# Between reduced powers and ultrapowers, II.

Ilijas Farah  Department of Mathematics and Statistics
York University
4700 Keele Street
North York, Ontario
Canada, M3J 1P3 and Matemativ̧ki Institut SANU, Kneza Mihaila 36, Belgrade
11001, Serbia https://ifarah.mathstats.yorku.ca<EMAIL_ADDRESS>and  Saharon
Shelah  The Hebrew University of Jerusalem
Einstein Institute of Mathematics
Edmond J. Safra Campus, Givat Ram
Jerusalem 919041, Israel and Department of Mathematics
Hill Center-Busch Campus
Rutgers, The State University of New Jersey
110 Frelinghuysen Road
Piscataway, NJ 08854-8019 USA<EMAIL_ADDRESS>http://shelah.logic.at/

###### Abstract.

We prove that, consistently with ZFC, no ultraproduct of countably infinite
(or separable metric, non-compact) structures is isomorphic to a reduced
product of countable (or separable metric) structures associated to the
Fréchet filter. Since such structures are countably saturated, the Continuum
Hypothesis implies that they are isomorphic when elementarily equivalent.

###### Key words and phrases:

Ultrapowers, reduced powers, saturated models, Proper Forcing Axiom, Cohen
model, Continuum Hypothesis, universal models, small basis.

###### 2010 Mathematics Subject Classification:

03C20, 03C98, 03E50, 03E35

First author’s research is partially supported by NSERC. Second author’s
research partially supported by Israel Science Foundation (ISF) grant no:
1838/19 and by NSF grant no: DMS 1833363. No. 1202 on Shelah’s list of
publications.

The trivializing effect of the Continuum Hypothesis (CH) to the structure of
the continuum has been known at least since the times of Sierpiński and Gödel
([24]). The particular instance of this phenomenon that we are concerned with
in the present paper is the existence of highly non-canonical isomorphisms
between massive quotient structures of cardinality
$\mathfrak{c}=2^{\aleph_{0}}$. The operation of taking a reduced
product111Reduced products can be defined with respect to a filter or with
respect to the dual ideal, the two definitions resulting in the same object;
see Remark 1.1. $\prod_{\mathcal{F}}A_{n}$ of a sequence $(A_{n})$ of first-
order structures often results in a countably saturated
structure.222Saturation is a model-theoretic property that enables transfinite
constructions of isomorphisms and automorphisms; see e.g., [5, Chapter 5]. It
will not be used in this paper. This is the case with the two most commonly
used reduced products: ultraproducts associated with nonprincipal ultrafilters
on ${\mathbb{N}}$ and reduced products associated with the Fréchet filter. If
each $A_{n}$ has the cardinality of at most $\mathfrak{c}$ (in particular, if
it is countable or separable333Our results apply both to classical, discrete,
structures and metric structures, as in [2].), then so does
$\prod_{\mathcal{F}}A_{n}$, and the CH implies that the latter structure is
saturated. By a classical theorem of Keisler, elementarily equivalent and
saturated first-order structures of the same cardinality are isomorphic (see
[5, Theorem 5.1.13]). Therefore CH implies that the isomorphism of such
reduced products reduces (no pun intended) to elementary equivalence.

In [13], this observation was combined with computation of the theory of the
structure ($K$ denotes the Cantor space)

(0.1) $C(K,A)=\\{f\colon K\to A\mid\text{ $f$ is continuous}\\}$

for a separable (or countable discrete) structure $A$ to prove that CH implies
($\operatorname{Fin}$ denotes the ideal of finite subsets of ${\mathbb{N}}$)

(0.2) $\textstyle\prod_{\mathcal{U}}C(K,A)\cong\prod_{\operatorname{Fin}}A$

for any nonprincipal ultrafilter $\mathcal{U}$ on ${\mathbb{N}}$ ([13, Theorem
E]).444In [13], $\prod_{\operatorname{Fin}}A$ was denoted $A^{\infty}$ and
$\prod_{\mathcal{U}}A$ was denoted $A^{\mathcal{U}}$, following the notation
favoured by operator-algebraists. In the present paper we adopt the notation
favoured by logicians and apologize to any stray operator algebraists; see
however Corollary D. This result is the basis for [13, Theorem B], asserting
that under CH there exists an ultrafilter $\mathcal{U}$ on ${\mathbb{N}}$ such
that the quotient map from $\prod_{\operatorname{Fin}}A$ to
$\prod_{\mathcal{U}}A$ has a right inverse for every countable (or separable
metric) structure $A$. In the case when $A$ is a $C^{*}$-algebra, this
simplifies some intricate arguments in Elliott’s classification program for
nuclear, separable $C^{*}$-algebras (see the upcoming [4], also [53] and [38]
for related applications of ultrapowers). Although the assumption of CH can be
removed from the applications of (0.2) to the Elliott classification programme
([13, Theorem A]), the question whether (0.2) can be proved in ZFC remained.

A well-known instance of the trivializing effect of CH is Parovičenko’s
theorem from general topology. Stated in the dual, Boolean-algebraic, form, it
asserts that CH implies that all atomless, countably saturated, Boolean
algebras of cardinality $\mathfrak{c}$ are isomorphic. In [51] it was proved
that the conclusion of Parovičenko’s theorem is equivalent to CH. An
alternative proof of this fact is given by the main result of [19] (or by
[30]), asserting that if CH fails then there are $2^{\mathfrak{c}}$
nonisomorphic ultrapowers of the countable atomless Boolean algebra associated
with nonprincipal ultrafilters on ${\mathbb{N}}$. By Łoś’s Theorem all of
these Boolean algebras are elementarily equivalent (which in this case reduces
to being atomless, see [5, Exercise 1.5.3]), and they are countably saturated
and of cardinality $\mathfrak{c}$, being ultrapowers associated with countably
incomplete ultrafilters. Clearly, at most one of these ultrapowers can be
isomorphic to $\mathcal{P}({\mathbb{N}})/\operatorname{Fin}$. Theorem A and
Theorem B below show that in two of the most popular models of ZFC in which CH
fails (models of forcing axioms and Cohen’s original model of ZFC), none of
these ultrapowers is isomorphic to
$\mathcal{P}({\mathbb{N}})/\operatorname{Fin}$.

######  Theorem A.

The Proper Forcing Axiom, PFA, implies that
$\mathcal{P}({\mathbb{N}})/\operatorname{Fin}$ is not isomorphic to an
ultraproduct of Boolean algebras associated with a nonprincipal ultrafilter on
${\mathbb{N}}$.

Each of our other main results applies to wider class of structures. The first
one is primarily concerned with a larger class of quotient algebras
$\mathcal{P}({\mathbb{N}})/{\mathcal{I}}$ associated with Borel ideals on
${\mathbb{N}}$ in place of $\mathcal{P}({\mathbb{N}})/\operatorname{Fin}$. The
study of quotient Boolean algebras of the form
$\mathcal{P}({\mathbb{N}})/{\mathcal{I}}$ for an ideal ${\mathcal{I}}$ on
${\mathbb{N}}$, dates back at least to Erdös and Ulam (see [8]). The space
$\mathcal{P}({\mathbb{N}})$ is identified with the Cantor space, and thus
equipped with a canonical compact metrizable topology. If ${\mathcal{I}}$
includes the ideal $\operatorname{Fin}$ of finite subsets of ${\mathbb{N}}$,
then $\mathcal{P}({\mathbb{N}})/{\mathcal{I}}$ is atomless and therefore
elementarily equivalent to $\mathcal{P}({\mathbb{N}})/\operatorname{Fin}$. We
will consider only ideals ${\mathcal{I}}$ that include $\operatorname{Fin}$.

The question of countable saturation of
$\mathcal{P}({\mathbb{N}})/{\mathcal{I}}$ is a bit subtler. In [26] it was
proved that $\mathcal{P}({\mathbb{N}})/{\mathcal{I}}$ is countably saturated
for every $F_{\sigma}$ ideal ${\mathcal{I}}$ that includes
$\operatorname{Fin}$. The essence of the Just–Krawczyk construction is
encapsulated in the concept of a _layered ideal_ in [11], where it was proved
that if ${\mathcal{I}}$ is a layered ideal that includes $\operatorname{Fin}$
then $\mathcal{P}({\mathbb{N}})/\operatorname{Fin}$ is countably saturated.
(This class of ideals properly includes that of the $F_{\sigma}$ ideals; for
example, for any additively indecomposable countable ordinal $\alpha$ the
ideal $\\{X\subseteq\alpha|$ the order type of $X$ is less than $\alpha\\}$ is
layered.) Since the quotient over an analytic ideal that includes
$\operatorname{Fin}$ necessarily has cardinality $\mathfrak{c}$, these results
show that all quotients over layered analytic ideals that include
$\operatorname{Fin}$ are isomorphic under CH. This conclusion extends to
reduced products of countable Boolean algebras, $\prod_{\mathcal{I}}A_{n}$,
associated with layered ideals ([21, Theorem 2.7]).

In [27, Corollary 1.4] it was proved that if an ideal ${\mathcal{I}}$ includes
$\operatorname{Fin}$ then $\mathcal{P}({\mathbb{N}})/{\mathcal{I}}$ is
countably saturated if and only if $\mathcal{P}({\mathbb{N}})/{\mathcal{I}}$
has no $(\aleph_{0},\aleph_{0})$-gaps. In [36] it was proved that
$\mathcal{F}$ on ${\mathbb{N}}$ has the property that every reduced product of
the form $\prod_{n}A_{n}/\mathcal{F}$ is countably saturated if and only if
the Boolean algebra $\mathcal{P}({\mathbb{N}})/\mathcal{F}$ is countably
saturated. Results analogous to the latter one have been established in [35]
for filters on $\aleph_{1}$ and in [39] for filters on an arbitrary cardinal.

Note that $\mathcal{P}({\mathbb{N}})/{\mathcal{I}}$ is canonically isomorphic
to the reduced product, $\prod_{\mathcal{I}}A$, where $A$ is taken to be the
2-element Boolean algebra. Also note that all of these reduced products are
_projectively definable_ in the sense that there is an $n\in{\mathbb{N}}$ and
$\mathbf{\Sigma}^{1}_{n}$-formulas $\varphi$, $\varphi_{\wedge}$,
$\varphi_{\vee}$, and $\varphi_{\setminus}$, such that the set
$\\{x\in\mathbb{R}|\varphi(x)\\}$ equipped with the operations defined by
$\varphi_{\wedge}$, $\varphi_{\vee}$, and $\varphi_{\setminus}$ is a Boolean
algebra.555We avoid using the simpler term projective Boolean algebras in
order to avoid confusion with Boolean algebras that are projective objects in
the category of Boolean algebras.

######  Theorem B.

In a model obtained by adding at least $\mathfrak{c}^{+}$ Cohen reals to a
model of ZFC the following holds. If $\mathfrak{B}$ is a projectively
definable Boolean algebra then $\mathfrak{B}$ is not isomorphic to an
ultraproduct of countable Boolean algebras associated with a nonprincipal
ultrafilter on ${\mathbb{N}}$.

In particular, for any analytic ideal ${\mathcal{I}}$ on ${\mathbb{N}}$, the
quotient $\mathcal{P}({\mathbb{N}})/{\mathcal{I}}$ is not isomorphic to an
ultraproduct of countable Boolean algebras associated with a nonprincipal
ultrafilter on ${\mathbb{N}}$.

The proof of Theorem B applies to a wider class of models, but not to the
Sacks models; see §8.

The next result applies to a yet wider range of structures. For the order
property (OP) and the robust order property see Definition 1.2. Any theory in
which the order property is witnessed by an atomic formula has the robust
order property. In particular, the classes of atomless Boolean algebras and
dense linear orderings have the robust OP. Another relevant property specific
to the theory of atomless Boolean algebras is that it is preserved by taking
arbitrary reduced products. (Theories with this property are axiomatized by
Horn sentences, see e.g., [5, Theorem 6.2.5’].) This is not shared by many
other theories (e.g., linear orders or fields), and it is because of this fact
that in the following result the theories of $A_{n}$ and $B_{n}$ are a priori
unrelated.

######  Theorem C.

There exists a forcing extension in which for every countable theory $T$ that
has the robust order property the following holds.

For every sequence $(A_{n})$ of countable structures in the language of $T$,
every sequence $(B_{n})$ of countable models of $T$, and every nonprincipal
ultrafilter $\mathcal{U}$ on ${\mathbb{N}}$, the following are true.

  1. (1)

The ultraproduct $\prod_{\mathcal{U}}B_{n}$ is not isomorphic to
$\prod_{\operatorname{Fin}}A_{n}$ or even to an elementary submodel thereof.

  2. (2)

If the order property of $T$ is witnessed by a quantifier-free formula then
$\prod_{\mathcal{U}}B_{n}$ does not embed into
$\prod_{\operatorname{Fin}}A_{n}$.

Since the original impetus for these results drew from the Elliott
classification program of $C^{*}$-algebras, we’ll explicitly state the
relevant corollary. If $A$ is a $C^{*}$-algebra, then the structure $C(K,A)$
as in (0.1) is isomorphic to the tensor product $A\otimes C(K)$, where $C(K)$
is the algebra of continuous complex-valued functions on $K$. By [13, Theorem
E], for a separable $C^{*}$-algebra $A$ and a nonprincipal ultrafilter
$\mathcal{U}$ on ${\mathbb{N}}$, CH implies that the ultrapower $(A\otimes
C(K))^{\mathcal{U}}$ is isomorphic to $A^{\infty}:=\ell_{\infty}(A)/c_{0}(A)$
(the latter algebra, known as the asymptotic sequence algebra, is the
continuous analog of the reduced power $\prod_{\operatorname{Fin}}A$) and the
isomorphism extends the identity on $A$ ($A$ is routinely identified with its
diagonal copies in $A^{\mathcal{U}}$ and $A^{\infty}$).

######  Corollary D.

There exists a forcing extension in which the following holds for every
separable $C^{*}$-algebra $A$ and every ultrafilter $\mathcal{U}$ on
${\mathbb{N}}$.

  1. (1)

$(A\otimes C(K))^{\mathcal{U}}$ is not isomorphic to $A^{\infty}$.

  2. (2)

$(A\otimes C(K))^{\mathcal{U}}$ is not isomorphic to a $C^{*}$-subalgebra of
$B^{\infty}$ for any separable $C^{*}$-algebra $B$.

The related conclusion, that $C(K)^{\mathcal{U}}$ is not isomorphic to a
$C^{*}$-subalgebra of $\ell_{\infty}/c_{0}$, is known to be relatively
consistent with ZFC and its variant (known as _Woodin’s condition_) plays an
important role in Woodin’s proof of automatic continuity for homomorphisms of
Banach algebras ([6]).

The question of the existence of a universal structure among the ultrapowers
of a fixed countable (or separable metric) structure is closely related to the
questions answered in theorems stated above. Partial answers to this question
(which are easy consequences of the earlier work of one of the authors) are
given in the brief §7.

### Notation

We write $X\Subset Y$ as a shorthand for ‘$X\subseteq Y$ and $X$ is finite’
(this is sometimes denoted by the formula $X\in[Y]^{<\aleph_{0}}$). The ideal
of finite subsets of a set $X$ is denoted $\operatorname{Fin}_{X}$ (some
authors prefer $[X]^{<\aleph_{0}}$). For the Fréchet ideal
$\operatorname{Fin}_{\aleph_{0}}$ we write $\operatorname{Fin}$.

### Rough outline

Our proofs use model theory (§1, §7) and set theory (§2, §4). In §1 we discuss
the order property (OP) of first-order theories, discrete and continuous.
Standard facts about partially ordered sets (posets) are recalled in §2, and
depletions of posets are introduced and studied in §3. In §4 we define a
functor $E\mapsto\mathbb{H}_{E}$ from the category of partial orderings into
the category of forcing notions. The material from §3 is used to prove that
the $\mathbb{H}_{E}$ forces that $E$ embeds into the reduced product
$\prod_{n\in{\mathbb{N}}}(n,<)$ ($n$ is identified with $\\{0,\dots,n-1\\}$)
in a particularly gentle way. Theorem C and Corollary D are proved in §5,
while Theorem A and Theorem B are proved in §6. In §7 we record results about
the existence of a universal model among the ultrapowers of countable models
of $T$ associated with ultrafilters on ${\mathbb{N}}$. Some concluding remarks
and questions can be found in §8.

### Acknowledgments

I.F. would like to thank Alan Dow for pointing our attention to [51]. The
anonymous referee bravely read through the original version of this paper and
provided a very helpful report. For this we are indebted to the referee, and
so should be all future readers of this paper.

##  1\. Reduced products, the order property, continuous logic

In this section we recall the pertinent definitions. It should be emphasized
that the first-order theory $T$ is not assumed to be complete.

###  1.1. Reduced products

We will use the following convention. Suppose that $A_{n}$ are structures of
the same countable language, $\bar{a}_{n}$ is a tuple in $A_{n}$ for all
$n\in{\mathbb{N}}$, and all of these tuples are of the same sort. (If the
language is single-sorted, then the sort of a tuple is simply its arity. Note
that the natural languages associated with the unbounded metric structures,
such as $C^{*}$-algebras, are multisorted, see [15].) Then $\bar{a}$ denotes
the tuple $(\bar{a}_{n})$ in $\prod_{n}A_{n}$ of the same sort.

If $\mathcal{F}$ is a filter on ${\mathbb{N}}$ and $A_{n}$, for
$n\in{\mathbb{N}}$, are structures of the same language $\mathcal{L}$, then
the reduced product $\prod_{\mathcal{F}}A_{n}$ is defined as follows. Its
domain is the quotient of $\prod_{n}A_{n}$ over the relation
$\bar{a}\sim_{\mathcal{F}}\bar{b}$ if $\\{n|a_{n}=b_{n}\\}\in\mathcal{F}$. The
function symbols in $\mathcal{L}$ are interpreted in the natural way (note
that $\sim_{\mathcal{F}}$ is a congruence). If $k\geq 1$ and
$R(x(0),\dots,x(k-1))$ is a $k$-ary relation symbol and
$\bar{a}(0),\dots,\bar{a}(k-1)$ is a $k$-tuple, then we let
$\prod_{\mathcal{F}}A_{n}\models R(\bar{a}(0),\dots,\bar{a}(k-1))$ if and only
if the set

$\\{n|A_{n}\models R(a_{n}(0),\dots,a_{n}(k-1))\\}$

belongs to $\mathcal{F}$. The image of $\bar{a}$ in the reduced product
$\prod_{\mathcal{F}}A_{n}$ under the quotient map is also denoted $\bar{a}$,
by a standard and innocuous abuse of notation.

If $\mathcal{F}$ is the Fréchet filter (i.e., the filter of cofinite subsets
of ${\mathbb{N}}$), then $\prod_{\mathcal{F}}A_{n}$ is denoted
$\prod_{\operatorname{Fin}}A_{n}$. (This is yet another standard and innocuous
abuse of notation; $\operatorname{Fin}$ denotes the ideal dual to the Fréchet
filter, and the reduced products are sometimes defined with respect to the
dual ideals.) If $\mathcal{U}$ is an ultrafilter (i.e., a proper filter
maximal with respect to the inclusion), then $\prod_{\mathcal{U}}A_{n}$ is
called the ultraproduct.

When all structures $A_{n}$ are equal to some $A$, the corresponding reduced
products (ultraproducts) are called reduced powers (ultrapowers).

######  Remark 1.1.

Reduced products are sometimes defined with respect to a filter $\mathcal{F}$,
and sometimes with respect to the dual ideal $\mathcal{F}^{*}$, as convenient.
The only difference is in the notation. In this paper we will use the two
notions—reduced products with respect to filters and reduced products with
respect to ideals—interchangeably.

###  1.2. The order properties

This combinatorial property of a first-order theory marks the watershed
between well-behaved and wild (see [40]).

######  Definition 1.2.

Suppose that $T$ is a first-order theory, not necessarily complete.

  1. (1)

If $\varphi(\bar{x},\bar{y})$ is an asymmetric formula (with $\bar{x}$ and
$\bar{y}$ of the same sort) in the language of $T$ consider the asymmetric
binary relation $\triangleleft_{\varphi}$ on a model $A$ of $T$, defined by

(1.1) $\bar{a}\triangleleft_{\varphi}\bar{b}\text{ if
}A\models\varphi(\bar{a},\bar{b}).$

Some $\bar{a}_{j}$, for $j<n$, in $A$ form a _$\triangleleft_{\varphi}$
-chain_ if for all $i\neq j$ we have
$\bar{a}_{i}\triangleleft_{\varphi}\bar{a}_{j}$ if and only if $i<j$.

  2. (2)

If every model of $T$ has an arbitrarily long finite
$\triangleleft_{\varphi}$-chain, we say that the pair _$(T,\varphi)$ has the
order property, OP_ ([40]).666One says that $\varphi$ has the order property
when $T$ is clear from the context.

  3. (3)

The pair $(T,\varphi)$ has the _robust order property_ if it has the order
property and in addition for models $A_{n}$, for $n\in{\mathbb{N}}$, of $T$
and $\bar{a}$ and $\bar{b}$ in $\prod_{\operatorname{Fin}}A_{n}$ we have
$\prod_{\operatorname{Fin}}A_{n}\models\varphi(\bar{a},\bar{b})$ if and only
if the set

$\\{n\mid A_{n}\not\models\varphi(\bar{a}_{n},\bar{b}_{n})\\}$

is finite. (Note that it is not required that
$\prod_{\operatorname{Fin}}A_{n}$ models $T$.)

  4. (4)

The pair $(T,\varphi)$ is said to have the _strict order property_ (SOP) if
the relation $\triangleleft_{\varphi}$ is a partial ordering on every model of
$T$.

The relation between the order property and the robust order property depends
on the analysis of the relation between the theories of $A_{n}$ and the theory
of $\prod_{\operatorname{Fin}}A_{n}$, as given by the Feferman–Vaught theorem
([22] and [23] for continuous logic, also see [12, §16.3]). We will need only
the following easy case.

######  Lemma 1.3.

If a pair $(T,\varphi)$ has the order property and $\varphi(\bar{x},\bar{y})$
is atomic, or a negation of an atomic formula, then the pair $(T,\varphi)$ has
the robust order property.

###### Proof.

Fix models $A_{n}\models T$ for $n\in{\mathbb{N}}$ and suppose $\varphi$ is an
atomic formula. If $\bar{a}_{n}$ and $\bar{b}_{n}$ are tuples of the
appropriate sort in $A_{n}$ such that
$A_{n}\models\varphi(\bar{a}_{n},\bar{b}_{n})$, then (writing $\bar{a}$ for
the element of the product that has the representing sequence
$(\bar{a}_{n})$), we have $\prod_{n}A_{n}\models\varphi(\bar{a},\bar{b})$ and
moreover for any filter $\mathcal{F}$ on ${\mathbb{N}}$ we have
$\prod_{\mathcal{F}}A_{n}\models\varphi(\bar{a},\bar{b})$. This also applies
to the negation of $\varphi$, and the assertion follows immediately. ∎

The proof of Lemma 1.3 uses only the fact that $\varphi$ satisfies the variant
of Łoś’s Theorem for reduced products asserting that
$\prod_{\mathcal{F}}A_{n}\models\varphi(\bar{a},\bar{b})$ if and only if
$\\{n\mid A_{n}\models\bar{a}_{n},\bar{b}_{n})\\}\in\mathcal{F}$. A larger
class of formulas with this preservation property, called h-formulas, has been
isolated in [37].

###  1.3. Continuous logic

For more details on continuous logic see [2] (see [15] or [12, §16] for
operator algebras). That said, this subsection is targeted at the readers
already familiar with continuous logic, and its aim is to convince these
readers that the proofs of the continuous versions of our main results are
analogous to the proofs in the discrete case.

The value of a formula $\varphi(\bar{x})$ evaluated in a model $M$, at a tuple
$\bar{a}$ of the appropriate sort, is denoted $\varphi(\bar{a})^{M}$ and
defined by recursion on the complexity of $\varphi$.

###  1.4. Reduced products in continuous logic

If $\mathcal{F}$ is a filter on ${\mathbb{N}}$, then the reduced product
$\prod_{\mathcal{F}}A_{n}$ of metric structures of the same language is
defined as follows.777Needless to say, the definition of a reduced product
with a filter on some other set is analogous. With
$\mathcal{F}^{*}:=\\{A\subseteq{\mathbb{N}}\mid{\mathbb{N}}\setminus
A\notin\mathcal{F}\\}$ (the coideal of all sets positive with respect to
$\mathcal{F}$), on $\prod_{n}A_{n}$ define pseudometric

$d_{\mathcal{F}}(\bar{a},\bar{b}):=\inf_{X\in\mathcal{F}^{*}}\sup_{m\in
X}d_{m}(a_{m},b_{m}).$

The universe $A$ of $\prod_{\mathcal{F}}A_{n}$ is the completion of the
quotient space with respect to $d_{\mathcal{F}}$.

Every predicate symbol $R(\bar{x})$ in the language is interpreted as a
function into $\mathbb{R}$, and the syntax rules of continuous logic require
that its interpretation in each $A_{n}$ respects the same modulus of uniform
continuity. For $\bar{a}\in\prod_{n}A_{n}$ let

$R(\bar{a}):=\inf_{X\in\mathcal{F}^{*}}\sup_{m\in X}R^{A_{m}}(\bar{a}_{m}).$

Then $d_{\mathcal{F}}(\bar{a},\bar{b})=0$ implies $R(\bar{a})=R(\bar{b})$, and
one defines the interpretation of $R$ in $A$ by $R^{A}([\bar{a}]):=R(\bar{a})$
(where $[\bar{a}]$ is the equivalence class of $\bar{a}$ in
$\prod_{\mathcal{F}}A_{n}$).

Function symbols are interpreted in the obvious manner (using the fact that
they are also required to respect the same modulus of uniform continuity in
all $A_{n}$).

For more details see e.g., [2, §5] (for ultraproducts) and [12, §16.2 and
Definition D.2.13] for the general case.

###  1.5. Order property in continuous logic

The following is the continuous analog of Definition 1.2.

######  Definition 1.4.

Suppose that $T$ is a theory in a continuous language, not necessarily
complete.

  1. (1)

If $\varphi(\bar{x},\bar{y})$ is an asymmetric formula (with $\bar{x}$ and
$\bar{y}$ of the same sort) in the language of $T$ consider the asymmetric
binary relation $\triangleleft_{\varphi}$ on a model $A$ of $T$, defined by

$\displaystyle\bar{a}\triangleleft_{\varphi}\bar{b}$ $\displaystyle\text{ if
}\varphi(\bar{a},\bar{b})^{A}=0\text{ and }\varphi(\bar{b},\bar{a})^{A}=1.$

Some $\bar{a}_{j}$, for $j<n$, in $A$ form a _$\triangleleft_{\varphi}$
-chain_ if for all $i\neq j$ we have
$\bar{a}_{i}\triangleleft_{\varphi}\bar{a}_{j}$ if and only if $i<j$.

  2. (2)

If every model of $T$ has an arbitrarily long finite
$\triangleleft_{\varphi}$-chain, we say that the pair _$(T,\varphi)$ has the
order property, OP_ ([17, Definition 5.2]).

  3. (3)

The pair $(T,\varphi)$ has the _robust order property_ if for models $A_{n}$,
for $n\in{\mathbb{N}}$, of $T$, and all $\bar{a}$ and $\bar{b}$ in
$\prod_{\operatorname{Fin}}A_{n}$ we have
$\prod_{\operatorname{Fin}}A_{n}\models\varphi(\bar{a},\bar{b})$ if and only
if for all sufficiently small $\varepsilon>0$ the set

$\\{n\mid\varphi^{A_{n}}(\bar{a}_{n},\bar{b}_{n})<\varepsilon\text{ and
}\varphi^{A_{n}}(\bar{b}_{n},\bar{a}_{n})>1-\varepsilon\\}$

is finite. (As before, it is not required that
$\prod_{\operatorname{Fin}}A_{n}$ models $T$.)

  4. (4)

The pair $(T,\varphi)$ is said to have the _strict order property_ (SOP) if
the relation $\triangleleft_{\varphi}$ is a partial ordering on every model of
$T$.

Therefore by replacing $\varphi$ with $f(\varphi)$ for a suitable piecewise
continuous function $f$, the order property of a continuous theory as well as
its robustness are witnessed by a discrete (i.e., 0-1 valued) formula. Because
of this, we will provide proofs of our results only in the case of discrete
theories, with understanding that they carry on virtually unchanged to the
continuous context. A proof of the following is analogous to the proof of
Lemma 1.3 and therefore omitted.

######  Lemma 1.5.

If $T$ is a continuous theory, a pair $(T,\varphi)$ has the order property,
and $\varphi(\bar{x},\bar{y})$ is an atomic formulas then the pair
$(T,\varphi)$ has the robust order property. ∎

##  2\. Background on posets

In this section we warm up by stating and proving some well-known results.
Consider the following two partial quasi-orderings on
${\mathbb{N}}^{{\mathbb{N}}}$ (by $\forall^{j}$ we denote the quantifier ‘for
all but finitely many $j\in{\mathbb{N}}$’):

$\displaystyle f\leq^{*}g$
$\displaystyle\Leftrightarrow(\forall^{\infty}j)f(j)\leq g(j)$ $\displaystyle
f<^{*}g$ $\displaystyle\Leftrightarrow(\forall^{\infty}j)f(j)<g(j).$

Any proper initial segment of $({\mathbb{N}}^{{\mathbb{N}}},\leq^{*})$ is
included in one of the form $(\\{f\in{\mathbb{N}}^{{\mathbb{N}}}\mid
f\leq\eta\\},\leq^{*})$ for some $\eta\in{\mathbb{N}}^{{\mathbb{N}}}$. Such an
initial segment is isomorphic to $(\prod_{k}\eta(k),\leq^{*})$ (if
$f\leq^{*}\eta$, then the pointwise minimum of $f$ and $\eta$ is an element of
$\prod_{k}\eta(k)$ equal to $f$ modulo finite) and these structures will be
our main focus. The following is essentially a bounded variant of [9,
Proposition 0.1].

######  Lemma 2.1.

There are $\eta\in{\mathbb{N}}^{\mathbb{N}}$ and
$\Phi\colon(\prod_{k}k,\leq^{*})\to(\prod_{k}\eta(k),<^{*})$ such that for all
$f$ and $g$, if $f\leq^{*}g$ and $g\nleq^{*}f$ then $\Phi(f)<^{*}\Phi(g)$.

A morphism $\Phi$ as guaranteed by Lemma 2.1 is called _strictly increasing_.

###### Proof.

Recursively define $\eta$ by $\eta(0):=1$ and $\eta(n+1):=\sum_{j\leq
n}j\eta(j)+1$ for $n\geq 0$. By rewriting the recursive definition of $\eta$,
one sees that for every $n\geq 1$ and every $m\geq 0$ we have

(2.1) $(m+1)\eta(n)>\sum_{j<n}j\eta(j)+m\eta(n).$

Fix $f\in\prod_{k}k$. Let $\Phi(f)(0):=0$ and for $n\geq 0$ let

$\Phi(f)(n+1):=\sum_{j\leq n}f(j)\eta(j).$

Then, since $f(j)<j$ for all $j$, we have $\Phi(f)(n+1)<\sum_{j\leq
n}j\eta(j)<\eta(n+1)$ for all $n\geq 1$, and therefore $\Phi(f)$ belongs to
$\prod_{k}\eta(k)$.

Suppose that $f$ and $g$ are in $\prod_{k}k$. If $n\geq 1$ is such that
$g(n)>f(n)$, then (2.1) with $m:=f(n)$ implies

$g(n)\eta(n)\geq(f(n)+1)\eta(n)>\sum_{j<n}j\eta(j)+f(n)\eta(n)\geq\Phi(f)(n+1)$

and therefore $\Phi(g)(n+1)>\Phi(f)(n+1)$. This implies that if $f$ and $g$
are in $\prod_{k}k$, then every $n\geq 1$ such that $f(n)<g(n)$ in addition
satisfies $\Phi(f)(n+1)<\Phi(g)(n+1)$.

It remains to prove that $f\leq^{*}g$ and $g\not\leq^{*}f$ together imply
$f(k)<g(k)$ for all sufficiently large $k$. Fix such $f$ and $g$, and let
$n\geq 1$ be such that $f(n)<g(n)$ and $f(k)\leq g(k)$ for all $k\geq n$.
Then, as we have just seen, $\Phi(f)(n+1)<\Phi(g)(n+1)$. Since
$\Phi(f)(k+1)=\Phi(f)(k)+f(k)\eta(k)$, for every $k\in{\mathbb{N}}$ the
conditions $\Phi(f)(k)<\Phi(g)(k)$ and $f(k+1)\leq g(k+1)$ together imply that
$\Phi(f)(k+1)<\Phi(g)(k+1)$. By induction on $k\geq n+1$ one proves that
$\Phi(f)(k)<\Phi(g)(k)$ for all $k\geq n+1$, as required. ∎

The universal structure obtained in Lemma 2.2 below is very similar to the
Rado graph, also known as the (countably infinite) random graph, and it ought
to be well-known. It was however easier to include a proof than to look for it
in the literature.

######  Lemma 2.2.

There exists an injectively universal countable structure $(C,\triangleleft)$
with an asymmetric binary relation $\triangleleft$. This universality property
is absolute between transitive models of a sufficiently large fragment of
$\operatorname{ZFC}$.

###### Proof.

Let $C:={\mathbb{N}}$ and define the relation $\triangleleft$ as follows. If
$m<n$ are in ${\mathbb{N}}$ and $n=\sum_{j}d_{j}(n)3^{j}$ is the ternary
expansion of $n$ (so that $d_{j}(n)\in\\{0,1,2\\}$ for all $j$) then let
$m\triangleleft n$ if $d_{m}(n)=1$, $n\triangleleft m$ if $d_{m}(n)=2$, and
let $m$ and $n$ unrelated if $d_{m}(n)=0$. The structure $(C,\triangleleft)$
has the following property resembling the random graph:

  1. (*)

If $F$ and $G$, are disjoint finite subsets of $C$, then there exists $n\in C$
such that $m\in F$ implies $m\triangleleft n$, $m\in G$ implies
$n\triangleleft m$, and $m\notin F\cup G$ implies that $m$ and $n$ are
unrelated.

To see this, let $n:=\sum_{m\in F}3^{m}+\sum_{m\in G}2\cdot 3^{m}$.

Given the property (*) of $(C,\triangleleft)$, every countable
$(A,\triangleleft^{\prime})$ can be isomorphically embedded into
$(C,\triangleleft)$ by recursion. Since (*) is a first-order property, it is
absolute between transitive models of a sufficiently large fragment of
$\operatorname{ZFC}$ (see e.g., [32, Lemma II.4.3]). ∎

##  3\. The depletion of a poset

The notion of the _depletion_ of a linear ordering given in Definition 3.1
appears implicitly in [9].

######  Definition 3.1.

Suppose that $I$ is a finite linear ordering whose elements are listed in the
increasing order as $\xi(i)$, for $i<m$ and $m=|I|\geq 2$.888If this appears
excessively pedantic, note that the conditions (2b) and (3b) below are
sensitive to leaving gaps in $I$. We note that $I$ is not necessarily a set of
ordinals. Also suppose $A$ and $F(\xi)$, for $\xi\in I$, are disjoint sets,
and $\leq$ is a partial ordering on a set that includes
$E:=A\cup\bigcup_{\xi\in I}F(\xi)$. Define a binary relation $\ll_{I}$ on $E$
as follows.

If $x$ and $y$ belong to $E$, we let $x\ll_{I}y$ if and only $x\leq y$ and in
addition one of the following applies.

  1. (1)

Both $x$ and $y$ belong to $A\cup F(\xi)$ for some $\xi\in I$.

  2. (2)

There are $i<j$ such that $x\in F(\xi(i))$ and $y\in F(\xi(j))$ and one of the
following holds.

    1. (a)

There exists $a\in A$ such that $x\leq a$ and $a\leq y$

    2. (b)

With $k=j-i$, there are $x_{l}\in F(\xi(i+l))$ for $0\leq l\leq k$ such that
$x_{0}=x$, $x_{k}=y$, and $x_{l}\leq x_{l+1}$ for all $l<k$.

  3. (3)

There are $i>j$ such that $x\in F(\xi(i))$ and $y\in F(\xi(j))$ and one of the
following holds.

    1. (a)

There exists $a\in A$ such that $x\leq a$ and $a\leq y$

    2. (b)

With $k=i-j$, there are $x_{l}\in F(\xi(j+l))$ for $0\leq l\leq k$ such that
$x_{0}=y$, $x_{k}=x$, and $x_{l}\geq x_{l+1}$ for all $l<k$.

The elements $x_{i}$, for $i\leq k$ as in (2b) or (3b) comprise an _$I$ -walk
between $x$ and $y$_, or an _$I$ -walk with endpoints $x$ and $y$_. The
relation $\ll_{I}$ is called the _depletion of $\leq$ given by $I$, $A$, and
$(F(\xi)|\xi\in I)$_. When $I$ is clear from the context, we write $\ll$ for
$\ll_{I}$.

Take note of the fact that a depletion depends on the order on $I$ in an
essential way. In addition, it clearly depends on $A$, the choice of $F(\xi)$
for $\xi\in I$, and the order on $A\cup\bigcup_{\xi\in I}F(\xi)$. Hence
writing $\ll_{(A,(F(\xi)|\xi\in I),\leq)}$ in place of $\ll_{I}$ may appear to
be a more reasonable choice. Fortunately, $A$ and $F(\xi)$, for $\xi$ in some
set including $I$, as well as the ordering $\leq$ on some set that includes
$A\cup\bigcup_{\xi}F(\xi)$, will (unlike $I$) be fixed and clear from the
context in any given instance in which a depletion is used.

It should be emphasized that in both (2b) and (3b) it is required that the
‘walk’ between $x$ and $y$ hits $F(\xi(k))$ for all $i\leq k\leq j$.

In order to help the reader internalize these definitions, we state two lemmas
whose easy proofs are omitted.

######  Lemma 3.2.

With the notation as in Definition 3.1, if $|I|\leq 2$, then $\ll_{I}$ agrees
with the restriction of $\leq$ to $A\cup\bigcup_{\xi\in I}F(\xi)$. ∎

######  Lemma 3.3.

With the notation as in Definition 3.1, if $s\subseteq t\subseteq I$, if
$\\{\min(s),\max(s)\\}=\\{\min(t),\max(t)\\}=\\{\xi,\eta\\}$, and $x(\zeta)$,
for $\zeta\in t$, is a $t$-walk, then $x(\zeta)$, for $\zeta\in s$, is an
$s$-walk with the same endpoints.

On the other hand, it is possible that $\xi(\zeta)$, for $\zeta\in s$, is an
$s$-walk but there is no $t$-walk with the same endpoints that extends it. ∎

The (admittedly rather dull) Lemma 3.4 will be instrumental in a critical
place in the proof of Theorem 4.13.

######  Lemma 3.4.

Suppose that $I$, $A$, $F(\xi)$, for $\xi\in I$, and an ordering $\leq$ on
$A\cup\bigcup_{\xi\in I}F(\xi)$ are as in Definition 3.1. The depletion $\ll$
of $\leq$ given by these parameters is a partial ordering whose graph is
included in the graph of $\leq$.

###### Proof.

Fix $A$, $I$ enumerated in the increasing order as $\xi(i)$, for $i<m$,
$F(\xi)$, for $i<m$, and an ordering $\leq$ on
$E:=A\cup\bigcup_{i<m}F(\xi(i))$.

We will write $\ll$ for $\ll_{s}$. It is clear from the definition that $x\ll
y$ implies $x\leq y$ and that $\ll$ and $\leq$ agree on $A\cup F(i)$ for every
$i$. Therefore $\ll$ is antisymmetric and reflexive, and it will suffice to
prove that it is transitive.

Towards this end, fix $x,y$, and $z$ such that $x\ll y$ and $y\ll z$. Then
$x\leq y$ and $y\leq z$, and therefore $x\leq z$. If $x$ and $z$ belong to
$A\cup F(\xi(i))$ for some $i$, then $x\ll z$ by (1). Therefore if at least
one of $x\in A$ or $z\in A$ holds then $x\ll z$, and we may assume

  1. ($xz$)

$x\in F(\xi(i))$ and $z\in F(\xi(j))$ for some distinct $i$ and $j$.

If $y\in A$ then $x\ll z$ by (2a). Similarly, if there exists $a\in A$ such
that $x\leq a$ and $a\leq y$, then $x\ll z$. Also, if there exists $a\in A$
such that $y\leq a$ and $a\leq z$, then $x\ll z$. We can therefore assume (in
addition to ($xz$) above) that

  1. ($y$)

$y\in F(\xi(n))$ for some $n$

and that both $x\ll y$ and $y\ll z$ are witnessed by instances of (2b).

The proof now reduces to the analysis of the ordering of the set
$\\{i,j,n\\}$. The following claim will help when discussing the possible
cases.

######  Claim 3.5.

Suppose that $i<m$, $0<k\leq m-i$, $x\in F(\xi(i))$ and $y\in F(\xi(i+k))$.

  1. (1)

Assume there is no $a\in A$ such that $x\leq a$ and $a\leq y$. Then $x\ll y$
if and only if there are $x_{l}\in F(\xi(i+l))$ for all $0\leq l\leq k$ such
that $x_{0}=x$, $x_{k}=y$, and $x_{l}\leq x_{l+1}$ for all $0\leq l<k$.

  2. (2)

Assume there is no $a\in A$ such that $y\leq a$ and $a\leq x$. Then $y\ll x$
if and only if there are $x_{l}\in F(\xi(i+l))$ for all $0\leq l\leq k$ such
that $x_{0}=x$, $x_{k}=y$, and $x_{l+1}\leq x_{l}$ for all $0\leq l<k$.

###### Proof.

(1) For the direct implication, note that the assumptions imply that (2b) of
Definition 3.1 applies. Let $x_{0}:=x$, $x_{k}:=y$, and for $0<l<k$ let
$x_{l}$ be a witness for (2b) of Definition 3.1. These objects are clearly as
required.

For the converse implication, assume that $x_{l}$ for $0\leq l\leq k$ are as
in the statement of the claim. Then clearly (2b) of Definition 3.1 applies.

The proof of (2) is analogous and therefore omitted. ∎

Back to our proof. With $i,j$, and $n$ as in ($xz$) and ($y$), if $i\leq n\leq
j$, then part (1) of Claim implies that $x\ll z$. If $i<j<n$, then the
witnessing sequence for $x\ll y$ contains $t\in F(\xi(j))$, such that $x\ll t$
and $t\ll y$. But then (since $\ll$ implies $\leq$) $t\leq z$, and $t\ll z$
since both $t$ and $z$ belong to $F(\xi(j))$. A proof in the case when $n<i<j$
is similar and uses part (2) of the Claim. This proves our claim in the case
when $i<j$.

The proof in the case when $i>j$ is analogous. ∎

The following lemma extends Lemma 3.3 and stands in contrast to the situation
described in Remark 3.7.

######  Lemma 3.6.

With the notation as in Definition 3.1 and $I$ not necessarily finite, if
$s\subseteq t\Subset I$, then for any two $x$ and $y$ in $A\cup\bigcup_{\xi\in
s}F(\xi)$ we have that $x\ll_{t}y$ implies $x\ll_{s}y$.

###### Proof.

Fix $x\ll_{t}y$ in $A\cup\bigcup_{\xi\in s}F(\xi)$. A glance at Definition 3.1
shows that we may assume that $x\in F(\xi)$ and $y\in F(\eta)$ for some
distinct $\xi$ and $\eta$ in $s$, since in any other situation $x\ll_{s}y$
follows immediately. If there is $z\in A$ such that $x\leq z\leq y$, then
clearly $x\ll_{w}y$ whenever $\\{\xi,\eta\\}\subseteq w\Subset I$. We may
therefore assume that there is a $t$-walk with endpoints $x$ and $y$. We
denote the ‘steps’ of $s$ by $x(\zeta)$, for $\zeta\in t$ and
$\xi\leq\zeta\leq\eta$. Lemma 3.3 implies that $x(\zeta)$, for $\zeta\in s$
and $\xi\leq\zeta\leq\eta$ is an $s$-walk with endpoints $x$ and $y$, hence
$x\ll_{s}y$. ∎

######  Remark 3.7.

Suppose that we are in the situation of Lemma 3.4 and $s$ is a proper subset
of $t$. The fact that in (2) and (3) the ‘walk’ between $x$ and $y$ is
required to hit every $F(\xi(k))$ implies that the graph of $\ll_{s}$ may be a
proper superset of the graph of the restriction of $\ll_{t}$ to the domain of
$\ll_{s}$.

Here is another easy lemma with an omitted proof that complements Remark 3.7.

######  Lemma 3.8.

Suppose that we are in the situation of Lemma 3.4 and $s\subseteq t$ is convex
(i.e., if $\xi<\eta<\zeta$ are in $t$ and $\\{\xi,\zeta\\}\subseteq s$ then
$\eta\in s$). Then $\ll_{s}$ agrees with the restriction of $\ll_{t}$ to
$A\cup\bigcup_{\xi\in I}F(\xi)$. ∎

Proposition 3.9 below is a relative to a result of Kurepa ([33]) and to [9,
Theorem 7.1]. The role that this proposition plays in the proof of our Theorem
4.13 is analogous to the role that [9, Theorem 7.1] had played in the proof of
[9, Theorem 9.1]. For reader’s convenience, we include a proof. If $I$ is a
linear ordering, then $I^{*}$ denotes the converse ordering.

######  Proposition 3.9.

Suppose that $\kappa$ is an uncountable cardinal, $A$ and $F(\xi)$, for
$\xi<\kappa$, are disjoint, and $\leq$ is a partial ordering of
$E:=A\cup\bigcup_{\xi<\kappa}F(\xi)$. In addition suppose that $A$ is
countable, all $F(\xi)$ are finite, and $E$ has neither $\kappa$ nor
$\kappa^{*}$-chains.

Then there exists a cofinal $X\subseteq\kappa$ such that for any two distinct
elements $\xi$ and $\eta$ of $X$ the following condition holds.

  1. $*_{(\xi,\eta)}$

there is $s\Subset\kappa$ such that $\\{\xi,\eta\\}=\\{\min(s),\max(s)\\}$ and
there is no $s$-walk with endpoints in $F(\xi)$ and $F(\eta)$.999We emphasize
that $s$ is not necessarily a subset of $X$.

###### Proof.

Let $\mathbb{P}$ be the poset of all $X\subseteq\kappa$ such that $0\in X$
which satisfy the condition $*_{(\xi,\eta)}$ for all distinct $\xi$ and $\eta$
in $X$, ordered by the inclusion. This poset is clearly closed under unions of
chains, and therefore Zorn’s lemma implies that it has a maximal element, $X$.
(Readers not fond of the Axiom of Choice will notice that since $\kappa$ is
well-ordered, the existence of $X$ can be proved by recursion in ZF.) We claim
that $X$ satisfies the requirements. Towards obtaining a contradiction,
suppose that $X$ is not cofinal in $\kappa$.

If $X$ has no maximal element, then let $\alpha:=\sup(X)$ and
$Y:=X\cup\\{\alpha\\}$. In order to verify that $Y$ belongs to $\mathbb{P}$,
it suffices to verify $*_{(\xi,\alpha)}$ for all $\xi\in X$. Fix $\xi\in X$.
We need to prove that there exists $t$ such that there is no $t$-walk with
endpoints in $F(\xi)$ and $F(\alpha)$. Let $\eta:=\min(X\setminus(\xi+1))$ and
(using $*_{(\xi,\eta)}$) let $s\Subset\kappa$ be such that
$\\{\xi,\eta\\}=\\{\min(s),\max(s)\\}$ and there is no $s$-walk with endpoints
$F(\xi)$ and $F(\eta)$. Let $t:=s\cup\\{\alpha\\}$. Then
$\\{\xi,\alpha\\}=\\{\min(t),\max(t)\\}$ and Lemma 3.3 implies there is no
$t$-walk with endpoints in $F(\xi)$ and $F(\alpha)$.

Now suppose that $X$ has a maximal element, $\xi$. Assume for a moment that
there are $\eta>\xi$ and $s\Subset\kappa$ which witnesses that
$*_{(\xi,\eta)}$ holds. We claim that $X\cup\\{\eta\\}$ belongs to
$\mathbb{P}$. Fix $\zeta\in X$ such that $\zeta<\xi$. Since $X\in\mathbb{P}$,
there is $t\Subset\kappa$ such that $\\{\zeta,\xi\\}=\\{\min(t),\max(t)\\}$
and there is no $t$-walk with endpoints in $F(\zeta)$ and $F(\xi)$. With
$w:=s\cup t$ we have $\\{\zeta,\eta\\}=\\{\min(w),\max(w)\\}$ and Lemma 3.3
implies that there is no $w$-walk with endpoints in $F(\zeta)$ and $F(\eta)$.
Therefore $*_{(\zeta,\eta)}$ holds. Since $\zeta$ was an arbitrary element of
$X$ below $\xi$ (and since $*_{(\xi,\eta)}$ holds), this implies
$X\cup\\{\eta\\}$ is an element of $\mathbb{P}$ strictly larger than $X$;
contradiction.

It will therefore suffice to find $\eta>\xi$ such that $*_{(\xi,\eta)}$ holds.
Assume that such $\eta$ does not exist. Then for every $\eta>\xi$ and every
$s\Subset\kappa$ which satisfies $\\{\xi,\eta\\}=\\{\min(s),\max(s)\\}$ there
is an $s$-walk with endpoints in $F(\xi)$ and $F(\eta)$. For each $s$ fix a
walk, $x(s,\zeta)$, for $\zeta\in s$, with endpoints in $F(\xi)=\min(s)$ and
$F(\eta)=\max(s)$.

Let $\mathcal{U}$ be an ultrafilter on $\operatorname{Fin}_{\kappa}$ which for
every $s\Subset\kappa$ includes the set $\\{t\Subset\kappa|s\subset t\\}$
(such $\mathcal{U}$ exists, since the family of sets of this form has the
finite intersection property). Fix $\xi\leq\zeta<\kappa$. Since $F(\zeta)$ is
finite, there exists a unique $y(\zeta)\in F(\zeta)$ such that
$\\{s|x(s,\zeta)=y(\zeta)\\}\in\mathcal{U}$. Clearly $y(\zeta)$, for
$\xi\leq\zeta<\kappa$, is a $\kappa$-chain or a $\kappa^{*}$-chain (as decided
by $\mathcal{U}$); contradiction.

Therefore there exist $\eta>\xi$ and $s\Subset\kappa$ such that
$\\{\xi,\eta\\}=\\{\min(s),\max(s)\\}$ such that there is no $s$-walk with
endpoints in $F(\xi)$ and $F(\eta)$. As already pointed out, this implies
$X\cup\\{\eta\\}\in\mathbb{P}$; contradiction.

Therefore any maximal element of $\mathbb{P}$ is cofinal in $\kappa$, as
required. ∎

##  4\. Gently embedding posets into reduced powers

In the present section we assume that the reader is familiar with the basics
of forcing as presented in e.g., [32] or [44]. The present section is largely
based on [9], and Theorem 4.13 is a close relative to [9, Theorem 9.1].

The category of partially ordered sets is considered with respect to the
order-embeddings, i.e., injections $f\colon E\to E^{\prime}$ such that
$a\leq_{E}b$ if and only if $f(a)\leq_{E^{\prime}}f(b)$. The category of
forcing notions is considered with respect to regular embeddings (also known
as complete embeddings, [32, Definition III.3.65]). If a forcing notion
$\mathbb{H}_{0}$ is a regular subordering of a forcing notion
$\mathbb{H}_{1}$, we then write $\mathbb{H}_{0}\lessdot\mathbb{H}_{1}$.
Notably, $\mathbb{H}_{0}\lessdot\mathbb{H}_{1}$ is equivalent to the assertion
that for every generic filter $G\subseteq\mathbb{H}_{1}$,
$G\cap\mathbb{H}_{0}$ is also generic. In other words, $\mathbb{H}_{1}$ can be
considered as a two-step iteration of $\mathbb{H}_{0}$ followed by the
naturally defined quotient forcing notion.

If $\kappa$ is an uncountable cardinal, a forcing notion $\mathbb{P}$ is said
to have _precaliber $\kappa$_ if every set of $\kappa$ conditions in
$\mathbb{P}$ has a subset of cardinality $\kappa$ such that each of its finite
subsets has a common lower bound. Precaliber $\aleph_{1}$ is a strong form of
the countable chain condition. For example, if $\mathbb{P}$ has precaliber
$\aleph_{1}$ then it is _productively ccc_ , in the sense that the product of
$\mathbb{P}$ with any ccc poset is ccc. (We will not need this fact.)

######  Theorem 4.1.

There is a functor from the category of partially ordered sets into the
category of forcing notions $E\mapsto\mathbb{H}_{E}$ with the following
properties.

  1. (1)

$\mathbb{H}_{E}$ has precaliber $\kappa$ for every uncountable regular
cardinal $\kappa$.

  2. (2)

$\mathbb{H}_{E}$ forces that there is an embedding $\Upsilon\colon
E\to(\prod_{k}k,\leq^{*})$ (thus for all $a$ and $b$ in $E$ we have
$a\leq_{E}b$ if and only if $\Upsilon(a)\leq^{*}\Upsilon(b)$).

  3. (3)

If $\kappa>\mathfrak{c}$ is a regular cardinal and neither $\kappa$ nor its
reverse $\kappa^{*}$ embed into $E$, then $\mathbb{H}_{E}$ forces that
$\kappa$ does not embed into $\prod_{\operatorname{Fin}}(A_{n},\triangleleft)$
for every sequence $(A_{n},\triangleleft_{n})$ of countable structures
equipped with an asymmetric binary relation.

###### Proof.

The proof of this theorem will occupy most of the present section. For
$\mathbb{H}_{E}$ see Definition 4.2, (1) is Lemma 4.4, (2) is Lemma 4.6, and
(3) is Theorem 4.13. ∎

In the Definition 4.2 and elsewhere, if
$\operatorname{dom}(f)\subseteq{\mathbb{N}}$ then $f\restriction m$ denotes
the restriction of $f$ to $m=\\{0,\dots,m-1\\}$.

######  Definition 4.2.

For a partially ordered set $E$, $\mathbb{H}_{E}$ is the forcing notion
defined as follows. The conditions of $\mathbb{H}_{E}$ are triples
$p=(D_{p},n_{p},f_{p})$, where $D_{p}\Subset E$, $n_{p}\in{\mathbb{N}}$, and
$f_{p}\colon D_{p}\to\prod_{m<n_{p}}m$.

The ordering is defined by letting $p\leq q$ ($p$ _extends_ $q$) if the
following conditions hold.

  1. (1)

$D_{p}\supseteq D_{q}$, $n_{p}\geq n_{q}$, $f_{p}(a)\restriction
n_{q}=f_{q}(a)$ for all $a\in D_{q}$, and

  2. (2)

for all $a$ and $b$ in $D_{q}$, if $a\leq_{E}b$ then $f_{p}(a)(j)\leq
f_{p}(b)(j)$ for all $j\in[n_{q},n_{p})$.

In order to relax the notation, if $(p_{\xi})$ is an indexed family of
conditions in $\mathbb{H}_{E}$ we write $p_{\xi}=(D_{\xi},n_{\xi},f_{\xi})$.

######  Lemma 4.3.

Suppose that $E$ is a poset, $R\Subset E$, $m\geq 2$ and $p_{i}$, for $i<m$,
are conditions in $\mathbb{H}_{E}$ such that the following holds whenever
$i\neq j$.

  1. (1)

We have $D_{i}\cap D_{j}=R$.

  2. (2)

All $a\in R$ satisfy
$f_{i}(a)\restriction\min(n_{i},n_{j})=f_{j}(a)\restriction\min(n_{i},n_{j})$.

Then some $q\in\mathbb{H}_{E}$ extends all $p_{i}$.101010We write $q\leq p$ if
$q$ extends $p$.

###### Proof.

Let $D_{q}:=\bigcup_{i<m}D_{i}$ and $n_{q}:=\max_{i<m}n_{i}$. If $i<m$ is such
that $n_{i}=n_{q}$, then for $a\in D_{i}$ let $f_{q}(a)=f_{i}(a)$. Then
$f_{q}(a)$ is well-defined for $a\in R$ by (2). For $i<m$ such that
$n_{i}<n_{q}$ and for $a\in D_{i}\setminus R$, let (with $\max\emptyset=0$)

$f_{q}(a)(j):=\max\\{f_{q}(b)(j)\mid b\in R,b\leq_{E}a\\}.$

for $n_{i}\leq_{E}j<n_{q}$. This defines $q\in\mathbb{H}_{E}$. We will prove
that $q\leq p_{i}$ for all $i<m$.

Clearly, $q$ and $p_{i}$ satisfy (1) of Definition 4.2 for all $i<m$. Fix
$i<m$.

If $n_{i}=n_{q}$ then (2) of Definition 4.2 is vacuous, hence
$q\leq_{E}p_{i}$.

Suppose $n_{i}<n_{q}$. To check that $q\leq p_{i}$, we need to verify (2) of
Definition 4.2. Fix $a$ and $b$ in $D_{i}$ such that $a\leq_{E}b$. If there is
no $c\in R$ such that $c\leq_{E}b$, then for all $j\in[n_{i},n_{q})$ we have
$f_{q}(a)(j)=f_{q}(b)(j)=0$. If there is $c\in R$ such that $c\leq_{E}b$, then
$\\{c\mid c\leq_{E}a\\}\subseteq\\{c\mid c\leq_{E}b\\}$ and by the definition
of $f_{q}$ we have $f_{q}(a)(j)\leq f_{q}(b)(j)$.

Thus (2) of Definition 4.2 holds, and $q\leq p_{i}$. ∎

######  Lemma 4.4.

The poset $\mathbb{H}_{E}$ has precaliber $\kappa$ for every uncountable
regular cardinal $\kappa$.

###### Proof.

Fix a family $p_{\xi}$, for $\xi<\kappa$, in $\mathbb{H}_{E}$. By the
$\Delta$-system lemma and passing to a subfamily of the same cardinality, we
may assume that there exists $R\Subset E$ such that $D_{\xi}\cap D_{\eta}=R$
for all distinct $\xi$ and $\eta$ below $\kappa$. By the pigeonhole principle
(using the assumption that $\kappa$ has uncountable cofinality), we may also
assume that there exists $n$ such that $n_{\xi}=n$ for all $\xi$. Also, since
there are only finitely many possibilities for $f_{\xi}(a)$, for $a\in R$, we
may assume that the functions $f_{\xi}$ agree on $R$ and therefore we are in
the situation of Lemma 4.3. Therefore, after this refining argument, Lemma 4.3
implies that every finite subset of $\\{p_{\xi}\mid\xi<\kappa\\}$ has a common
lower bound. ∎

A proof of the following lemma is straightforward and therefore omitted.

######  Lemma 4.5.

For any poset $E$ the following holds.

  1. (1)

For every $n$ and every $a\in E$, the set

$\mathcal{D}(\mathbb{H}_{E},n,a):=\\{p\in\mathbb{H}_{E}\mid n_{p}\geq n,a\in
F_{p}\\}$

is dense in $\mathbb{H}_{E}$.

  2. (2)

If $b\nleq_{E}a$ in $E$, then for every $n\in{\mathbb{N}}$ the set

${\mathcal{E}}(\mathbb{H}_{E},n,a,b):=\\{p\in\mathbb{H}_{E}\mid a\in
F_{p},b\in F_{p},\text{ and }(\exists k\geq n)f_{p}(a)(k)<f_{p}(b)(k)\\}$

is dense in $\mathbb{H}_{E}$. ∎

######  Lemma 4.6.

If $E$ is a poset and $G\subseteq\mathbb{H}_{E}$ is a generic filter, then

$\Upsilon_{G}(a)(j):=f_{p}(a)(j)$

for $p\in G$ defines a strictly increasing function $\Upsilon_{G}\colon
E\to(\prod_{k}k,\leq^{*})$.

###### Proof.

By genericity, $G$ intersects all dense sets defined in Lemma 4.5 and
therefore $\Upsilon$ is a strictly increasing map from $E$ into
$(\prod_{k}k,\leq^{*})$. ∎

If $E$ is a subordering of $E^{\prime}$ then every $p\in\mathbb{H}_{E}$ is
(literally) a condition in $\mathbb{H}_{E^{\prime}}$. We will therefore
identify $\mathbb{H}_{E}$ with a subordering of $\mathbb{H}_{E^{\prime}}$.

######  Lemma 4.7.

If $E^{\prime}$ is a poset and $E$ is a subposet of $E^{\prime}$, then
$\mathbb{H}_{E}$ is a regular subordering of $\mathbb{H}_{E^{\prime}}$.

###### Proof.

The identity map from $\mathbb{H}_{E}$ into $\mathbb{H}_{E^{\prime}}$ is
clearly an order-embedding. It suffices to prove that there exists a
_reduction_ (or _projection_)
$\pi\colon\mathbb{H}_{E^{\prime}}\to\mathbb{H}_{E}$: A map such that for every
$p\in\mathbb{H}_{E^{\prime}}$ we have $p\leq\pi(p)$ and every
$q\in\mathbb{H}_{E}$ such that $q\leq\pi(p)$ is compatible with $p$ ([32,
Lemma III.3.72]). Let

$\pi_{E}(p):=(D_{p}\cap E,n_{p},f_{p}\restriction(D_{p}\cap E)).$

Clearly, $p\leq\pi_{E}(p)$. If $q\leq\pi(p)$, then $D_{q}\cap D_{p}=D_{p}\cap
E$ and $f_{p}(a)(j)=f_{q}(a)(j)$ for all $a\in D_{p}\cap D_{q}$ and all
$j<n_{p}$. By Lemma 4.3, $p$ and $q$ are compatible. ∎

In the situation when $E$ is a subordering of $E^{\prime}$, as in Lemma 4.7,
we will need a description of the quotient forcing
$\mathbb{H}_{E^{\prime}}/\dot{G}$, for a generic $G\subseteq\mathbb{H}_{E}$.
If for some $k\in{\mathbb{N}}$ we have $s\in\prod_{n<k}n$ and
$f\in\prod_{k}k$, then

$s\sqsubset f\text{ stands for }s=f\restriction k.$

######  Definition 4.8.

If $E\subseteq E^{\prime}$ are partial orderings and $\Upsilon\colon
E\to(\prod_{k}k,\leq^{*})$ is a strictly increasing function, a forcing notion
$\mathbb{H}_{E^{\prime}}(E,\Upsilon)$ is defined as follows. The conditions in
$\mathbb{H}_{E^{\prime}}(E,\Upsilon)$ are the triples $p=(D_{p},n_{p},f_{p})$,
where $D_{p}\Subset E^{\prime}$, $n_{p}\in{\mathbb{N}}$, $f_{p}\colon
D_{p}\to\prod_{j<n}n$, and for $a\in E$ we have
$f_{p}(a)\sqsubset\Upsilon(a)$.

The ordering is inherited from $\mathbb{H}_{E^{\prime}}$. Therefore $p\leq q$
($p$ _extends_ $q$) if the following conditions hold.

  1. (1)

$D_{p}\supseteq D_{q}$, $n_{p}\geq n_{q}$, $f_{p}(a)\restriction
n_{q}=f_{q}(a)$ for all $a\in D_{q}$, and

  2. (2)

for all $a$ and $b$ in $D_{q}$, if $a\leq_{E^{\prime}}b$ then $f_{p}(a)(j)\leq
f_{p}(b)(j)$ for all $j\in[n_{q},n_{p})$.

Thus $\mathbb{H}_{E^{\prime}}(E,\Upsilon)$ is a subordering of
$\mathbb{H}_{E^{\prime}}$ consisting of those conditions that ‘agree’ with
$\Upsilon$ on $E$ and $\mathbb{H}_{E^{\prime}}(E,\Upsilon)$ generically adds
an embedding from $E^{\prime}\setminus E$ into $\prod_{k}k$. Note that
$\mathbb{H}_{E^{\prime}}(E,\Upsilon)$ is not necessarily separative; this will
not cause any issues.

The proofs of the two parts of Lemma 4.10 below are virtually identical to the
proofs of [9, Theorem 4.2] and [9, Lemma 4.3], respectively.

######  Definition 4.9.

For a poset $E^{\prime}$ and $a\in E^{\prime}$ let

$L(a):=\\{b\in E^{\prime}\mid b\leq_{E^{\prime}}a\\}$

and

$R(a):=\\{b\in E^{\prime}\mid b\geq_{E^{\prime}}a\\}.$

######  Lemma 4.10.

Suppose $E^{\prime}$ is a poset, $E$ is a subposet of $E^{\prime}$, and
$\dot{G}$ is the canonical name for the $\mathbb{H}_{E}$-generic filter.

  1. (1)

With the projection $\pi_{E}\colon\mathbb{H}_{E^{\prime}}\to\mathbb{H}_{E}$ as
in the proof of Lemma 4.7, the map $p\mapsto(\pi_{E}(p),p)$ from
$\mathbb{H}_{E}$ into
$\mathbb{H}_{E^{\prime}}*\mathbb{H}_{E^{\prime}}/\dot{G}$ is a dense
embedding.

  2. (2)

$\mathbb{H}_{E}$ forces that $\mathbb{H}_{E^{\prime}}/\dot{G}$ is forcing-
equivalent to $\mathbb{H}_{E^{\prime}}(E,\Upsilon_{\dot{G}})$
($\Upsilon_{\dot{G}}$ is the generic embedding, see Lemma 4.6).

  3. (3)

If $X\subseteq E$ is such that for every $a\in E^{\prime}\setminus E$ the set
$X\cap L(a)$ is cofinal in $L(a)$ and the set $X\cap R(a)$ is coinitial in
$R(a)$, then $\mathbb{H}_{E}$ forces that
$\mathbb{H}_{E^{\prime}}(E,\Upsilon_{\dot{G}})$ and
$\mathbb{H}_{E^{\prime}\setminus(E\setminus X)}(E\cap
X,\Upsilon_{\dot{G}}\restriction X)$ are forcing-equivalent. ∎

The following is [9, Lemma 5.1] (see also [3, Lemma 2.5]).

######  Lemma 4.11.

Suppose $\mathbb{P}_{0}$ and $\mathbb{P}_{1}$ are forcing notions and
$\dot{f}_{j}$ is a $\mathbb{P}_{j}$-name for an element of $\prod_{k}k$ for
$j<2$. If
$\mathbb{P}_{0}\times\mathbb{P}_{1}\Vdash\dot{f}_{0}\leq^{*}\dot{f}_{1}$ then
the set of all $p\in\mathbb{P}_{0}\times\mathbb{P}_{1}$ such that there exist
$m\in{\mathbb{N}}$ and $h\in\prod_{k}k$ which satisfy
$p\Vdash\dot{f}_{0}\leq^{m}\check{h}$ and
$p\Vdash\check{h}\leq^{m}\dot{f}_{1}$ is dense in
$\mathbb{P}_{0}\times\mathbb{P}_{1}$. ∎

In combination with Lemma 4.11, the following lemma will be used in a crucial
place in the proof of Theorem 4.13.

######  Lemma 4.12.

Suppose $(E,\leq)$ is a poset and $A,B$, and $D$ are subsets of $E$ such that
$E=A\cup B$, $D=A\cap B$, and for every $a\in A$ and every $b\in B$ the
following conditions hold.

  1. (1)

$a\leq b$ if and only if $a\leq d$ and $d\leq b$ for some $d\in D$, and

  2. (2)

$a\geq b$ if and only if $a\geq d$ and $d\geq b$ for some $d\in D$.

Then $\mathbb{H}_{D}$ forces that ($\dot{G}$ is the canonical name for the
generic filter in $\mathbb{H}_{D}$) $\mathbb{H}_{E}(D,\Upsilon_{\dot{G}})$ and
$\mathbb{H}_{A}(D,\Upsilon_{\dot{G}})\times\mathbb{H}_{B}(D,\Upsilon_{\dot{G}})$
are forcing equivalent.

With the assumptions of Lemma 4.12 it can be proved that the function

$\Xi\colon\mathbb{H}_{E}(D,\Upsilon_{\dot{G}})\to\mathbb{H}_{A}(D,\Upsilon_{\dot{G}})\times\mathbb{H}_{B}(D,\Upsilon_{\dot{G}})$

defined by $\Xi(p):=(\pi_{A}(p),\pi_{B}(p))$ is a dense embedding, but we will
not need this fact.

###### Proof.

We use the notation from Lemma 4.10 and write $\dot{G}(X)$ for the canonical
name for the generic filter for $\mathbb{H}_{X}$ (or
$\mathbb{H}_{X}(Y,\Upsilon)$ for some $Y$ and $\Upsilon$) where $X$ is
$A,B,D$, or $E$.

By Lemma 4.10 (1) with $E$ and $A$ in place of $E^{\prime}$ and $E$,
$\mathbb{H}_{E}$ is forcing equivalent to
$\mathbb{H}_{A}*\mathbb{H}_{E}(A,\Upsilon_{\dot{G}(A)})$. By the same lemma
with $A$ and $D$ in place of $E^{\prime}$ and $E$, $\mathbb{H}_{A}$ is forcing
equivalent to $\mathbb{H}_{D}*\mathbb{H}_{A}(D,\Upsilon_{\dot{G}(D)})$.
Therefore $\mathbb{H}_{E}$ is forcing equivalent to the iteration

(4.1)
$\mathbb{H}_{D}*\mathbb{H}_{A}(D,\Upsilon_{\dot{G}(D)})*\mathbb{H}_{E}(A,\Upsilon_{\dot{G}(A)}).$

The assumptions imply that $L(b)\cap D$ is cofinal in $L(b)$ and $R(b)\cap D$
is coinitial in $R(b)$, for every $b\in B$. Therefore by Lemma 4.10 (3)
applied with $\mathbb{H}_{E}(A,\Upsilon_{\dot{G}(D)})$, $E$, $A$, and $D$ in
place of $\mathbb{H}_{E^{\prime}}(E,\Upsilon_{\dot{G}})$, $E^{\prime}$, $E$,
and $X$, we conclude that $\mathbb{H}_{A}$ forces that
$\mathbb{H}_{E}(A,\Upsilon_{\dot{G}(A)})$ is forcing equivalent to
$\mathbb{H}_{B}(D,\Upsilon_{\dot{G}(D)})$ (also recall that $\mathbb{H}_{D}$
is a regular subordering of $\mathbb{H}_{A}$, and that $\Upsilon_{G(A)}$
extends $\Upsilon_{G(D)}$). Since $\mathbb{H}_{B}(D,\Upsilon_{\dot{G}(D)})$
does not depend on $\dot{G}(A)$, but only on its intersection with
$\mathbb{H}_{D}$, the iteration in (4.1) is forcing equivalent to

$\mathbb{H}_{D}*(\mathbb{H}_{B}(D,\Upsilon_{\dot{G}(D)})\times\mathbb{H}_{A}(A,\Upsilon_{\dot{G}(D)}))$

and $\mathbb{H}_{D}$ forces that $\mathbb{H}_{E}(A,\Upsilon_{\dot{G}(A)})$ is
forcing equivalent to the product of $\mathbb{H}_{B}(D,\Upsilon_{\dot{G}(D)})$
and $\mathbb{H}_{A}(A,\Upsilon_{\dot{G}(D)})$, as claimed. ∎

In the proof of Theorem 4.13 below, for $f$ and $g$ in $C^{\mathbb{N}}$ (with
$(C,\triangleleft)$ as guaranteed by Lemma 2.2) we will write
$f\triangleleft^{n}g$ if $f(j)\triangleleft g(j)$ for all $j\geq n$. A proof
of Theorem 4.13 is analogous to, but shorter than, the proof of [9, Theorem
9.1] (a baroque writeup of this proof with an ample supply of limiting
examples and all sorts of digressions (many of which were warranted) can be
found in [9]).

######  Theorem 4.13.

Suppose $\kappa$ is a regular cardinal such that $\lambda^{\aleph_{0}}<\kappa$
for every cardinal $\lambda<\kappa$ and $E$ is a partial ordering such that
neither $\kappa$ nor $\kappa^{*}$ embeds into $E$. Then $\mathbb{H}_{E}$
forces that $\prod_{\operatorname{Fin}}(A_{n},\triangleleft_{n})$ has no
$\kappa$-chains for any sequence $(A_{n},\triangleleft_{n})$, for
$n\in{\mathbb{N}}$, of countable sets with asymmetric binary relations.

###### Proof.

Since $C$ as in Lemma 2.2 is universal, $\mathbb{H}_{E}$ forces that
$\prod_{\operatorname{Fin}}(A_{n},\triangleleft_{n})$ has a $\kappa$-chain for
some sequence $(A_{n},\triangleleft_{n})$ (not necessarily in the ground
model) if and only if $\mathbb{H}_{E}$ forces that
$(C^{\mathbb{N}},\triangleleft^{*}):=\prod_{\operatorname{Fin}}(C,\triangleleft)$
has a $\kappa$ chain. It will therefore suffice to prove the theorem with the
additional assumption that $(A_{n},\triangleleft_{n})=(C,\triangleleft)$ for
all $n$.

Assume that $\dot{f}_{\xi}$, for $\xi<\kappa$, is a name for a $\kappa$-chain
in $(C^{\mathbb{N}},\triangleleft^{*})$. (We emphasize that this means that
for all $\xi<\eta$, $\mathbb{H}_{E}$ forces both
$\dot{f}_{\xi}\triangleleft^{*}\dot{f}_{\eta}$ and
$\dot{f}_{\eta}\ntriangleleft^{*}\dot{f}_{\xi}$.) The ccc-ness of
$\mathbb{H}_{E}$ implies that for every limit ordinal $\xi$ there exists a
countable $E(\xi)\subseteq E$ such that $\dot{f}_{\xi}$, $\dot{f}_{\xi+1}$,
and $\dot{f}_{\xi+2}$ are $\mathbb{H}_{E(\xi)}$-names. Since $\kappa$ is
regular and $\lambda^{\aleph_{0}}<\kappa$ for all $\lambda<\kappa$, the
$\Delta$-system lemma for countable sets implies that any family of $\kappa$
countable sets includes a $\Delta$-system of cardinality $\kappa$. By passing
to a subfamily, we may assume that the sets $E(\xi)$ form a $\Delta$-system
with a countable root $A$.

For every limit ordinal $\xi$ fix $q_{\xi}\in\mathbb{H}_{E}$ and
$n\in{\mathbb{N}}$ such that

(4.2)
$q_{\xi}\Vdash_{\mathbb{H}_{E}}\dot{f}_{\xi}\triangleleft^{n}\dot{f}_{\xi+1}\triangleleft^{n}\dot{f}_{\xi+2}.$

By the pigeonhole principle and passing to a subfamily if necessary, we may
assume that $n$ as in (4.2) is the same for all $\xi$. Lemma 4.7 implies that
$\mathbb{H}_{E(\xi)}$ is a regular subordering of $\mathbb{H}_{E}$. We may
therefore assume that $q_{\xi}\in\mathbb{H}_{E(\xi)}$ and we have

$q_{\xi}\Vdash_{\mathbb{H}_{E(\xi)}}\dot{f}_{\xi}\triangleleft^{n}\dot{f}_{\xi+1}\triangleleft^{n}\dot{f}_{\xi+2}.$

Writing $q_{\xi}=(D_{\xi},n_{\xi},f_{\xi})$, let $F(\xi):=D_{\xi}\setminus A$.
Note that each $F(\xi)$ is finite.

Fix $\xi<\kappa$ for a moment and fix a generic filter
$G\subseteq\mathbb{H}_{A\cup F(\xi)}$ such that $q_{\xi}\in G$. Recall that
the domain of $C$ is ${\mathbb{N}}$, and is therefore equipped with a well-
ordering (as a matter of fact, the first well-ordering known to man). For
$j\in{\mathbb{N}}$ we can let $h_{\xi}(j)$ be the least element of $C$ (in
this well-ordering) such that

$r\Vdash\dot{f}_{\xi+1}(j)=\check{c}$

for some $r$ in the quotient $\mathbb{H}_{E}(A\cup F(\xi),\Upsilon_{G})/G$
(see Lemma 4.10).

This defines $h_{\xi}\in C^{\mathbb{N}}$ in $V[G]$. Use the Maximality
Principle ([32, Theorem IV.7.1]) to choose an $\mathbb{H}_{A\cup F(\xi)}$-name
$\dot{h}_{\xi}$ for this function.

######  Claim 4.14.

The condition $q_{\xi}$ forces that
$\dot{f}_{\xi}\triangleleft^{n}\dot{h}_{\xi}\triangleleft^{n}f_{\xi+2}$.

###### Proof.

If there are $r\leq q_{\xi}$ in $\mathbb{H}_{E}$ and $j\geq n$ such that
$r\Vdash\dot{f}_{\xi}(j)\ntriangleleft\dot{h}_{\xi}(j)$, fix a generic filter
$G$ in $\mathbb{H}_{E}$ containing $r$. Then in $V[G]$ we have
$\dot{f}_{\xi}\ntriangleleft^{n}\dot{f}_{\xi+1}$, although $q_{\xi}\in G$;
contradiction. An analogous argument gives that there is no $j\geq n$ such
that some $r\leq_{E}q_{\xi}$ forces that
$\dot{h}_{\xi}(j)\ntriangleleft\dot{f}_{\xi+2}(j)$. ∎

The pairs $(q_{\xi},\dot{h}_{\xi})$ are indexed by limit ordinals below
$\kappa$. We re-enumerate them preserving the order and obtain conditions
$q_{\xi}$ and names $\dot{h}_{\xi}$ for $\xi<\kappa$. Since $\mathbb{H}_{E}$
has the ccc, some condition $q\in\mathbb{H}_{E}$ forces that $\kappa$ of the
$q_{\xi}$’s belong to the generic filter. Therefore $q$ forces that the family
of all $h_{\xi}$ such that $q_{\xi}$ belongs to the generic filter is a
$\kappa$-chain in $(C^{\mathbb{N}},\triangleleft^{*})$.

By Proposition 3.9, there exists a cofinal $X\subseteq\kappa$ such that for
any two distinct elements $\xi<\eta$ of $X$ there is $s\Subset\kappa$ such
that $\\{\xi,\eta\\}=\\{\min(s),\max(s)\\}$ and there is no $s$-walk with
endpoints in $F(\xi)$ and $F(\eta)$.

Fix $\xi\in X$, let $\xi^{\prime}:=\min(X\setminus(\xi+1))$, and fix
$s\Subset\kappa$ such that $\\{\xi,\xi^{\prime}\\}=\\{\min(s),\max(s)\\}$ and
there is no $s$-walk with endpoints in $F(\xi)$ and $F(\xi^{\prime})$. We will
analyze the relation between the names $\dot{h}_{\xi}$ and
$\dot{h}_{\xi^{\prime}}$.

Let $\xi(j)$, for $j<n$,111111This $n$ is unrelated to the $n$ appearing in
4.2. No danger of confusion here. be an increasing enumeration of $s$, so that
in particular $\xi(0)=\xi$ and $\xi(n-1)=\xi^{\prime}$. Consider the depletion
$\ll_{s}$ of $\leq_{E}$ given by $A$ and $F(\eta)$, for $\eta\in s$. By Lemma
3.4, $\ll_{s}$ is a partial ordering on

$A^{\prime}:=A\cup\bigcup_{j<n}F(\xi(j)).$

For $i<j<n$ let

$A(i,j):=A\cup F(\xi(i))\cup F(\xi(j)),$

ordered by $\ll_{s}$. Note that if $|i-j|>1$ then $A(i,j)$ is ordered by a
restriction of the depletion associated with $s$ which can differ from the
natural depletion $\ll_{(\xi(i),\xi(j))}$ (Remark 3.7). However, for $i<n-1$,
Lemma 3.8 implies that the restriction of $\ll_{s}$ to $A(i,i+1)$ agrees with
the depletion $\ll_{\\{\xi(i),\xi(i+1)\\}}$, which by Lemma 3.2 agrees with
the ordering induced from $E$. By Lemma 4.7, we have
$\mathbb{H}_{A(i,i+1)}\lessdot\mathbb{H}_{E}$.

Since $A^{\prime}$ is ordered by $\ll_{s}$, which possibly disagrees with the
ordering of $A^{\prime}$ inherited from $E$, the posets
$\mathbb{H}_{A^{\prime}}$ and $\mathbb{H}_{E}$ are possibly unrelated. On the
other hand, each $A(i,j)$ is a subordering of $A^{\prime}$ and therefore Lemma
4.7 implies $\mathbb{H}_{A(i,j)}\lessdot\mathbb{H}_{A^{\prime}}$.

Since $\dot{h}_{\xi(i)}$ and $\dot{h}_{\xi(j)}$ are
$\mathbb{H}_{A(i,j)}$-names and $\mathbb{H}_{E}$ forces
$\dot{h}_{\xi(i)}\triangleleft^{*}\dot{h}_{\xi(i+1)}$ for all $i<n-1$, we have
that $\mathbb{H}_{A^{\prime}}$ forces
$\dot{h}_{\xi(i)}\triangleleft^{*}\dot{h}_{\xi(i+1)}$ for all $i<n-1$. By
transitivity, $\mathbb{H}_{A^{\prime}}$ forces
$\dot{h}_{\xi(0)}\triangleleft^{*}\dot{h}_{\xi(n-1)}$. Since
$\dot{h}_{\xi(0)}$ and $\dot{h}_{\xi(n-1)}$ are $\mathbb{H}_{A(0,n-1)}$-names
and $\mathbb{H}_{A(0,n-1)}\lessdot\mathbb{H}_{A^{\prime}}$, we have that
$\mathbb{H}_{A(0,n-1)}$ forces
$\dot{h}_{\xi(0)}\triangleleft^{*}\dot{h}_{\xi(n-1)}$.

Since there is no $s$-walk whose endpoints are some $x\in F(\xi(0))$ and some
$y\in F(\xi(n-1))$, we have that $x\ll_{s}y$ implies there is $a\in A$ such
that $x\leq a\leq y$, and that $y\ll_{s}x$ imples there is $a\in A$ such that
$y\leq a\leq x$. This means that the assumptions of Lemma 4.12 with $E$, $A$,
$B$, and $D$ replaced with $A\cup F(\xi(0))\cup F(\xi(n-1))$, $A\cup
F(\xi(0))$, $A\cup F(\xi(n-1))$, and $A$, respectively, (sorry!) are
satisfied. Therefore if $G\subseteq\mathbb{H}_{A}$ is a generic filter then
the quotient $\mathbb{H}_{A\cup A(0,n-1)}/G$ is forcing-equivalent to the
product

$\mathbb{H}_{A\cup F(\xi(0)}(A,\Upsilon_{G})\times\mathbb{H}_{A\cup
F(\xi(n-1))}(A,\Upsilon_{G}).$

Most importantly, the names $\dot{h}_{\xi(0)}$ and $\dot{h}_{\xi(n-1)}$ are
added by the two factors of this product. By Lemma 4.11, there exist a
condition $p_{\xi}\in\mathbb{H}_{A}$ and an $\mathbb{H}_{A}$-name
$\dot{g}_{\xi}$ (recall that $\xi=\xi(0)$) such that

$p_{\xi}\Vdash\dot{h}_{\xi(0)}\leq^{*}\dot{g}_{\xi}\leq^{*}\dot{h}_{\xi(n-1)}.$

Since $\mathbb{H}_{A}$ is countable, there is $q\in\mathbb{H}_{A}$ such that
$Y=\\{\xi\in X|p_{\xi}=q\\}$ is a cofinal subset of $X$ (and of $\kappa$).
Therefore $q$ forces that $\mathbb{H}_{A}$ adds a strictly increasing
$\kappa$-chain $\dot{g}_{\xi}$, for $\xi<\kappa$, to
$({\mathbb{N}}^{{\mathbb{N}}},\leq^{*})$. Since $A$ is countable,
$\mathbb{H}_{A}$ cannot add more than $\mathfrak{c}$ reals; contradiction. ∎

The robustness of the robust order property (Definition 1.2) is used in the
following proposition.

######  Proposition 4.15.

Suppose that the pair $(T,\varphi)$ has the robust order property and $E$ is
any poset. Then $\mathbb{H}_{E}$ forces the following.

  1. (1)

The poset $E$ embeds into
$\prod_{\operatorname{Fin}}(A_{n},\triangleleft_{\varphi})$ for every sequence
$(A_{n})$ of models of $T$.

  2. (2)

For any nonprincipal ultrafilter $\mathcal{U}$ on ${\mathbb{N}}$ there is
strictly increasing map from $E$ into
$\prod_{\mathcal{U}}(A_{n},\triangleleft_{\varphi})$ whose range is linearly
ordered by $\triangleleft_{\varphi}$.

###### Proof.

The first part is almost obvious, but proving it in some detail will also
provide a proof of the second part.

By Lemma 2.1, there are $\eta\in{\mathbb{N}}^{{\mathbb{N}}}$ and
$\Phi\colon\prod_{\operatorname{Fin}}(n,\leq^{*})\to\prod_{\operatorname{Fin}}(\eta(n),<^{*})$
such that $f\leq^{*}g$ and $g\nleq^{*}f$ implies
$(\forall^{\infty}n)\Phi(f)(n)<\Phi(g)(n)$. Since $A_{n}$ is a model of $T$,
there exists a $\triangleleft_{\varphi}$-chain $C_{n}$ of length $\eta(n)$ in
$A_{n}$. By identifying this chain with $(\eta(n),\leq)$, we obtain
$\Phi\colon\prod_{\operatorname{Fin}}(n,\leq^{*})\to\prod_{\operatorname{Fin}}(A_{n},\triangleleft_{\varphi})$
such that $f\leq^{*}g$ and $g\nleq^{*}f$ implies
$\Phi(f)(n)\triangleleft_{\varphi}\Phi(g)(n)$ and
$\Phi(g)(n)\ntriangleleft_{\varphi}\Phi(f)(n)$ for all but finitely many $n$.
By composing the embedding of $E$ into
$\prod_{\operatorname{Fin}}(n,\leq^{*})$ provided by Theorem 4.1 with $\Phi$,
we obtain an $\mathbb{H}_{E}$-name for an embedding of $\Xi\colon
E\to\prod_{\operatorname{Fin}}(A_{n},\triangleleft_{\varphi})$ (this proves
the first part; read on for the proof of the second part) that in addition has
the property that $a<_{E}b$ implies

$(\forall^{\infty}n)(\Xi(f)(n)\triangleleft_{\varphi}\Xi(g)(n)\text{ and
}\Xi(g)(n)\ntriangleleft_{\varphi}\Xi(f)(n)).$

Let $\mathcal{U}$ be a nonprincipal ultrafiter on ${\mathbb{N}}$ and let
$\pi_{\mathcal{U}}$ denote the quotient map from
$\prod_{\operatorname{Fin}}A_{n}$ to $\prod_{\mathcal{U}}A_{n}$. Then the
displayed formula implies that the restriction of $\pi_{\mathcal{U}}$ to
$\Xi[E]$ is strictly increasing. The range of this map is the ultraproduct of
the $\triangleleft_{\varphi}$-chains $C_{n}$, and therefore linearly ordered
by Łoś’s Theorem. This proves the second part. ∎

##  5\. Proofs of Theorem C and Corollary D

The proof of Theorem C will use the following result (see [9, Theorem 3.2] for
a proof) .

######  Theorem (Galvin).

For every uncountable cardinal $\kappa$ there exists a partial ordering
$E_{\kappa}$ such that $E_{\kappa}$ has no infinite chains but for every
linear ordering $\mathcal{L}$ such that there are neither $\kappa$-chains nor
$\kappa^{*}$-chains in $\mathcal{L}$ there is no strictly increasing map
$\Phi\colon E\to\mathcal{L}$. ∎

###### Proof of Theorem C.

Fix a theory $T$ with the robust order property. We will prove that the Levy
collapse of the continuum to $\aleph_{1}$ followed by $\mathbb{H}_{E}$ for $E$
provided by Galvin’s theorem forces both conclusions of Theorem C.

These proofs have a common initial segment that we now present.

In the extension by the Levy collapse of the continuum to $\aleph_{1}$, let
$\kappa>\mathfrak{c}$ be a regular cardinal ($\kappa=\aleph_{2}$ will do). Let
$E$ be the poset as guaranteed by Galvin’s theorem stated at the beginning of
this section. We will prove that the Levy collapse followed by
$\mathbb{H}_{E}$ is the forcing notion as promised in the statement of Theorem
C.

Fix an ultrafilter $\mathcal{U}$ on ${\mathbb{N}}$, a sequence $(A_{n})$ of
countable structures in the language of $T$, and a sequence $(B_{n})$ of
countable models of $T$. Proposition 4.15 implies that $\mathbb{H}_{E}$ adds a
strictly increasing map from $E$ into
$\prod_{\mathcal{U}}(B_{n},\triangleleft_{\varphi})$ whose range is linearly
ordered by $\triangleleft_{\varphi}$. By the choice of $E$, there exists a
$\kappa$-$\triangleleft_{\varphi}$-chain or a
$\kappa^{*}$-$\triangleleft_{\varphi}$-chain in
$\prod_{\mathcal{U}}(B_{n},\triangleleft_{\varphi})$.

On the other hand, by Theorem 4.1 there are neither $\kappa$-chains nor
$\kappa^{*}$-chains in
$\prod_{\operatorname{Fin}}(A_{n},\triangleleft_{\varphi})$.

From this point on the proofs of (1) and (2) differ.

(1) We need to prove that $\prod_{\mathcal{U}}B_{n}$ is not isomorphic to an
elementary submodel of $\prod_{\operatorname{Fin}}A_{n}$. Since elementary
embeddings preserve $\triangleleft_{\varphi}$, this is immediate from the fact
that the former contains a $\kappa$\- or
$\kappa^{*}$-$\triangleleft_{\varphi}$-chain and the latter does not.

(2) Suppose in addition that $\varphi$ is quantifier-free. Then all embeddings
preserve $\triangleleft_{\varphi}$, and $\prod_{\mathcal{U}}B_{n}$ cannot be
isomorphic to a submodel of $\prod_{\operatorname{Fin}}A_{n}$. ∎

###### Proof of Corollary D.

Suppose that $A$ is a separable $C^{*}$-algebra and $\mathcal{U}$ is an
ultrafilter on ${\mathbb{N}}$. If $\mathcal{U}$ is principal, then $(A\otimes
C(K))^{\mathcal{U}}$ is isomorphic to $A\otimes C(K)$ while $A^{\infty}$ is
nonseparable. We may therefore assume that $A$ is infinite-dimensional and
that $\mathcal{U}$ is nonprincipal.

The theory of infinite-dimensional $C^{*}$-algebras has the order property
witnessed by an atomic formula ([16, Lemma 5.3]). Therefore the theory of
$A\otimes C(K)$ has the robust order property, and Theorem C (2) implies that
$(A\otimes C(K))^{\mathcal{U}}$ does not embed into $B^{\infty}$ for any
$C^{*}$-algebra $B$. ∎

##  6\. Proofs of Theorem A and Theorem B: Tie points

The contents of this section is rather accurately described by its title.

######  Definition 6.1.

Suppose $X$ is a compact Hausdorff space. A point $x\in X$ is a _tie point_ if
there are closed subsets $A$ and $B$ of $X$ such that $A\cup B=X$ and $A\cap
B=\\{x\\}$ (in symbols, $A\bowtie_{x}B$).

Two subsets ${\mathcal{I}}$ and ${\mathcal{J}}$ of a Boolean algebra
$\mathfrak{B}$ are _orthogonal_ if $a\wedge b=0_{\mathcal{B}}$ for all
$a\in{\mathcal{I}}$ and all $b\in{\mathcal{J}}$. The following is proved by
parsing the definitions.

######  Proposition 6.2.

Suppose $\mathfrak{B}$ is a Boolean algebra. The following are equivalent for
an ultrafilter $\mathcal{U}$ on $\mathfrak{B}$.

  1. (1)

The complement of $\mathcal{U}$ is equal to the union of two orthogonal
ideals.

  2. (2)

$\mathcal{U}$ is a tie-point in the Stone space of $\mathfrak{B}$. ∎

######  Definition 6.3.

By analogy with true P-points, an ultrafilter $\mathcal{U}$ in a Boolean
algebra is called a _true tie point_ if the ideals as in Proposition 6.2 (2)
can be chosen so that each one of them is generated by a linearly ordered
subset.

The salient point of the proof of the following is the observation that true
tie points are $\Sigma_{1}$-definable, but the reader may choose to ignore
this remark and read the proof instead.

######  Lemma 6.4.

Every ultraproduct of countable atomless Boolean algebras has a true tie
point.

###### Proof.

Every ultrafilter in a countable atomless Boolean algebra is a true tie point,
since the generating sets of order type $\omega$ can be chosen by recursion.
Suppose $\prod_{\mathcal{U}}C_{n}$ is an ultraproduct of countable atomless
Boolean algebras. If $\mathcal{U}$ is principal, then $\prod_{n}C_{n}$ is
isomorphic to one of the $C_{n}$’s and the assertion follows from the first
sentence of this proof.

Now assume $\mathcal{U}$ is nonprincipal. For every $C_{n}$ fix a true tie
point $p_{n}$ and linearly ordered generating sets ${\mathcal{A}}_{n}$ and
$\mathcal{B}_{n}$ for the ideal $C_{n}\setminus p_{n}$. Then
$(C_{n},{\mathcal{A}}_{n},\mathcal{B}_{n})$ is an expansion of $C_{n}$ to the
language with two additional unary predicates. Each one of these structures
satisfies the following: Both ${\mathcal{A}}_{n}$ and $\mathcal{B}_{n}$ are
linearly ordered, $A\wedge B=\emptyset$ for all $A\in{\mathcal{A}}_{n}$ and
$B\in\mathcal{B}_{n}$, and for every $X\in C_{n}$ either $X$ or its complement
belongs to ${\mathcal{A}}_{n}\cup\mathcal{B}_{n}$. These are all first-order
statements, and they imply that the complement of
${\mathcal{A}}_{n}\cup\mathcal{B}_{n}$ is an ultrafilter.

The ultraproduct
$\prod_{\mathcal{U}}(C_{n},{\mathcal{A}}_{n},\mathcal{B}_{n})$ is an expansion
of $\prod_{n}C_{n}$ and by Łoś’s Theorem the sets
${\mathcal{A}}:=\prod_{\mathcal{U}}{\mathcal{A}}_{n}$ and
$\mathcal{B}:=\prod_{\mathcal{U}}\mathcal{B}_{n}$ generate ideals of
$\prod_{\mathcal{U}}C_{n}$ whose complement is a true tie point. ∎

###### Proof of Theorem A.

We need to prove that PFA implies
$\mathcal{P}({\mathbb{N}})/\operatorname{Fin}$ is not isomorphic to an
ultraproduct of Boolean algebras associated with a nonprincipal ultrafilter on
${\mathbb{N}}$. By [50] (see [7, Corollary 1.9]), PFA implies that there are
no tie points in $\mathcal{P}({\mathbb{N}})/\operatorname{Fin}$, while there
are tie points in an ultraproduct of countable atomless Boolean algebras by
Lemma 6.4. ∎

The following will be used in the proof of Theorem B.

######  Lemma 6.5.

The poset for adding at least $\mathfrak{c}^{+}$ Cohen reals forces that every
projectively definable atomless Boolean algebra $\mathfrak{B}$ has no true tie
points.

###### Proof.

Suppose that $\kappa\geq\mathfrak{c}^{+}$ and let $\mathbb{C}_{\kappa}$ denote
the poset for adding $\kappa$ Cohen reals. Fix an $n\in{\mathbb{N}}$ and
$\mathbf{\Sigma}^{1}_{n}$-formulas $\varphi$, $\varphi_{\wedge}$,
$\varphi_{\vee}$, and $\varphi_{\setminus}$, which define $\mathfrak{B}$. We
will only need the $\mathbf{\Delta}^{1}_{n+1}$ formula $\varphi_{<}$, that
defines the relation $a<b$ in $\mathfrak{B}$. By passing to an intermediate
forcing extension, without a loss of generality we may assume that the reals
coding these formulas belong to the ground model.

By genericity, no nonprincipal ultrafilter on ${\mathbb{N}}$ in the forcing
extension is generated by fewer than $\kappa$ subsets of ${\mathbb{N}}$. (This
is well-known, but here is a sketch of the proof: After adding $\kappa$ Cohen
reals, for every ${\mathcal{X}}\subseteq\mathcal{U}$ of cardinality less than
$\kappa$ there is a Cohen real $Y$ generic over $V[{\mathcal{X}}]$. For every
infinite $X\subseteq{\mathbb{N}}$, the set of all $Y\subseteq{\mathbb{N}}$
such that $X\cap U$ and $X\setminus Y$ are both infinite is comeager.
Therefore ${\mathcal{X}}$ does not ‘decide’ whether $Y\in\mathcal{U}$ or
${\mathbb{N}}\setminus Y\in\mathcal{U}$.) Assume $p$ is a true tie point in
$\mathfrak{B}$ and let ${\mathcal{A}}$ and $\mathcal{B}$ be the linearly
ordered (modulo ${\mathcal{I}}$) sets whose complements generate
$\mathfrak{B}\setminus p$. By genericity, at least one of ${\mathcal{A}}$ and
$\mathcal{B}$ has cofinality greater than $\mathfrak{c}$. By interchanging
${\mathcal{A}}$ and $\mathcal{B}$, we may assume that the cofinality of
${\mathcal{A}}$ is $\kappa>\mathfrak{c}$.

The proof is completed by Kunen’s isomorphism of names argument ([31]; see §8
for disambiguation) that we now sketch.

Suppose that $\dot{f}_{\xi}$, for $\xi<\kappa$, is a
$\mathbb{C}_{\kappa}$-name for a strictly increasing chain cofinal in
${\mathcal{A}}$. In particular, $f_{\xi}$, for $\xi<\mathfrak{c}^{+}$, is a
name for a strictly increasing chain in ${\mathcal{A}}/{\mathcal{I}}$. This
will suffice to obtain a contradiction. Since each $\dot{f}_{\xi}$ is a name
for a real, it is coded by a sequence of antichains and the union, denoted
$D_{\xi}$, of the supports of all conditions in these antichains is a
countable subset of $\kappa$. Since $\mathfrak{c}^{+}$ is regular and
$\lambda<\mathfrak{c}^{+}$ implies $\lambda^{\aleph_{0}}<\mathfrak{c}^{+}$, by
passing to a cofinal subset we may assume that the sets $D_{\xi}$ form a
$\Delta$-system with root $R$. By a counting argument and passing to a cofinal
subset again we may assume that the restrictions of $\dot{f}_{\xi}$ to $R$
agree, and that $\dot{f}_{\xi}$ and $\dot{f}_{\eta}$ are isomorphic for all
$\xi$ and $\eta$. This means that for $\xi<\eta$ there is an automorphism
$\Phi_{\xi\eta}$ of $\mathbb{C}_{\kappa}$ that sends $\dot{f}_{\xi}$ to
$\dot{f}_{\eta}$, for any two $\xi<\eta<\mathfrak{c}^{+}$. However, since
$\mathbb{C}_{\kappa}$ forces that $\varphi_{<}(\dot{f}_{\xi},\dot{f}_{\xi})$
is true and the real coding the asymmetric formula $\varphi_{<}$ is in the
ground model, this is a contradiction that completes the proof. ∎

###### Proof of Theorem B.

If ${\mathcal{I}}$ does not include the Frèchet filter then
$\mathcal{P}({\mathbb{N}})/{\mathcal{I}}$ has atoms, and therefore cannot be
isomorphic to $\mathcal{P}({\mathbb{N}})/\operatorname{Fin}$. We may therefore
assume that ${\mathcal{I}}$ incudes the Frèchet filter.

In the model obtained by adding at least $\mathfrak{c}^{+}$ Cohen reals to a
model of ZFC, suppose that $\mathfrak{B}$ is a projectively definable Boolean
algebra. By Lemma 6.5, there are no true tie points in $\mathfrak{B}$. By
Lemma 6.4, in every model of $\operatorname{ZFC}$ there is a true tie point in
any ultraproduct of countable atomless Boolean algebras. ∎

##  7\. The existence of universal ultrapowers

Hitherto unbeknownst to the junior (!) author, some questions closely related
to those resolved in our main results have easy answers, collected in this
section.

Suppose that $T$ is a theory in a countable (or separable) language and let

$\displaystyle\operatorname{Ult}_{T}$
$\displaystyle=\\{A^{\mathcal{U}}|A\models T,A\text{ is countable (separable),
and }\mathcal{U}\in\beta{\mathbb{N}}\setminus{\mathbb{N}}\\},$
$\displaystyle\operatorname{Mod_{\mathfrak{c}}}(T)$
$\displaystyle=\\{A|A\models T,|A|=\mathfrak{c}\\}.$

CH implies that all $M\in\operatorname{Ult}_{T}$ are saturated, and therefore
isomorphic. This conclusion is by [17, Theorem 5.6] (also [19]) equivalent to
CH. In some applications it suffices to know that among the ultrapowers of a
model $A$ of $T$ there exists one which is (injectively) universal. We’ll say
that a set of models has a $\preceq$-_universal element_ if it has an element
universal under elementary embeddings, and that it has a
$\hookrightarrow$-_universal element_ if it has an element universal under not
necessarily elementary embeddings.

As pointed out in [47, p. 181], this sort of nitpicking (distinguishing
between $\hookrightarrow$ and $\preceq$) is in general unnecessary, since one
can make the difference disappear by expanding the language by Skolem
functions. We nevertheless nitpick because the existence of Skolem functions
in continuous logic is a delicate problem.

######  Proposition 7.1.

Suppose that $T$ is a first-order theory. Then $\operatorname{Ult}_{T}$ has a
$\preceq$-universal ($\hookrightarrow$-universal) element if and only if
$\operatorname{Mod_{\mathfrak{c}}}(T)$ does.

###### Proof.

It is well-known that every model of $T$ of cardinality $\mathfrak{c}$ is
isomorphic to an elementary submodel of an ultrapower of a countable model of
$T$. This follows from the results of [40, Chapter VI.5] or [19]. Therefore
any $M\in\operatorname{Ult}_{T}$ which is $\preceq$-universal is also
$\preceq$-universal for $\operatorname{Mod_{\mathfrak{c}}}(T)$, and similarly
for $\hookrightarrow$-universality. ∎

######  Proposition 7.2.

If $T$ is a stable theory in a countable language, then
$\operatorname{Ult}_{T}$ has a $\preceq$-universal, and therefore a
$\hookrightarrow$-universal, model (in ZFC).

###### Proof.

In [17, Theorem 5.6] it was proved that if $T$ is stable then all ultrapowers
$A^{\mathcal{U}}$ for a countable (or separable) $A$ and
$\mathcal{U}\in\beta{\mathbb{N}}\setminus{\mathbb{N}}$ are saturated, and
therefore isomorphic. ∎

We can therefore assume that $T$ is not stable, or equivalently, that it has
the order property (for the continuous case, this equivalence is in [17,
Theorem 5.5], generalizing classical result of the second author, [40]).

######  Question 7.3.

Suppose that $T$ is a theory in a countable language with the order property
and CH fails. Can $\operatorname{Mod_{\mathfrak{c}}}(T)$ (equivalently,
$\operatorname{Ult}_{T}$) have a $\preceq$-universal, or a
$\hookrightarrow$-universal, model?

We give some partial answers to this question. The strict order property (SOP)
of $T$ is the strengthening of the order property in which the witnessing
formula $\varphi$ is required to define a (partial) ordering on every model of
$T$.

######  Proposition 7.4.

Suppose that $T$ is a theory in a countable language with the SOP. If there
exists a cardinal $\kappa$ such that
$\kappa^{+}<\mathfrak{c}=\operatorname{cf}(\mathfrak{c})<2^{\kappa}$ and $T$
is complete, then $\operatorname{Ult}_{T}$ does not have a $\preceq$-universal
model.

If the SOP is witnessed by a quantifier-free formula, then even if $T$ is not
complete, $\operatorname{Ult}_{T}$ does not even have a
$\hookrightarrow$-universal model.

###### Proof.

The existence of $\kappa$ as stated implies there is no
$\hookrightarrow$-universal linear order of cardinality $\mathfrak{c}$ by [28,
Theorem 3.10]. By [28, Theorem 5.5], this implies that
$\operatorname{Mod_{\mathfrak{c}}}(T)$ has no $\preceq$-universal element. As
explained in [28], if the SOP is witnessed by a quantifier-free formula, then
$\operatorname{Mod_{\mathfrak{c}}}(T)$ has no $\hookrightarrow$-universal
element. ∎

The SOP4 ([43, Definition 2.5]) is a technical weakening of the strict order
property, hence the following is a strengthening of Proposition 7.4.

######  Proposition 7.5.

Suppose that $T$ is a theory in a countable language with the SOP4. If there
exists a cardinal $\kappa$ such that
$\kappa^{+}<\mathfrak{c}=\operatorname{cf}(\mathfrak{c})<2^{\kappa}$ and $T$
is complete, then $\operatorname{Ult}_{T}$ does not have a $\preceq$-universal
model.

If the SOP4 is witnessed by a quantifier-free formula, then even if $T$ is not
complete, $\operatorname{Ult}_{T}$ does not even have a
$\hookrightarrow$-universal model.

###### Proof.

This is [43, Theorem 2.12]. ∎

The _olive property_ is a collection of properties of a first-order theory
introduced in [46, Definition 1.8 and Definition 2.1]. One talks about the
$(\Delta,\eta,k,m)$-olive property, but for our purposes, $\Delta$ is the set
of all formulas in the language of $T$ if $T$ is complete or the set of all
quantifier-free formulas otherwise. The parameter $m$ is the arity of the
tuples witnessing the order property and can be suppressed. The roles of
$\eta$ and $k$ are laid out in [46, Definition 2.1], and we will say that ‘$T$
has the olive property’ if it has the $(\eta,k)$-olive property for some
$\eta\in\\{0,1\\}^{n}$ and $k$. We only remark that by [46, Theorem 3.1] the
theory of groups has the olive property (by [49] the theory of groups fails
SOP4), hence (since the olive property for groups is witnessed by quantifier-
free formulas), the following applies to it.

######  Proposition 7.6.

Suppose that $T$ is a theory in a countable language with the olive property.
If there exists a cardinal $\kappa$ such that
$\kappa^{+}<\mathfrak{c}=\operatorname{cf}(\mathfrak{c})<2^{\kappa}$ and $T$
is complete, then $\operatorname{Ult}_{T}$ does not have a $\preceq$-universal
model.

If the olive property is witnessed by a quantifier-free formulas, then even if
$T$ is not complete, $\operatorname{Ult}_{T}$ does not even have a
$\hookrightarrow$-universal model.

###### Proof.

This is [46, Theorem 2.9 (i) and (ii)]. For the quantifier-free case, use part
(ii) and the $\lambda-(\eta,k,m)$-olive property from [46, Definition 2.3
(ii)]. ∎

With a strengthened cardinal arithmetic assumption one can say more. For a
cardinal $\mu$, a set of models ${\mathcal{A}}$ is said to have a _$\preceq$
-basis of cardinality $\mu$_ if there is $\mathbb{B}\subseteq{\mathcal{A}}$ of
cardinality $\mu$ such that every $A\in{\mathcal{A}}$ elementarily embeds into
some element of $\mathbb{B}$. It is said to have a _$\hookrightarrow$ -basis
of cardinality $\mu$_ if there is $\mathbb{B}\subseteq{\mathcal{A}}$ of
cardinality $\mu$ such that every $A\in{\mathcal{A}}$ embeds (not necessarily
elementarily) into some element of $\mathbb{B}$.

######  Proposition 7.7.

Suppose that $T$ is a theory in a countable language with the SOP, SOP4, or
the olive property. If there exists a cardinal $\kappa$ such that
$\kappa^{+}<\mathfrak{c}=\operatorname{cf}(\mathfrak{c})$ and
$\mathfrak{c}^{+}<2^{\kappa}$ and $T$ is complete, then
$\operatorname{Mod_{\mathfrak{c}}}$ does not have a $\preceq$-basis of
cardinality less than $2^{\kappa}$.

If the SOP, the SOP4, or the olive property, is witnessed by quantifier-free
formulas, then even if $T$ is not complete,
$\operatorname{Mod_{\mathfrak{c}}}(T)$ does not have a $\hookrightarrow$-basis
of cardinality less than $2^{\kappa}$.

###### Proof.

We will prove the $\preceq$-case, starting with the following.

######  Claim 7.8.

For every $\mathbb{A}\subseteq\operatorname{Mod_{\mathfrak{c}}}(T)$ with
$|\mathbb{A}|=2^{\kappa}$ there are
$M_{A}\in\operatorname{Mod_{\mathfrak{c}}}(T)$ such that $A\prec M_{A}$ for
$A\in\mathbb{A}$ and for every choice of
$N_{A}\in\operatorname{Mod_{\mathfrak{c}}}(T)$ with $M_{A}\preceq N_{A}$ for
$A\in\mathbb{A}$ there exists $X\subseteq\mathbb{A}$ of cardinality
$2^{\kappa}$ such that $N_{A}$ does not embed into $N_{B}$ for all distinct
$A$ and $B$ in $X$.

###### Proof.

With $\kappa$ as in the assumptions, a function
$\operatorname{inv}\colon\operatorname{Mod_{\mathfrak{c}}}(T)\to[\mathcal{P}(\kappa)]^{\mathfrak{c}}$
with the following properties exists.

  1. (1)

If $M_{0}$ and $M_{1}$ are in $\operatorname{Mod_{\mathfrak{c}}}(T)$ and
$M_{0}$ is elementarily embeddable into $M_{1}$ then
$\operatorname{inv}(M_{0})\subseteq\operatorname{inv}(M_{1})$.

  2. (2)

If the property in question is witnessed by quantifier-free formulas, $M_{0}$
and $M_{1}$ are in $\operatorname{Mod_{\mathfrak{c}}}(T)$, and $M_{0}$ is
embeddable (not necessarily elementarily) into $M_{1}$, then
$\operatorname{inv}(M_{0})\subseteq\operatorname{inv}(M_{1})$.

  3. (3)

If $M_{0}\in\operatorname{Mod_{\mathfrak{c}}}(T)$ and $S\subseteq\kappa$ then
there exists $M_{1}\in\operatorname{Mod_{\mathfrak{c}}}(T)$ such that
$M_{0}\prec M_{1}$ and $S\in\operatorname{inv}(M_{1})$.

For SOP, $\operatorname{inv}$ is $\operatorname{INV}(M,\bar{C})$ for a fixed
$\kappa$-scale $\bar{C}$ (see [28, §3(4)] and [28, Lemma 3.7]). For SOP4, this
is $\operatorname{INV}_{\varphi}(M,\bar{C})$, where $\varphi$ witnesses SOP4
and $\bar{C}$ is a club system (see [43, Definition 2.13 (b)]). For the olive
property, see [46, Remark 1.9].

Fix $\mathbb{A}\subseteq\operatorname{Mod_{\mathfrak{c}}}(T)$ with
$|\mathbb{A}|=2^{\kappa}$. Let $S_{A}$, for $A\in\mathbb{A}$, be distinct
subsets of $\kappa$. By a realizing types argument and (3), there are
$M_{A}\in\operatorname{Mod_{\mathfrak{c}}}(T)$ such that $A\prec M_{A}$ and
$S_{\xi}\in\operatorname{inv}(M_{A})$. Fix $N_{A}$ such that $M_{A}\preceq
N_{A}$. Since $|\operatorname{inv}(N_{A})|=\mathfrak{c}$ when
$|A|=\mathfrak{c}$, by Hajnal’s free subset theorem ([25, Theorem 1] applied
with $m=2^{\kappa}$ and $n=\mathfrak{c}$ to the function
$A\mapsto\operatorname{inv}(N_{A})$) there exists $X\subseteq\mathbb{A}$ of
cardinality $2^{\kappa}$ such that $S_{A}\notin\operatorname{inv}(N_{B})$ for
all distinct $A$ and $B$ in $X$, and therefore $N_{A}$, for $A\in X$, are as
required. ∎

Suppose towards contradiction that $\operatorname{Mod_{\mathfrak{c}}}$ has a
$\preceq$-basis $\mathbb{B}$ of cardinality less than $2^{\kappa}$. Since
$\operatorname{Ult}_{T}$ has $2^{\mathfrak{c}}$ elements ([19]), we can fix
$\mathbb{A}\subseteq\operatorname{Ult}_{T}$ of cardinality $2^{\kappa}$. With
$M_{A}$, for $A\in\mathbb{A}$, as provided by Claim 7.8, for every $A$ there
is $N_{A}\in\mathbb{B}$ such that $M_{A}\preceq N_{A}$. Since
$|\mathbb{B}|<2^{\kappa}$, the conclusion of Claim 7.8 fails; contradiction.

A proof of the $\hookrightarrow$-case of Proposition is analogous, using the
modification of the Claim in which it is allowed that $N_{\xi}\hookrightarrow
N_{\xi}^{\prime}$. ∎

##  8\. Concluding remarks and questions

The question that initiated the research reported here remains open

######  Question 8.1.

Suppose that there is a nontrivial countable (or separable) structure $A$
whose theory has the order property and $\prod_{\mathcal{U}}A$ is isomorphic
to $\prod_{\operatorname{Fin}}A$ for some
$\mathcal{U}\in\beta{\mathbb{N}}\setminus{\mathbb{N}}$. Does it follow that
the CH holds?

Our main results show that in some models of ZFC in which CH fails the premise
of Question 8.1 fails as well. The methods of [45], [42], and [41] may be
relevant to the possibility of giving negative answer to this question.

Our proof of Theorem B uses the well-known technique introduced by Kunen in
the proof of [31, Theorem 12.7]. It may be worth pointing out that, although
the proof is well-known, the actual statement of the theorem isn’t quite as
well-known as it should be. This theorem asserts that in the standard model
for adding $\kappa>\mathfrak{c}$ Cohen reals no well-ordering of $\mathbb{R}$
belongs to the $\sigma$-algebra generated by arbitrary rectangles on
$\mathbb{R}$. The conclusion of this result is equivalent to the assertion
that there are no $\kappa$-chains in any Borel ordering on a Polish space, and
it is often misstated as the weaker assertion that there are no
$\kappa$-chains in ${\mathbb{N}}^{{\mathbb{N}}}/\operatorname{Fin}$.

The proof of Theorem C uses a forcing notion related to the forcing
$\mathbb{C}_{\kappa}$ for adding $\kappa$ side-by-side Cohen reals and an
analysis of names which is to some extent similar to Kunen’s. (This forcing
belongs to the class of _semicohen_ forcing notions, see [29].) The two
results are however different, since the forcing $\mathbb{H}_{E}$ used in the
proof of Theorem C can add an $\omega_{2}$-chain to some Borel poset
$({\mathbb{N}}^{{\mathbb{N}}},\rho)$ without adding an $\omega_{2}$-chain to
$({\mathbb{N}}^{{\mathbb{N}}},\leq^{*})$ (this has been proved for a close
relative of $\mathbb{H}_{E}$ in [9, Theorem 2.1]).

The argument of the proof of Theorem B works for many other forcings that add
more than $\mathfrak{c}$ reals, as long as one can uniformize the names and
there are no ultrafilters on ${\mathbb{N}}$ with small generating sets in the
extension. The latter does not apply to the Sacks forcing. As a matter of
fact, after adding $\mathfrak{c}^{+}$ Sacks reals to a model of CH with
countable supports (by either countable support product or countable support
iteration), there exists a selective $\aleph_{1}$-generated ultrafilter on
${\mathbb{N}}$, and it is a true tie point ([1]). It is therefore not clear
whether in some of the Sacks models
$\mathcal{P}({\mathbb{N}})/\operatorname{Fin}$ is isomorphic to an
ultraproduct of countable atomless Boolean algebras. If so, then this would
have to be an $\aleph_{1}$-generated ultrafilter. The most obvious choice
would be an ultrafilter generated by a ground-model selective ultrafilter
(there are $2^{\aleph_{1}}$ such ultrafilters by [1]). As all of these
ultrafilters ‘look the same’ (see [54] for an interpretation of this
assertion) this suggests the following test question.

######  Question 8.2.

Suppose that in either one of the Sacks models $\mathcal{U}$ and $\mathcal{V}$
are $\aleph_{1}$-generated selective ultrafilters. Is it true that
$({\mathbb{N}},\leq)^{\mathcal{U}}\cong({\mathbb{N}},\leq)^{\mathcal{V}}$?

One could ask an analogous question for countable models of other countable
first-order theories with the order property; $({\mathbb{N}},\leq)$ just
appears to provide the simplest interesting instance of this question. The
ideas from [48, §2 and §4] may be relevant to this problem in the case of
Boolean algebras.

Question 7.3 on the existence universal ultrapowers in the absence of CH
tackled in §7 also remains open. See [47] for the bigger picture.

We conclude with a few words on ‘definable’ reduced products
$\prod_{\mathcal{F}}A_{n}$. If $\mathcal{F}$ is an analytic filter on
${\mathbb{N}}$ (i.e., one that is analytic as a subset of
$\mathcal{P}({\mathbb{N}})$, given its Cantor-set topology) that extends the
Fréchet filter, then the restriction of $\mathcal{F}$ to any
$\mathcal{F}$-positive set is not an ultrafilter (because all analytic sets,
unlike the nonprincipal ultrafilters, have the universal property of Baire.)
Therefore the Feferman–Vaught theorem ([22], and for the metric case [23] or
[12, §16.3]) implies that if all $A_{n}$ are elementarily equivalent, and if
$\mathcal{F}$ is analytic and extends the Fréchet filter then
$\prod_{\mathcal{F}}A_{n}$ is elementarily equivalent to
$\prod_{\operatorname{Fin}}A_{n}$. Many (but not all) of the reduced products
$\prod_{\mathcal{F}}A_{n}$ are countably saturated121212A sufficient condition
for countable saturation of $\prod_{\mathcal{F}}A_{n}$ was isolated in [11,
Definition 6.5]. and therefore isomorphic to $\prod_{\operatorname{Fin}}A_{n}$
if CH holds. The following question is somewhat vague—the answer is clearly a
function of the theories of the $A_{n}$s, among other things— but see the
discussion in the paragraph following it for clarification.

######  Question 8.3.

Given a sequence $A_{n}$, for $n\in{\mathbb{N}}$, of structures of the same
language, for what Borel filters $\mathcal{F}$ on ${\mathbb{N}}$ is
$\prod_{\mathcal{F}}A_{n}\cong\prod_{\operatorname{Fin}}A_{n}$ provable in ZFC
for every sequence of countable (separable) models $A_{n}$?

In the case when each $A_{n}$ is the two-element Boolean algebra—i.e., the
case of the quotients of the form
$\mathcal{P}({\mathbb{N}})/{\mathcal{I}}$—the isomorphism is provable if and
only if there is a continuous
$f\colon\mathcal{P}({\mathbb{N}})\to\mathcal{P}({\mathbb{N}})$ that lifts such
an isomorphism ([20]) and in many (conjecturally, all) cases this is
equivalent to the Rudin–Keisler isomorphism of the underlying ideals ([10,
Corollary 3.4.2] and [20, Corollary 3]). For current state of the art in this
subject see [14].

In the case when all $A_{n}$ are Boolean algebras, Question 8.3 is really a
question about abelian $C^{*}$-algebras. This is because the category of
Boolean algebras is, via the Stone duality, equivalent to the category of
compact, zero-dimensional, Hausdorff spaces and the latter category is, by the
Gelfand–Naimark duality, equivalent to the category of unital, abelian,
$C^{*}$-algebras (see [12, §1.3]). By this observation and the main result of
[18], PFA implies that two such reduced products are isomorphic if and only if
there is an (appropriately defined) ‘trivial’ isomorphism between them. For
example, PFA implies that
$\prod_{\operatorname{Fin}}B\not\cong\mathcal{P}({\mathbb{N}})/\operatorname{Fin}$
if $B$ is the atomless countable Boolean algebra. The ultimate extension of
the result of [18] to the coronas of arbitrary separable $C^{*}$-algebras was
proved in [34] and [52]; see also the survey [14].

## References

  * [1] J.E. Baumgartner and R. Laver, _Iterated perfect-set forcing_ , Ann. Math. Logic 17 (1979), no. 3, 271–288.
  * [2] I. Ben Yaacov, A. Berenstein, C.W. Henson, and A. Usvyatsov, _Model theory for metric structures_ , Model Theory with Applications to Algebra and Analysis, Vol. II (Z. Chatzidakis et al., eds.), London Math. Soc. Lecture Notes Series, no. 350, London Math. Soc., 2008, pp. 315–427.
  * [3] J. Brendle and T. LaBerge, _Forcing tightness in products of fans_ , Fund. Math. 150 (1996), no. 3, 211–226.
  * [4] J. Carrión, J. Gabe, C. Schafhauser, A. Tikuisis, and S. White, _Classifying *-homomorphisms_ , Manuscript(s) in preparation.
  * [5] C. C. Chang and H. J. Keisler, _Model theory_ , third ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam, 1990.
  * [6] H.G. Dales and W.H. Woodin, _An introduction to independence for analysts_ , London Math. Soc. Lecture Note Series, vol. 115, Cambridge University Press, 1987.
  * [7] Alan Stewart Dow and Saharon Shelah, _Asymmetric tie-points and almost clopen subsets of $\mathbb{N}^{*}$_, Comment. Math. Univ. Carolin. 59 (2018), no. 4, 451–466, arXiv: 1801.02523. MR 3914712
  * [8] P. Erdös, _My Scottish Book ‘problems’_ , The Scottish Book (D. Mauldin, ed.), Springer, 2015, pp. 27–33.
  * [9] I. Farah, _Embedding partially ordered sets into $\omega^{\omega}$_, Fund. Math. 151 (1996), 53–95.
  * [10] by same author, _Analytic quotients: theory of liftings for quotients over analytic ideals on the integers_ , Memoirs AMS, vol. 148, no. 702, 2000.
  * [11] by same author, _How many Boolean algebras $\mathcal{P}({\mathbb{N}})/{\mathcal{I}}$ are there?_, Illinois Journal of Mathematics 46 (2003), 999–1033.
  * [12] by same author, _Combinatorial set theory and $C^{*}$-algebras_, Springer Monographs in Mathematics, Springer, 2019.
  * [13] by same author, _Between reduced powers and ultrapowers_ , to appear in J. Eur. Math. Soc. (2020).
  * [14] I. Farah, S. Ghasemi, A. Vaccaro, and A. Vignati, _Corona rigidity_ , arXiv preprint arXiv:2201.11618 (2022).
  * [15] I. Farah, B. Hart, M. Lupini, L. Robert, A. Tikuisis, A. Vignati, and W. Winter, _Model theory of $C^{*}$-algebras_, Mem. Amer. Math. Soc. 271 (2021), no. 1324, viii+127. MR 4279915
  * [16] I. Farah, B. Hart, and D. Sherman, _Model theory of operator algebras I: Stability_ , Bull. London Math. Soc. 45 (2013), 825–838.
  * [17] by same author, _Model theory of operator algebras II: Model theory_ , Israel J. Math. 201 (2014), 477–505.
  * [18] I. Farah and P. McKenney, _Homeomorphisms of Čech–Stone remainders: the zero-dimensional case_ , Proc. Amer. Math. Soc. 146 (2018), no. 5, 2253–2262.
  * [19] Ilijas Farah and Saharon Shelah, _A dichotomy for the number of ultrapowers_ , J. Math. Log. 10 (2010), no. 1-2, 45–81, arXiv: 0912.0406. MR 2802082
  * [20] by same author, _Trivial automorphisms_ , Israel J. Math. 201 (2014), no. 2, 701–728, arXiv: 1112.3571. MR 3265300
  * [21] by same author, _Rigidity of continuous quotients_ , J. Inst. Math. Jussieu 15 (2016), no. 1, 1–28, arXiv: 1401.6689. MR 3427592
  * [22] S. Feferman and R. Vaught, _The first order properties of products of algebraic systems_ , Fundamenta Mathematicae 47 (1959), no. 1, 57–103.
  * [23] S. Ghasemi, _Reduced products of metric structures: a metric Feferman–Vaught theorem_ , J. Symb. Log. 81 (2016), no. 3, 856–875.
  * [24] K. Gödel, _What is Cantor’s continuum problem?_ , The American Mathematical Monthly 54 (1947), no. 9, 515–525.
  * [25] A. Hajnal, _Proof of a conjecture of S. Ruziewicz_ , Fund. Math 50 (1961), 123–128.
  * [26] W. Just and A. Krawczyk, _On certain Boolean algebras ${\mathcal{P}}(\omega)/I$_, Trans. Amer. Math. Soc. 285 (1984), 411–429.
  * [27] W. Just and Ž. Mijajlović, _Separation properties of ideals over $\omega$_, Mathematical Logic Quarterly 33 (1987), no. 3, 267–276.
  * [28] Menachem Kojman and Saharon Shelah, _Nonexistence of universal orders in many cardinals_ , J. Symbolic Logic 57 (1992), no. 3, 875–891, arXiv: math/9209201. MR 1187454
  * [29] Sabine Koppelberg and Saharon Shelah, _Subalgebras of Cohen algebras need not be Cohen_ , Logic: from foundations to applications (Staffordshire, 1993), Oxford Sci. Publ., Oxford Univ. Press, New York, 1996, arXiv: math/9610227, pp. 261–275. MR 1428008
  * [30] Linus Kramer, Saharon Shelah, Katrin Tent, and Simon Thomas, _Asymptotic cones of finitely presented groups_ , Adv. Math. 193 (2005), no. 1, 142–173, arXiv: math/0306420. MR 2132762
  * [31] K. Kunen, _Inaccessibility properties of cardinals._ , Ph.D. thesis, Stanford, 1969.
  * [32] by same author, _Set theory_ , Studies in Logic, vol. 34, College Publications, London, 2011.
  * [33] G. Kurepa, _L’hypothèse du continu et le problème de Souslin_ , Publ. Inst. Math. Belgrade 2 (1948), 26–36.
  * [34] P. McKenney and A. Vignati, _Forcing axioms and coronas of $C^{*}$-algebras_, J. Math. Logic 21 (2021), no. 02, 2150006.
  * [35] L. Pacholski, _On countably compact reduced products, III_ , Coll. Math., vol. 23, Institute of Mathematics Polish Academy of Sciences, 1971, pp. 5–15.
  * [36] L. Pacholski and C. Ryll-Nardzewski, _On countably compact reduced product I_ , Fund. Math 67 (1970), no. 1, 155–161.
  * [37] B.A. Palyutin, _Categorical Horn classes. I_ , Algebra and Logic 19 (1980), no. 5, 377–400.
  * [38] M. Rørdam, _Classification of nuclear $C^{*}$-algebras_, Encyclopaedia of Math. Sciences, vol. 126, Springer-Verlag, Berlin, 2002.
  * [39] Saharon Shelah, _For what filters is every reduced product saturated?_ , Israel J. Math. 12 (1972), 23–31. MR 0304157
  * [40] by same author, _Classification theory and the number of nonisomorphic models_ , 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR 1083551
  * [41] by same author, _Vive la différence. I. Nonisomorphism of ultrapowers of countable models_ , Set theory of the continuum (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., vol. 26, Springer, New York, 1992, arXiv: math/9201245, pp. 357–405. MR 1233826
  * [42] by same author, _Vive la différence. II. The Ax-Kochen isomorphism theorem_ , Israel J. Math. 85 (1994), no. 1-3, 351–390, arXiv: math/9304207. MR 1264351
  * [43] by same author, _Toward classifying unstable theories_ , Ann. Pure Appl. Logic 80 (1996), no. 3, 229–255, arXiv: math/9508205. MR 1402297
  * [44] by same author, _Proper and improper forcing_ , 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206
  * [45] by same author, _Vive la différence. III_ , Israel J. Math. 166 (2008), 61–96, arXiv: math/0112237. MR 2430425
  * [46] by same author, _No universal group in a cardinal_ , Forum Math. 28 (2016), no. 3, 573–585, arXiv: 1311.4997. MR 3510831
  * [47] by same author, _Divide and conquer: Dividing lines and universality_ , Theoria 87 (2021), no. 2, 259–348.
  * [48] Saharon Shelah and Juris Steprāns, _Non-trivial automorphisms of $\mathcal{P}(\mathbb{N})/[\mathbb{N}]^{<\aleph_{0}}$ from variants of small dominating number_, Eur. J. Math. 1 (2015), no. 3, 534–544. MR 3401904
  * [49] Saharon Shelah and Alexander Usvyatsov, _Banach spaces and groups—order properties and universal models_ , Israel J. Math. 152 (2006), 245–270, arXiv: math/0303325. MR 2214463
  * [50] E.K. van Douwen, K. Kunen, and J. van Mill, _There can be $C^{*}$-embedded dense proper subspaces in $\beta\omega\setminus\omega$_, Proc. Amer. Math. Soc. 105 (1989), no. 2, 462–470.
  * [51] E.K. van Douwen and J. van Mill, _Parovičenko’s characterization of $\beta\omega-\omega$ implies CH_, Proc. Amer. Math. Soc. 72 (1978), no. 3, 539–541.
  * [52] A. Vignati, _Rigidity conjectures_ , Annales scientifiques de l’École normale supérieure (to appear).
  * [53] W. Winter, _Structure of nuclear $C^{*}$-algebras: from quasidiagonality to classification and back again_, Proceedings of the International Congress of Mathematicians (ICM 2018), World Scientific, 2019, pp. 1801–1823.
  * [54] J. Zapletal, _Terminal notions_ , Bull. Symb. Logic 5 (1999), no. 4, 470–478.