_{C}(Y,v).$

This completes the induction. ∎

The sequence $\mathcal{I}$ in the above proof is called a _Morley sequence_ in
type $\operatorname{tp}(a_{0}/C)$. A proof of the following fact for first
order logic using Morley sequences can be found, for example, in [Ad09]. The
same proof works for continuous logic (see also [EG12]).

######  Fact 6.12.

Every stable independence relation $\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern 2.85764pt}}$
has the following additional properties:

  * •

Base Monotonicity: If $C\subseteq D\subseteq B$ and
$A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}B$, then $A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{D}B$.

  * •

Normality: If $A\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}B$ then $AC\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}B$.

  * •

Anti-reflexivity: If $a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}a$ then $a$ belongs to the algebraic closure of $C$.

######  Lemma 6.13.

Suppose $T$ is stable and $\mathcal{I}$ is an indiscernible set. Then for any
set $B\subseteq\mathfrak{M}$ there is a set $X\subseteq\mathcal{I}$ such that
$|X|\leq|B|+\aleph_{1}$, and $\mathcal{I}\setminus X$ is $(B\cup\bigcup
X)$-indiscernible.

###### Proof.

The argument is exactly the same as the proof of Corollary III.3.5 of [Sh78],
but applied to continuous rather than first order logic. ∎

######  Lemma 6.14.

Suppose $T$ is stable, $\mathcal{M}\prec\mathfrak{M}$ is small and
$\aleph_{1}$-saturated, and every indiscernible set
$\mathcal{I}\subseteq\mathcal{M}$ of cardinality $\aleph_{0}$ can be extended
to an indiscernible set $\mathcal{H}\subseteq\mathcal{M}$ of cardinality
$\kappa$. Then $\mathcal{M}$ is $\kappa$-saturated.

###### Proof.

The proof is similar to the proof of Lemma III.3.10 in [Sh78]. Suppose
$\mathcal{M}$ is not $\kappa$-saturated. Then $\kappa>\aleph_{1}$, and there
is a set $B\subseteq M$ of cardinality $|B|<\kappa$ and an element
$a\in\mathfrak{M}$ such that no element of $M$ realizes
$\operatorname{tp}(a/B)$. By Strong Local Character and Base Monotonicity,
there is a countable algebraically closed set $C\subseteq M$ such that
$a\mathop{\mathchoice{\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\kern 5.71527pt\hbox
to0.0pt{\hss$\textstyle\mid$\hbox to0.0pt{$$}\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\kern 4.00069pt\hbox
to0.0pt{\hss$\scriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower 2.71246pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 4.00069pt}{\kern 2.85764pt\hbox
to0.0pt{\hss$\scriptscriptstyle\mid$\hbox to0.0pt{$$}\hss}\lower
1.93747pt\hbox to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
2.85764pt}}_{C}M$.

By Lemma 6.11, there is a Morley sequence $\mathcal{I}=\langle
a_{h}\rangle_{h\in\mathbb{N}}$ in $\operatorname{tp}(a/M)$, with $a_{0}=a$. By
Corollary 6.10 (i), each $a_{h}\in\mathcal{I}$ realizes
$\operatorname{Av}(\mathcal{I},M)$. Let $\mathcal{I}_{e}=\\{a_{h}\colon
h\mbox{ is even}\\}$ and $\mathcal{I}_{o}=\\{a_{h}\colon h\mbox{ is odd}\\}$.
Since $\mathcal{M}$ is $\aleph_{1}$-saturated and $C$ is countable, there is a
sequence $\mathcal{I}^{\prime}$ of elements of $M$ such that
$(\mathcal{I}_{e},\mathcal{I}_{o})\equiv_{C}(\mathcal{I}_{e},\mathcal{I}^{\prime})$.
Then $\mathcal{I}_{e}\cup\mathcal{I}^{\prime}$ is $C$-indiscernible. By
Corollary 6.10 (ii),

$\operatorname{Av}(\mathcal{I},M)=\operatorname{Av}(\mathcal{I}_{e},M)=\operatorname{Av}(\mathcal{I}^{\prime},M).$

By hypothesis, $\mathcal{I}^{\prime}$ can be extended to an indiscernible set
$\mathcal{H}\subseteq M$ of cardinality $\kappa$. By Lemma 6.13, there is a
set $X\subseteq\mathcal{H}$ such that $|X|\leq|BC|+\aleph_{1}<\kappa$ and
$\mathcal{H}^{\prime}=\mathcal{H}\setminus X$ is $BC$-indiscernible. Then
$\mathcal{H}^{\prime}$ is infinite. By Corollary 6.10 (ii),

$\operatorname{Av}(\mathcal{I}^{\prime},M)=\operatorname{Av}(\mathcal{H},M)=\operatorname{Av}(\mathcal{H}^{\prime},M),$

so

$\operatorname{Av}(\mathcal{I},BC)=\operatorname{Av}(\mathcal{H}^{\prime},BC).$

Pick an element $a^{\prime}\in\mathcal{H}^{\prime}$. Then $a^{\prime}\in M$.
By Corollary 6.10 (i), $a$ realizes $\operatorname{Av}(\mathcal{I},BC)$ and
$a^{\prime}$ realizes $\operatorname{Av}(\mathcal{H}^{\prime},BC)$. But then
$a\equiv_{BC}a^{\prime}$, which contradicts the fact that no element of $M$
realizes $\operatorname{tp}(a/B)$. So $\mathcal{M}$ is $\kappa$-saturated
after all. ∎

The following definition is from Shelah [Sh78].

######  Definition 6.15.

Let $\mathcal{D}$ be a regular ultrafilter. The _lower cofinality_
$\operatorname{lcf}(\mathcal{D})$ is the least cardinal $\kappa$ such that
there is a co-initial set $C$ of $\kappa$ infinite elements of the ultrapower
$(\mathbb{N},\leq)_{\mathcal{D}}$. That is, $C$ is a set of infinite elements
of $(\mathbb{N},\leq)_{\mathcal{D}}$, $|C|=\kappa$, and for each infinite
element $e\in(\mathbb{N},\leq)_{\mathcal{D}}$, we have
$(\mathcal{N},\leq)_{\mathcal{D}}\models c\leq e$ for some $c\in C$.

######  Definition 6.16.

Let $\mathcal{D}$ be a regular ultrafilter. The _lower cardinality_
$\operatorname{lca}(\mathcal{D})$ is the least cardinality of an infinite
initial segment of the ultrapower $(\mathbb{N},\leq)_{\mathcal{D}}$. That is,
the least infinite cardinal $\lambda$ such that for some
$a\in(\mathbb{N},\leq)_{\mathcal{D}}$,
$\lambda=|\\{b\colon(\mathbb{N},\leq)_{\mathcal{D}}\models b\leq a\\}|$.

Note that for every regular ultrafilter $\mathcal{D}$ over $I$, we have

$\aleph_{0}<\operatorname{lcf}(\mathcal{D})\leq\operatorname{lca}(\mathcal{D})\leq|2^{I}|.$

######  Fact 6.17.

(Theorem VI.3.12 in [Sh78]). If
$\aleph_{0}<\kappa\leq\mu=\mu^{\aleph_{0}}\leq|2^{I}|$, then there is a
regular ultrafilter $\mathcal{D}$ over $I$ such that
$\operatorname{lcf}(\mathcal{D})=\kappa$ and
$\operatorname{lca}(\mathcal{D})=\mu$.

######  Definition 6.18.

Let $\varepsilon\in[0,1]$, and $\Delta$ be a finite set of formulas with free
variables included in $\vec{y}$ that contains at least the formula
$d(y_{0},y_{1})$. A sequence $\mathcal{I}\subseteq\mathcal{M}$ is
$(\Delta,\varepsilon)$-_indiscernible_ in $\mathcal{M}$ if for all proper
$|\vec{y}|$-tuples $\vec{b},\vec{c}$ of elements of $\mathcal{I}$ and every
$\varphi(\vec{y})\in\Delta$, we have

$|\varphi(\vec{b})^{\mathcal{M}}-\varphi(\vec{c})^{\mathcal{M}}|\leq\varepsilon.$

$\mathcal{I}$ is $r$-_separated_ if $d^{\mathcal{M}}(u,v)\geq r$ for some
$u,v\in\mathcal{I}$.

We make some easy observations about $(\Delta,\varepsilon)$-indiscernibility.
Let $1\geq r>\varepsilon$. If $\mathcal{I}$ is $r$-separated, then any
superset of $\mathcal{I}$ in $\mathcal{M}$ is $r$-separated. Since
$d(y_{0},y_{1})$ is in $\Delta$, if $\mathcal{I}$ is
$(\Delta,\varepsilon)$-indiscernible in $\mathcal{M}$ and $r$-separated, then
two elements $u,v\in\mathcal{I}$ are distinct if and only if
$d^{\mathcal{M}}(u,v)\geq r-\varepsilon$.

For each $k\in\mathbb{N}$, there is a single formula $\varphi(\vec{z})$ with
$|\vec{z}|=k$ such that for each $k$-tuple $\mathcal{H}$ in $\mathcal{M}$,
$\varphi^{\mathfrak{M}}(\mathcal{H})$ holds if and only if $\mathcal{H}$ is
$(\Delta,\varepsilon)$-indiscernible in $T$ and $r$-separated. $\mathcal{I}$
is indiscernible in $T$ if and only if it is $(\Delta,1/n)$-indiscernible in
$T$ for all non-empty finite $\Delta$ and all $0<n\in\mathbb{N}$. The property
of $(\Delta,\varepsilon)$-indiscernibility gets stronger as $\Delta$ gets
larger and $\varepsilon$ gets smaller.

The proof of the next lemma is similar to the argument for Theorem VI.5.1 (2)
in [Sh78], but is simpler because the argument in [Sh78] also dealt with three
other more difficult cases at the same time.

######  Lemma 6.19.

If $T$ is stable, $\mathcal{D}$ is a regular ultrafilter, and
$\mathcal{M}\models T$, then $\mathcal{M}_{\mathcal{D}}$ is
$\operatorname{lca}(\mathcal{D})$-saturated.

###### Proof.

By Remark 2.26, $\mathcal{M}_{\mathcal{D}}$ is $\aleph_{1}$-saturated. By
Lemma 6.14, it suffices to prove that every countably infinite indiscernible
set $\mathcal{I}=\\{a_{h}\colon h\in\mathbb{N}\\}$ in
$\mathcal{M}_{\mathcal{D}}$ can be extended to an indiscernible set
$\mathcal{H}$ in $\mathcal{M}_{\mathcal{D}}$ of cardinality
$\geq\operatorname{lca}(\mathcal{D})$.

$\mathcal{I}$ is $2r$-separated for some $1\geq r>0$. Let
$\mathcal{M}^{\prime}=(\mathcal{M},S,\in)$ be the two-sorted real-valued
structure where one sort is $M$, the other sort is the set $S$ of finite
subsets of $M$, and $\in$ is the $\in$ relation on $M\times S$.
($\mathcal{M}^{\prime}$ is not a metric structure.) Let $V$ and $V^{\prime}$
be the vocabularies of $\mathcal{M}$ and $\mathcal{M}^{\prime}$. Let
$\boldsymbol{z}$ be a variable of sort $S$. For every finite set
$\Delta(\vec{y})$ of strict $V$-formulas containing $d(y_{0},y_{1})$ and every
$1/r<n\in\mathbb{N}$, there is a $V^{\prime}$-formula
$\varphi_{\Delta,n,r}(\boldsymbol{z})$ saying that the set $\boldsymbol{z}$ is
$(\Delta,1/n)$-indiscernible and $r$-separated. By Remark 2.26, the ultrapower
$\mathcal{M}^{\prime}_{\mathcal{D}}$ is $\aleph_{1}$-saturated. The set of
$V^{\prime}$-formulas

$\Gamma(\boldsymbol{z})=\\{a_{h}\in\boldsymbol{z}\colon
h\in\mathbb{N}\\}\cup\\{\varphi_{\Delta,n,r}(\boldsymbol{z})\colon\Delta\mbox{
finite and strict},0<n\in\mathbb{N}\\}$

is finitely satisfiable in $\mathcal{M}^{\prime}_{\mathcal{D}}$, because each
finite subset $\Gamma_{0}\subseteq\Gamma$ is satisfied by the element
$b_{0\mathcal{D}}$ of $S_{\mathcal{D}}$ where for each $t\in I$, $b_{0}[t]$ is
the set of $a_{h}[t]$ such that $a_{h}$ occurs in $\Gamma_{0}$.
$\Gamma(\boldsymbol{z})$ is countable because there are only countably many
strict $V$-formulas. Therefore $\Gamma(\boldsymbol{z})$ is satisfied in
$\mathcal{M}^{\prime}_{\mathcal{D}}$ by some element $b_{\mathcal{D}}$ of sort
$S$ in $\mathcal{M}^{\prime}_{\mathcal{D}}$. It follows that the set

$\mathcal{H}=\\{a\in
M_{\mathcal{D}}\colon\mathcal{M}^{\prime}_{\mathcal{D}}\models a\in b\\}$

contains $\mathcal{I}$ and is indiscernible in $\mathcal{M}_{\mathcal{D}}$.
Since $\mathcal{H}$ is infinite, we have

$|\mathcal{H}|=\left|\prod_{\mathcal{D}}b[t]\right|\geq\operatorname{lca}(\mathcal{D}),$

as required. ∎

######  Lemma 6.20.

Suppose $\mathcal{D}$ is a regular ultrafilter, and the ultrapower
$\mathcal{M}_{\mathcal{D}}$ is
$\operatorname{lcf}(\mathcal{D})^{+}$-saturated. Then
$\operatorname{Th}(\mathcal{M})$ is stable.

###### Proof.

Suppose $\varphi(\vec{x},\vec{y})$ is an unstable formula in
$\operatorname{Th}(\mathcal{M})$. By Theorem 3.5, we may assume that
$\mathcal{M}\prec\mathfrak{M}$. There are dyadic rationals $r>s$ in $[0,1]$,
and a sequence $\langle(\vec{a}_{k},\vec{b}_{k})\rangle_{k\in\mathbb{N}}$ of
tuples in $\mathcal{M}$ such that whenever $h<j<k$ in $\mathbb{N}$ we have
$\varphi^{\mathcal{M}}(\vec{a}_{j},\vec{b}_{h})\geq r$ and
$\varphi^{\mathcal{M}}(\vec{a}_{j},\vec{b}_{k})\leq s$. For simplicity we give
the proof in the case that $\vec{x},\vec{y}$ are $1$-tuples, and that
$\varphi^{\mathcal{M}}(a_{j},b_{h})\geq r$ and
$\varphi^{\mathcal{M}}(a_{j},b_{k})\leq s$ whenever $h\leq j<k$. (The proof in
general is essentially the same). We may take the $b_{k}$ to be distinct.
Consider the countable first order structure $(K,\leq^{K})$ where
$K=\\{b_{m}\colon m\in\mathbb{N}\\}$ and $\leq^{K}=\\{(b_{m},b_{n})\colon
m\leq n\\}$. Let $f$ be the unique isomorphism from $(\mathbb{N},\leq)$ onto
$(K,\leq^{K})$, and $f_{\mathcal{D}}$ be the corresponding isomorphism from
$(\mathbb{N},\leq)_{\mathcal{D}}$ onto $(K,\leq^{K})_{\mathcal{D}}$. Note that
$K\subseteq M$, so $K_{\mathcal{D}}\subseteq M_{\mathcal{D}}$. Let
$\kappa=\operatorname{lcf}(\mathcal{D})$. There is a co-initial set $C$ of
infinite elements of $(\mathbb{N},\leq)_{\mathcal{D}}$ such that $|C|=\kappa$.
Let $\Sigma(x)$ be the set of formulas

$\Sigma(x)=\\{r\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle\leq$}\hss}\raise 4.30554pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle\leq$}\hss}\raise 4.30554pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle\leq$}\hss}\raise 3.01389pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle\leq$}\hss}\raise 2.15277pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern
1.94441pt}}\varphi(x,f_{\mathcal{D}}(n))\colon
n\in\mathbb{N}\\}\cup\\{\varphi(x,f_{\mathcal{D}}(c))\mathbin{\mathchoice{\kern
2.77774pt\hbox to0.0pt{\hss\hbox{$\displaystyle\leq$}\hss}\raise
4.30554pt\hbox to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern
2.77774pt\hbox to0.0pt{\hss\hbox{$\textstyle\leq$}\hss}\raise 4.30554pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle\leq$}\hss}\raise 3.01389pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle\leq$}\hss}\raise 2.15277pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}s\colon c\in C\\}.$

(By Remark 2.8, we are using $\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle\leq$}\hss}\raise 4.30554pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle\leq$}\hss}\raise 4.30554pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle\leq$}\hss}\raise 3.01389pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle\leq$}\hss}\raise 2.15277pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}$ as an alternate
notation for $\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}$). Then $\Sigma(x)$ is
finitely satisfiable in $\mathcal{M}_{\mathcal{D}}$ and has cardinality
$\kappa$.

We claim that $\Sigma(x)$ is not satisfiable in $\mathcal{M}_{\mathcal{D}}$.
To see this, suppose $a$ satisfies $\Sigma(x)$ in $\mathcal{M}_{\mathcal{D}}$.
Then for some $X\in\mathcal{D}$, for every $t\in X$ there exists
$e[t]\in\mathbb{N}$ such that $\varphi^{\mathcal{M}}(a[t],b_{e[t]})\geq r$ and
$\varphi^{\mathcal{M}}(a[t],b_{e[t]+1})\leq s$. Then
$\varphi^{\mathcal{M}_{\mathcal{D}}}(a,f_{\mathcal{D}}(e))\geq r$ and
$\varphi^{\mathcal{M}_{\mathcal{D}}}(a,f_{\mathcal{D}}(e+1))\leq s$. Moreover,
$e$ is an infinite element of $(\mathbb{N},\leq)_{\mathcal{D}}$, so
$c\leq_{\mathcal{D}}e$ for some $c\in C$. Then
$\varphi^{\mathcal{M}_{\mathcal{D}}}(a,f_{\mathcal{D}}(c))\geq r$, which
contradicts the assumption that $a$ satisfies $\Sigma(x)$ in
$\mathcal{M}_{\mathcal{D}}$. This shows that $\Sigma(x)$ is not satisfiable in
$\mathcal{M}_{\mathcal{D}}$, so $\mathcal{M}_{\mathcal{D}}$ is not
$\kappa^{+}$-saturated. ∎

######  Definition 6.21.

$LCA$ is the class of regular ultrafilters $\mathcal{D}$ over sets $I$ such
that $|I|<\operatorname{lca}(\mathcal{D})$.

$LCF$ is the class of regular ultrafilters $\mathcal{D}$ over sets $I$ such
that $|I|<\operatorname{lcf}(\mathcal{D})$.

By Fact 6.17, $LCF$ is a proper subclass of $LCA$. In fact, whenever
$\aleph_{1}<2^{|I|}$, there is a regular ultrafilter $\mathcal{D}$ over $I$
such that $\mathcal{D}\in LCA\setminus LCF$.

######  Theorem 6.22.

  * (i)

$T$ is stable if and only if $\operatorname{Sat}(T)\supseteq LCA$.

  * (ii)

$T$ is unstable if and only if $\operatorname{Sat}(T)\subseteq LCF$.

  * (iii)

If $T$ is stable and $U$ is unstable, then $T\vartriangleleft U$.

###### Proof.

Let $\mathcal{M}$ be a model of $T$.

(i) Forward: Suppose $T$ is stable. By Lemma 6.19, $\mathcal{M}_{\mathcal{D}}$
is $\operatorname{lca}(\mathcal{D})$-saturated. If $\mathcal{D}\in LCA$, then
$|I|<\operatorname{lca}(\mathcal{D})$, so $\mathcal{M}_{\mathcal{D}}$ is
$|I|^{+}$-saturated and $\mathcal{D}\in\operatorname{Sat}(T)$. Therefore
$\operatorname{Sat}(T)\supseteq LCA$.

(ii) Suppose $T$ is unstable. By Lemma 6.20, $\mathcal{M}_{\mathcal{D}}$ is
not $\operatorname{lcf}(\mathcal{D})^{+}$-saturated. If
$\mathcal{D}\in\operatorname{Sat}(T)$, then $\mathcal{M}_{\mathcal{D}}$ is
$|I|^{+}$-saturated, so $|I|<\operatorname{lcf}(\mathcal{D})$ and
$\mathcal{D}\in LCF$. Thus $\operatorname{Sat}(T)\subseteq LCF$. Now suppose
$\operatorname{Sat}(T)\subseteq LCF$. Then $\operatorname{Sat}(T)\supseteq
LCA$ fails, so $T$ is unstable by (i) forward,

(i) Reverse: Suppose $T$ is unstable. Take $\mathcal{D}\in LCA\setminus LCF$.
By (ii), $\mathcal{D}\notin\operatorname{Sat}(T)$, so
$\operatorname{Sat}(T)\supseteq LCA$ fails.

(iii) follows by (i), (ii), and Fact 6.17. ∎

######  Corollary 6.23.

There are regular ultrafilters $\mathcal{D}$ that saturate all stable metric
theories but no unstable metric theories. The class of all such $\mathcal{D}$
is $LCA\setminus LCF$.

###### Proof.

Let $\operatorname{stb}_{\mathbb{M}}$ be the set of
$\trianglelefteq$-equivalence classes of stable metric theories in
$\mathbb{M}$, and $\operatorname{stb}_{\mathbb{F}}$ be the set of
$\trianglelefteq$-equivalence classes of stable first order theories in
$\mathbb{F}$. By Fact 6.17,

$\\{\mathcal{D}\colon\operatorname{Sat}_{\mathbb{F}}(\mathcal{D})=\operatorname{stb}_{\mathbb{F}}\\}=LCA\setminus
LCF\neq\emptyset.$

By Theorem 6.22, we have

$\\{\mathcal{D}\colon\operatorname{Sat}_{\mathbb{F}}(\mathcal{D})=\operatorname{stb}_{\mathbb{F}}\\}=\\{\mathcal{D}\colon\operatorname{Sat}_{\mathbb{M}}(\mathcal{D})=\operatorname{stb}_{\mathbb{M}}\\}.$

∎

##  7\. $\trianglelefteq$-minimal theories

Theorem 7.10 below gives a characterization of $\trianglelefteq$-minimal
metric theories. Theorem 7.12 shows that there are exactly two
$\trianglelefteq$-classes of stable metric theories, and that they are the
lowest two $\trianglelefteq$-equivalence classes. We are indebted to James
Hanson for pointing out an error in the definition of NFCP and the proof of
Theorem 7.10 in an earlier version of this section. To correct that error we
modified the definition of NFCP from the earlier version.

For first order logic, in [Sh78], Section VI, Shelah identified the first two
classes in the partial ordering $(\mathbb{F},\trianglelefteq)$.

######  Definition 7.1.

We denote the set of first order theories that are stable but not
$\trianglelefteq$-minimal by $\operatorname{stb\neg min}_{\mathbb{F}}$. We
denote the set of metric theories that are stable but not
$\trianglelefteq$-minimal by $\operatorname{stb\neg min}_{\mathbb{M}}$.

######  Fact 7.2.

(Theorem VI.5.9 in [Sh78]),

(i) $\operatorname{stb\neg min}_{\mathbb{F}}\in\mathbb{F}$.

(ii) For each $H\in\mathbb{F}$, either $H=\min_{\mathbb{F}}$,
$H=\operatorname{stb\neg min}_{\mathbb{F}}$, or $\operatorname{stb\neg
min}_{\mathbb{F}}\vartriangleleft H$.

In Theorem 7.12 and Corollary 7.13 below, we will prove the analogous results
for metric theories. In this section, $T$ will be a complete metric theory and
$\mathcal{M}$ will denote an $\aleph_{1}$-saturated model of $T$.

######  Definition 7.3.

Let $\varepsilon\in[0,1]$ and $\Sigma(\vec{y})$ be a set of formulas with
parameters in $\mathcal{M}$ (where $\vec{y}$ is a possibly infinite sequence
of variables). We say that $\Sigma(\vec{y})$ is $\varepsilon$-_satisfiable_ in
$\mathcal{M}$ if there is a $|\vec{y}|$-sequence of elements $\vec{a}$ of
$\mathcal{M}$ such that $\psi^{\mathcal{M}}(\vec{a})\leq\varepsilon$ for all
$\psi\in\Sigma(\vec{y})$.

######  Corollary 7.4.

Let $\Sigma(\vec{y})$ be a set of formulas with parameters in $\mathcal{M}$.

(i) $\Sigma(\vec{y})$ is $\varepsilon$-satisfiable in $\mathcal{M}$ if and
only if the set of formulas

$\\{\psi(\vec{y})\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern
1.94441pt}}\varepsilon\colon\psi\in\Sigma\\}$

is satisfiable in $\mathcal{M}$.

(ii) If $\Sigma(\vec{y})$ is finite, $\Sigma(\vec{y})$ is
$\varepsilon$-satisfiable in $\mathcal{M}$ if and only if the sentence
$(\inf_{\vec{y}})\max(\Sigma(\vec{y}))\mathbin{\mathchoice{\kern
2.77774pt\hbox to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 1.94441pt}}\varepsilon$ holds in
$\mathcal{M}$.

(iii) If $\Sigma(\vec{y})$ is countable and every finite subset of
$\Sigma(\vec{y})$ is $\varepsilon$-satisfiable in $\mathcal{M}$, then
$\Sigma(\vec{y})$ is $\varepsilon$-satisfiable in $\mathcal{M}$.

###### Proof.

(i) is a restatement of the definition, and (ii) follows easily from the fact
that $\mathcal{M}$ is $\aleph_{1}$-saturated.

(iii): By (i), the countable set of formulas

$\\{\psi(\vec{y})\mathbin{\mathchoice{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox
to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox
to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.27774pt\hbox
to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox
to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.27774pt}{\kern 1.94441pt\hbox
to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox
to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern
1.94441pt}}\varepsilon\colon\psi(\vec{y})\in\Sigma(\vec{y})\\}$

is finitely satisfiable in $\mathcal{M}$. Since $\mathcal{M}$ is
$\aleph_{1}$-saturated, that set of formulas is satisfiable in $\mathcal{M}$.
By (i), $\Sigma(\vec{y})$ is $\varepsilon$-satisfiable in $\mathcal{M}$. ∎

######  Definition 7.5.

Let $\vec{x}$ be an $n$-tuple of distinct variables, $B\subseteq\mathcal{M}$,
and $\Delta(\vec{x})$ be a finite set of formulas with free variables among
$\vec{x}$. By an $n$-_tuple over_ $(\vec{y},B)$ we mean an $n$-tuple
$(u_{0},\ldots,u_{n-1})$ where each $u_{i}$ is in $\vec{y}\cup B$, and no
variable or parameter appears more than once in $\vec{u}$. By a _$\Delta$
-formula over $(\vec{y},B)$_ we mean a formula $\psi(\vec{u})$ obtained by a
formula $\psi(\vec{x})\in\Delta$ by replacing $\vec{x}$ by an $n$-tuple
$\vec{u}$ over $(\vec{y},B)$.

In the case that $B=M$, we sometimes say “over $(\vec{y},\mathcal{M})$”
instead of “over $(\vec{y},B)$”.

######  Definition 7.6.

$T$ has the _non-finite cover property_ (NFCP) in $\mathcal{M}$ (a better name
would be “finite satisfaction property”) if $T$ is stable, and for each non-
empty finite set $\Delta(\vec{x})$ of formulas and $\varepsilon\in(0,1]$,
there is an $m(\Delta,\varepsilon)\in\mathbb{N}$ such that for every finite
set $\Sigma(\vec{y})$ of $\Delta$-formulas over $(\vec{y},\mathcal{M})$, if
every subset of $\Sigma(\vec{y})$ of cardinality $\leq m(\Delta,\varepsilon)$
is satisfiable in $\mathcal{M}$, then $\Sigma(\vec{y})$ is
$\varepsilon$-satisfiable in $\mathcal{M}$.

$T$ has the NFCP if $T$ has the NFCP in every $\aleph_{1}$-saturated model of
$T$.

We will see in Corollary 7.11 below that a first order theory has the NFCP if
and only if it does not have the finite cover property as defined in [Ke67].

######  Lemma 7.7.

If $T$ has the NFCP in some $\aleph_{1}$-saturated model of $T$, then $T$ has
the NFCP.

###### Proof.

Suppose $T$ has the NFCP in $\mathcal{M}$, and $\mathcal{N}$ is another
$\aleph_{1}$-saturated model of $T$. For every countable sequence
$\vec{b}\subseteq\mathcal{M}$ there is a countable sequence $\vec{c}\subseteq
N$ such that $(\mathcal{M},\vec{b})\equiv(\mathcal{N},\vec{c})$. By Corollary
7.4, a countable set of formulas $\Sigma(\vec{y},\vec{b})$ is
$\varepsilon$-satisfiable in $\mathcal{M}$ if and only if
$\Sigma(\vec{y},\vec{c})$ is $\varepsilon$-satisfiable in $\mathcal{N}$. It
follows that $T$ has the NFCP in $\mathcal{N}$ as well as in $\mathcal{M}$. ∎