# Absolute model companionship, forcibility, and the continuum problem

Matteo Viale

###### Abstract.

Absolute model companionship (AMC) is a strict strengthening of model
companionship defined as follows: For a theory $T$, $T_{\exists\vee\forall}$
denotes the logical consequences of $T$ which are boolean combinations of
universal sentences. $S$ is the AMC of $T$ if it is model complete and
$T_{\exists\vee\forall}=S_{\exists\vee\forall}$.

We use AMC to study the continuum problem and to gauge the expressive power of
forcing. We show that (a definable version of) $2^{\aleph_{0}}=\aleph_{2}$ is
the unique solution to the continuum problem which can be in the AMC of a
_partial Morleyization_ of the $\in$-theory $\mathsf{ZFC}+$_there are class
many supercompact cardinals_.

We also show that (assuming large cardinals) forcibility overlaps with the
apparently weaker notion of consistency for any mathematical problem $\psi$
expressible as a $\Pi_{2}$-sentence of a (very large fragment of) third order
arithmetic (CH, the Suslin hypothesis, the Whitehead conjecture for free
groups are a small sample of such problems $\psi$).

Finally we characterize a strong form of Woodin’s axiom $(*)$ as the assertion
that the first order theory of $H_{\aleph_{2}}$ as formalized in a certain
natural signature is model complete.

The author acknowledges support from INDAM through GNSAGA and from the
project: _PRIN 2017-2017NWTM8R Mathematical Logic: models, sets,
computability._ MSC: _03C10, 03E57._ Keywords: _Model Companionship, Generic
Absoluteness, Forcing Axioms, Large Cardinals._

This paper is divided in two parts: the first part introduces two model
theoretic concepts which are then used in the second part to analyze and gauge
the complexity of the axiomatization of set theory given by $\mathsf{ZFC}$
(eventually enriched with large cardinal axioms). The key theme of the present
work is to combine the ideas of Robinson around the notions of model
companionship, model completeness, and existentially closed models for a first
order theory with those arising in set theory from the analysis of the forcing
method. Specifically we will show that a natural strengthening of the notion
of model companionship is particularly fit to analyze $\mathsf{ZFC}$: on the
one hand it can be used to infer that forcibility overlaps with consistency at
least when dealing with a large family of interesting mathematical problems,
on the other hand it gives a viable model theoretic tool to tackle the
continuum problem and provides an argument to assert that
$2^{\aleph_{0}}=\aleph_{2}$. Furthermore it provides the means to give an
elegant model theoretic formulation of a natural strong form of Woodin’s axiom
$(*)$.

We now briefly outline the model theoretic content and the set theoretic
content of the paper trying to avoid technicalities111The model theory part of
this paper is independent from its set theory part and can be read by anyone
familiar with the basic facts about model companionship and model
completeness. The set theory part depends on the model theory part and has two
types of arguments: there are basic results whose proofs leverage on classical
theorems covered in any master course on the subject (essentially Levy
absoluteness, no knowledge of forcing required); there are also advanced
results whose proofs require a strong background in forcing axioms and
Woodin’s work on axiom $(*)$ and take advantage of Asperó and Schindler’s
recent breakthrough [3]. .

### Model theory

#### Absolute model companionship

A first new model theoretic concept in this paper is that of absolute model
companionship (AMC), which to our knowledge hasn’t been explicitly stated yet.
Given a first order theory $T$ in a signature $\tau$, a $\tau$-structure
$\mathcal{M}$ is $T$-existentially closed ($T$-ec) if and only if it is a
substructure of a $\tau$-model of $T$ and it is a $\Sigma_{1}$-elementary
substructure of any $\tau$-superstructure which models $T$. A standard example
is given by $\mathsf{Fields}$ (the $\left\\{+,\cdot,0,1\right\\}$-theory of
fields), with the $\mathsf{Fields}$-ec models being exactly the algebraically
closed fields.

The way algebraically closed fields sits inside the
$\left\\{+,\cdot,0,1\right\\}$-structures which are fields is described by
Robinson’s notion of model companionship: a $\tau$-theory $S$ is the model
companion of a $\tau$-theory $T$ if the elementary class given by the
$\tau$-models of $S$ consists exactly of the $T$-ec models. The
$\left\\{+,\cdot,0,1\right\\}$-theory of algebraically closed fields
$\mathsf{ACF}$ is the model companion of the
$\left\\{+,\cdot,0,1\right\\}$-theory $\mathsf{Fields}$; however for an
arbitrary theory $T$ the $T$-ec models may not form an elementary class (e.g.
the existentially closed models for the $\left\\{\cdot,1\right\\}$-theory of
groups are not an elementary class, hence this axiomatization of groups has no
model companion). A very special case occurs when $T$ is its own model
companion (e.g. the models of $T$ are exactly the $T$-ec models), in which
case $T$ is _model complete_.

A basic observation is that a $\tau$-structure $\mathcal{M}$ is $T$-ec if and
only if it is $T_{\forall}$-ec (where $T_{\forall}$ consists of the universal
consequences of $T$ in signature $\tau$) if and only if it is
$T_{\forall\vee\exists}$ (the latter being the boolean combinations of
universal sentences which follow from $T$). Another non-trivial fact is that
for a _complete_ theory $T$ a $T$-ec structure $\mathcal{M}$ realizes any
$\Pi_{2}$-sentence which holds true in some model of $T_{\forall\vee\exists}$.
A third non-trivial remark is that if $S$ is the model companion of $T$, $S$
is axiomatized by its $\Pi_{2}$-consequences.

Combining these three observations one is led to the speculation that the
model companion $S$ of a $\tau$-theory $T$ (if it exists) could be axiomatized
by the family of $\Pi_{2}$-sentences which holds in some model of
$T_{\forall\vee\exists}$. This is an assertion which is slightly too bold and
holds true in case $T$ is a _complete_ , model companionable theory. However
it can fail for non-complete model companionable theories; the standard
counterexample being $\mathsf{ACF}$ versus $\mathsf{Fields}$: $\forall
x\neg(x^{2}+1=0)$ is a $\Pi_{2}$-sentence (it is actually $\Pi_{1}$) which is
not an axiom of $\mathsf{ACF}$ and holds in the field $\mathbb{Q}$, being
therefore consistent with the universal and existential fragment of the theory
of fields.

This brings us to introduce the notion of absolute model companionship (AMC):
a $\tau$-theory $T$ has an AMC if the class of $T$-ec models is axiomatized by
the $\Pi_{2}$-sentences which are consistent with the universal and
existential fragments of any completion of $T$ (see Def. 1.3, Thm. 1.4).
Complete first order theories are model companionable if and only if they
admit an AMC, but there are non-complete theories admitting a model companion
but not an AMC (e.g. the $\left\\{+,\cdot,0,1\right\\}$-theory of fields).

#### The AMC-spectrum of a first order theory

A second model theoretic concept we introduce in this paper is that of AMC-
spectrum (and model companionship-spectrum) of a mathematical theory $T$.
Model theory is extremely successful in classifying the complexity of a
mathematical theory according to its “structural properties” and has produced
a variety of dividing lines to separate the so called “tame” mathematical
theories from the others: typically a mathematical theory is considered “wild”
or “unclassifiable” if it can code in itself first order arithmetic, hence the
$\in$-theory $\mathsf{ZFC}$ is considered unclassifiable. On the other hand
several tools have been developed to “classify” mathematical theories, for
example stability, simplicity, NIP are structural properties of “tame”
mathematical theories.

It is a matter of fact that most mathematical theories admit many different
first order axiomatizations in many distinct signatures. A common
characteristic of “tameness” properties such as stability, simplicity, NIP is
that they are _signature invariant_ : more precisely if we take a
$\tau$-theory $T$ and we consider its Morleyization $T^{*}$ in the signature
$\tau^{*}$ which adds predicate symbols and axioms granting that all
$\tau$-formulae are equivalent to atomic $\tau^{*}$-formulae (see Notation
1.6), $T$ is stable (NIP, simple) if and only if so is $T^{*}$. In contrast
Robinson’s notion of model companionship is a useful property of a first order
theory, but is not signature invariant: for example $\mathsf{ACF}$ is the
model companion of $\mathsf{Fields}$ in signature
$\tau=\left\\{+,\cdot,0,1\right\\}$, but if we consider their Morleyizations
$\mathsf{ACF}^{*}$ and $\mathsf{Fields}^{*}$ in signature $\tau^{*}$, it no
longer holds true that $\mathsf{ACF}^{*}$ is the model companion of
$\mathsf{Fields}^{*}$. Conversely a $\tau$-theory $R$ may not have a model
companion (e.g. the $\sigma=\left\\{\cdot,1\right\\}$-theory of groups) but
its Morleyization $R^{*}$ in signature $\sigma^{*}$ is its own model
companion.

In this paper we consider the dependence of the existence of a model companion
from the signature by itself a useful structural property of a mathematical
theory. Specifically we introduce the notion of _partial Morleyzation_ of a
$\tau$-theory $T$: given a set $A$ of $\tau$-formulae, we consider the
signature $\tau_{A}$ obtained by expanding $\tau$ exactly with predicate
symbols for the formulae in $A$ and the $\tau_{A}$-theory $T_{\tau,A}$ with
axioms asserting that every formula in $A$ is logically equivalent to the
corresponding predicate of $\tau_{A}\setminus\tau$ (see again222Actually we
give in this introduction a simpliefied version of the Morleyization
procedures we consider; the full details can be found in Notation 1.6.
Notation 1.6). The AMC-spectrum of a $\tau$-theory $T$ (Def. 1.7) is given by
those sets $A$ of $\tau$-formulae for which $T+T_{\tau,A}$ admits an absolute
model companion (mutatis mutandis we can define the model companionship
spectrum of $T$).

A category theoretic perspective on this approach can be enlightening: given a
$\tau$-theory $T$, consider the category $\mathcal{C}_{T}$ given by the
elementary class of its $\tau$-models as objects and the $\tau$-morphisms
(e.g. maps which preserve the _atomic_ $\tau$-formulae) between them as
arrows. By taking $A$ a subset of the $\tau$-formulae we can pass to the
category $\mathcal{C}_{T+T_{\tau,A}}$ whose objects are $\tau_{A}$-models of
$T+T_{\tau,A}$ and whose arrows are the $\tau_{A}$-morphisms (in accordance
with Notation 1.6). There is a natural identification of the objects of
$\mathcal{C}_{T}$ and those of $\mathcal{C}_{T+T_{\tau,A}}$, but the arrows of
$\mathcal{C}_{T+T_{\tau,A}}$ are now a possibly much narrower subfamily of the
arrows of $\mathcal{C}_{T}$. This paper (by considering the case of $T$ being
set theory) shows that important structural information on a $\tau$-theory $T$
is given (at least from a categorial point of view) by classifying which sets
of $\tau$-formulae $A$ produce an elementary class of $\tau_{A}$-models for
which the $T+T_{\tau,A}$-ec models constitute themselves an elementary class.
The model companionship spectrum of $T$ gives exactly this structural
information on the arrows of $\mathcal{C}_{T}$. Similar considerations apply
for the AMC-spectrum of a theory. On the other hand simplicity, stability, NIP
provide fundamental structural informations on the class of objects of
$\mathcal{C}_{T}$, but it is not transparent whether they also convey
information on its class of arrows.

### Set theory

#### What is the right signature for set theory?

The $\in$-signature is certainly sufficient to give by means of $\mathsf{ZFC}$
a first order axiomatization of set theory (with eventually other extra
hypothesis such as large cardinal axioms), but we can see rightaway that it is
not efficient to formalize many basic set theoretic concepts. Consider for
example the notion of ordered pair: on the board we write $x=\langle
y,z\rangle$ to mean that _$x$ is the ordered pair with first component $y$ and
second component $z$_. In set theory this concept is formalized by means of
Kuratowski’s trick stating that
$x=\left\\{\left\\{y\right\\},\left\\{y,z\right\\}\right\\}$. However the
“natural” $\in$-formula in free variables $x,y,z$ formalizing the above is:

$\exists t\exists u\;[\forall w\,(w\in x\leftrightarrow w=t\vee
w=u)\wedge\forall v\,(v\in t\leftrightarrow v=y)\wedge\forall v\,(v\in
u\leftrightarrow v=y\vee v=z)].$

It is clear that the meaning of this $\in$-formula is hardly decodable with a
rapid glance (unlike $x=\langle y,z\rangle$), moreover just from the point of
view of its syntactic complexity it is already $\Sigma_{2}$. On the other hand
we do not regard the notion of ordered pair as a complex or doubtful concept
(as is the case for the notion of uncountability, or many of the properties of
the continuum such as its correct place in the hierarchy of uncountable
cardinals, etc…). Similarly other very basic notions such as: being a
function, a binary relation, the domain or the range of a function, etc.. are
formalized by rather complicated $\in$-formulae, both from the point of view
of readability for human beings, and from the mere computation of their
syntactic complexity according to the Levy hierarchy.

The standard solution adopted by set theorists (e.g. [14, Chapter IV, Def.
3.5]) is to regard as elementary all those properties which can be formalized
using $\in$-formulae all of whose quantifiers are bounded to range over the
elements of some set, i.e. the so called $\Delta_{0}$-formulae. We adopt this
point of view in stating our main set theoretic results, and we maintain it
for the remainder of this paper, considering $\in_{\Delta_{0}}$ the correct
basic signature where to formalize set theory, where the latter is the
signature obtained with the partial Morleyization induced by the
$\Delta_{0}$-predicates (and slightly more, see Section 1.2 and Notation 3.1.1
for details).

#### Existentially closed fragments of the set theoretic universe versus AMC

We now give a non-exhaustive list outlining on the one hand that for certain
regular cardinals $\kappa$ $H_{\kappa}$ is not that far from being an
existentially closed structure for set theory as formalized in the appropriate
signature (e.g. Levy absoluteness, Shoenfield’s absoluteness, $\mathsf{BMM}$,
$\mathsf{BMM}^{++}$) or from having a model complete theory333Recall that a
$\tau$-theory $T$ is model complete if and only if the substructure relation
between its models overlaps with the elementary substructure relation. In
particular the mentioned results are weak form of “model completeness” for the
theory of $H_{\aleph_{i}}$ for $i=1,2$. (e.g. Woodin’s absoluteness,
$\text{{\sf MM}}^{+++}$). The reader needs not be familiar with these results,
as they only serve as motivation for what we aim to do in the sequel.

Below we denote a $\tau$-structure $(M,R^{M}:R\in\tau)$ by $(M,\tau^{M})$,
$\sqsubseteq$ denotes the substructure relation, $\prec$ the elementary
substructure relation, $\prec_{1}$ the $\Sigma_{1}$-elementary substructure
relation (see Notation 2.0.1 for details). Recall the following results:

Levy absoluteness:

(see Lemma 3.1 below) Whenever $\kappa$ is a regular uncountable cardinal,

$(H_{\kappa},\in_{\Delta_{0}}^{V})\prec_{1}(V,\in_{\Delta_{0}}^{V}).$

Shoenfield’s absoluteness:

(see [25, Lemma 1.2] for the apparently weaker formulation we give here)
Whenever $G$ is $V$-generic for some forcing notion in $V$,

$(H_{\omega_{1}},\in_{\Delta_{0}}^{V})\prec_{1}(V[G],\in_{\Delta_{0}}^{V[G]}).$

Woodin’s absoluteness:

(see [25, Lemma 3.2] for the weak form of Woodin’s result we give here)
Whenever $G$ is $V$-generic for some forcing notion in $V$ (and there are
class many Woodin cardinals in $V$),

$(H_{\omega_{1}}^{V},\in_{\Delta_{0}}^{V})\prec(H_{\omega_{1}}^{V[G]},\in_{\Delta_{0}}^{V[G]}).$

Bounded Martin’s Maximum ($\mathsf{BMM}$):

(see [6]) Whenever $G$ is $V$-generic for some stationary set preserving
forcing notion in $V$,

$(H_{\omega_{2}},\in_{\Delta_{0}}^{V})\prec_{1}(V[G],\in_{\Delta_{0}}^{V[G]}).$

$\mathsf{BMM}^{++}$:

(see [27, Def. 10.91]) Whenever $G$ is $V$-generic for some stationary set
preserving forcing notion in $V$,

$(H_{\omega_{2}},\in_{\Delta_{0}}^{V},\mathbf{NS}_{\omega_{1}}^{V})\prec_{1}(V[G],\in_{\Delta_{0}}^{V[G]},\mathbf{NS}_{\omega_{1}}^{V[G]}),$

where $\mathbf{NS}_{\omega_{1}}$ is a unary predicate symbol interpreted by
the non-stationary ideal on $\omega_{1}$.

Bounded category forcing axioms, $\text{{\sf MM}}^{+++}$,
$\mathsf{RA}_{\omega}(\text{{\sf SSP}})$:

(see444We omit a definition of these axioms since this demands a detour from
the main focus of the present paper. [4, 5, 24]) Whenever $V$ and $V[G]$ are
models of $\text{{\sf MM}}^{+++}$ ($\mathsf{RA}_{\omega}(\text{{\sf SSP}})$,
$\mathsf{BCFA}(\text{{\sf SSP}})$) and $G$ is $V$-generic for some stationary
set preserving forcing notion in $V$,

$(H_{\omega_{2}}^{V},\in_{\Delta_{0}}^{V})\prec(H_{\omega_{2}}^{V[G]},\in_{\Delta_{0}}^{V[G]}).$

We also want to mention:

Absoluteness between transitive models:

(see [12, Lemma 14.21]) whenever $G$ is $V$-generic for some forcing notion in
$V$,

$(V,\in_{\Delta_{0}}^{V})\sqsubseteq(V[G],\in_{\Delta_{0}}^{V[G]}).$

This latter property entails that forcing preserves the meaning of the basic
concepts of set theory555We ignored for the moment any consideration regarding
universally Baire sets. These sets will play an essential role in the proof of
Thm. 1.10., e.g. those formalized by atomic $\in_{\Delta_{0}}$-formulae;
furthermore Shoenfield’s and Levy’s absoluteness grant that
$\Pi_{1}$-sentences for $\in_{\Delta_{0}}$ do not change truth value in
forcing extensions.

### Main set theoretic results

The set theoretic results of the present paper will be outlined in details in
Section 1 and systematize the above considerations. They can be informally
summarized as follows:

  * •

The theories of the various $H_{\lambda}$ for $\lambda$ an uncountable regular
cardinal provide the prototypes of model companions for $\mathsf{ZFC}$ in some
signature $\left\\{\in\right\\}_{A}$ with $A$ in the model companionship
spectrum of666As a side remark we note that Hirschfeld [10] has proved that
$\mathsf{ZF}$ has an AMC for the $\in$-signature; however he himself
acknowledges that his result is not that informative on the properties of set
theory, since the $\in$-AMC of $\mathsf{ZF}$ is given by a theory which is a
small perturbation of the theory of dense linear orders (it interprets the
$\in$-relation as an irreflexive transitive relation which defines a dense and
strict preorder). Hirschfeld does not argue that $\mathsf{ZF}$ has an AMC but
he shows that it decides any universal $\in$-sentence and has a model
companion; under these assumptions the model companion of $\mathsf{ZF}$ is its
AMC. We believe that a signature in the AMC-spectrum of set theory is
meaningful if it allows to prove i of Thm. 1.9, e.g. if it is able to show
that the Goedel operations are well defined and that a large amount of
replacement holds in the AMC of set theory according to the signature. This
brings almost automatically to consider signatures which are more or less
expansions of $\in_{\Delta_{0}}$: they must be able to express the concept of
function, relation, domain, codomain, etc by means of terms or of quantifier
free formulae. $\mathsf{ZFC}$. More precisely: for any definable cardinal
$\kappa$, there is a set of formulae $A_{\kappa}$ in the AMC-spectrum of
$\mathsf{ZFC}$ such that the theory of $H_{\kappa^{+}}$ is the model companion
of set theory for $\left\\{\in\right\\}_{A_{\kappa}}$ (see the second item of
Thm. 1.9 and the first part of Thm. 1.10); note that for each such $\kappa$
$\left\\{\in\right\\}_{A_{\kappa}}$ extends $\in_{\Delta_{0}}$. Furthermore
any existentially closed structure for set theory in some signature extending
$\in_{\Delta_{0}}$ looks like an $H_{\lambda}$ for some regular uncountable
$\lambda$ (see the first item of Thm. 1.9).

  * •

Forcing suffices to produce the interesting models of a very large fragment of
the mathematical universe: for statements of second or third order arithmetic
formalizable by $\Pi_{2}$-sentences of $\in_{\Delta_{0}}$ (among which for
example the negation of the continuum hypothesis) their consistency (if at all
possible) can already be obtained by forcing (by the first part of Thm. 1.10).

  * •

The above results provide an argument for $2^{\aleph_{0}}=\aleph_{2}$ by on
the basis of the following assertions:

    1. (1)

Large cardinal axioms are part of the accepted axioms of set theory.

    2. (2)

The mathematical universe realizes the $\Pi_{2}$-sentences for third order
arithmetic which can hold in some model of set theory enriched with large
cardinal axioms.

Note that assertion (2) above is the standard argument used to motivate
forcing axioms and reproduces in the set theoretic framework the creation
process of new numbers bringing from the universal
$\left\\{+,\cdot,0,1\right\\}$-theory $T$ of semirings without zero divisors
(holding for the natural numbers) to the theory of algebraically closed fields
(holding for the algebraic numbers): the algebraic numbers are obtained in a
$\left\\{+,\cdot,0,1\right\\}$-structure which realizes $T$ and all the
“interesting” $\Pi_{2}$-sentences which can be individually made consistent
with $T$, e.g. for each $n$

$\forall a_{0},\dots,a_{n}\exists x\,(\sum_{i=0}^{n}a_{i}x^{i}=0).$

On the basis of assertion 1, assertion 2 (as well as
$2^{\aleph_{0}}=\aleph_{2}$) can be validated as follows:

    * –

Model theory (specifically the notion of AMC) gives us the means to formulate
elegantly in precise mathematical terms assertion 2 (by Thm. 1.4).

    * –

$2^{\aleph_{0}}=\aleph_{2}$ is the unique solution of the continuum problem
which falls in the model companion of set theory enriched with large cardinals
for at least one $\left\\{\in\right\\}_{A}$ with $A$ in the AMC-spectrum of
set theory (by Theorems 1.10, 1.11, 1.12).

    * –

Furthermore the $\Pi_{2}$-consequences for $H_{\aleph_{2}}$ of
$\mathsf{BMM}^{++}$, Woodin’s axiom $(*)$, etc as formalized in an appropriate
signature for third order arithmetic of the form $\left\\{\in\right\\}_{B}$
(for a natural set $B$ of $\in$-formulae) form the AMC of set theory enriched
with large cardinal axioms for $\left\\{\in\right\\}_{B}$; moreover this AMC
exactly overlaps with the forcible $\Pi_{2}$-sentences for $H_{\aleph_{2}}$
for this signature (see Thm. 1.10, Thm. 5.6) and has among its
$\Pi_{2}$-axioms one which entails that $2^{\aleph_{0}}=\aleph_{2}$ is
witnessed by a definable well order (for example by the results of [7, 18,
21]).

    * –

Finally it can also be shown that the assertion that the theory of
$H_{\aleph_{2}}$ in (a natural expansion of signature
$\left\\{\in\right\\}_{B}$) is model complete (equivalently it is the AMC of
set theory as formalized in this signature) gives a characterization of the
strong form (here denoted as $(*)\text{-}\mathsf{UB}$) of Woodin’s axiom $(*)$
which predicates the existence of a generic filter for
$\mathbb{P}_{\mathrm{max}}$ meeting all the dense subsets which are
universally Baire in the codes (cfr. Def. 5.5, Thm. 5.6).

These results outline the role of forcing axioms in detecting a “natural” AMC
for set theory describing the theory of $H_{\aleph_{2}}$. Key to their proof
is the breakthrough [3] by Asperó and Schindler that $\text{{\sf MM}}^{++}$
implies $(*)\text{-}\mathsf{UB}$.

##  1\. Main results

We now give a precise and brief list of the main results of this paper (the
proofs are deferred to later sections). We introduce hastily all new concepts
and terminology needed to phrase them. Our results are of three types:

Basic model theoretic results:

we introduce AMC (a strengthening of model companionship) and we analyze its
main properties. We also introduce the notion of partial Morleyization and we
show how to combine it with AMC in order to get a useful classification tool
for first order theories777This is elementary material accessible to anyone
familiar with the notion of model companionship..

Basic set theoretic results:

we outline the general properties of theories which can be the model companion
of partial Morleyizations of $\mathsf{ZFC}$ leveraging on classical set
theoretic results888A basic knowledge of set theory at the level of [11]
suffices to get through this part of the paper..

Advanced set theoretic result:

we show that set theory enriched with large cardinal axioms has an AMC with
respect to a natural signature for third order arithmetic999This signature
extends $\in_{\Delta_{0}}$ with predicates for the lightface definable
universally Baire sets, a constant for $\omega_{1}$, and a unary predicate for
the nonstationary ideal on $\omega_{1}$.. Furthermore this AMC is given by the
$\Pi_{2}$-sentences of a very large fragment of third order arithmetic which
are forcible. We also analyze why this result provides an argument in favour
of $2^{\aleph_{0}}=\aleph_{2}$ and gives a model theoretic characterization of
a strengthening of Woodin’s axiom101010This part requires familiarity with
forcing axioms, stationary tower forcing,
$\mathbb{P}_{\mathrm{max}}$-technology, the theory of universally Baire sets
assuming large cardinals. $(*)$.

The model theory part is completely independent of the other two parts, while
each of the set theoretic parts requires only the model theory part as a
background.

###  1.1. Absolute model companionship, partial Morleyizations, and the AMC-
spectrum of a theory

We introduce a strengthening of model companionship which to our knowledge
hasn’t been explicitly stated yet. We refer the reader to [23] or [8, Section
3.5], [20, Sections 3.1-3.2] for a detailed account of model
companionship111111The reader will find a detailed analysis of the concepts
and results we introduce here in Section 2 (which could be read rightaway
after section 1.1 if one has no interest in the set-theoretic content of this
article).

######  Notation 1.1.

Let $\tau$ be a signature and $T$ be a $\tau$-theory.

  * •

$\tau_{\forall}$ is the set of universal sentences for $\tau$.
$\tau_{\forall\vee\exists}$ is the set of boolean combinations of sentences in
$\tau_{\forall}$.

  * •

$T_{\forall}$ (respectively $T_{\forall\vee\exists}$) is the set of sentences
in $\tau_{\forall}$ (respectively $\tau_{\forall\vee\exists}$) which are
logical consequences of $T$.

######  Example 1.1.1.

In signature $\left\\{+,\cdot,0,1\right\\}$ let $\mathsf{ACF}$ be the theory
of algebraically closed fields and $\mathsf{Fields}$ be the theory of fields;
we have that $\mathsf{ACF}_{\forall}=\mathsf{Fields}_{\forall}$ while
$\mathsf{ACF}_{\forall\vee\exists}\supsetneq\mathsf{Fields}_{\forall\vee\exists}$:
$\exists x\,(x^{2}+1=0)$ is in the former but not in the latter.

####  1.1.1. Absolute model companionship

It is well known that if $T,S$ are $\tau$-theories such that
$T_{\forall}=S_{\forall}$, then any model of $T$ is a substructure of a model
of $S$ and conversely. However the following holds as well (and to our
knowledge hasn’t been explicitly stated):

######  Lemma 1.2.

Let $\tau$ be a signature and $T,S$ be $\tau$-theories. TFAE:

  1. (1)

$T_{\forall\vee\exists}\supseteq S_{\forall\vee\exists}$;

  2. (2)

Every $\tau$-model $\mathcal{M}$ of $T$ is a $\tau$-substructure of a
$\tau$-model $\mathcal{N}$ of $S$ such that $\mathcal{M}$ and $\mathcal{N}$
satisfy exactly the same sentences in $\tau_{\forall}$ (or equivalently in
$\tau_{\forall\vee\exists}$).

Recall that a theory $T$ is _model complete_ if the substructure relation
between its models overlaps with the elementary substructure relation
(equivalently if $T$ is its own model companion); recall also that $T$ is the
_model companion_ of $S$ if and only if $T_{\forall}=S_{\forall}$ and $T$ is
model complete. We introduce the following strengthening of model
companionship:

######  Definition 1.3.

Let $\tau$ be a signature and $T,S$ be $\tau$-theories. $T$ is the _absolute
model companion_ (AMC) of $S$ if the following conditions are met:

  1. (1)

$T_{\forall\vee\exists}=S_{\forall\vee\exists}$;

  2. (2)

$T$ is model complete.

Note that:

  * •

$\mathsf{ACF}$ is the model companion of $\mathsf{Fields}$ (since
$\mathsf{ACF}$ is model complete and
$\mathsf{ACF}_{\forall}=\mathsf{Fields}_{\forall}$) but not its absolute model
companion (as $\exists
x(x^{2}+1=0)\in\mathsf{ACF}_{\exists\vee\forall}\setminus\mathsf{Fields}_{\exists\vee\forall}$).

  * •

A model complete theory $T$ is the model companion of $T_{\forall}$ and the
AMC of $T_{\forall\vee\exists}$.

  * •

The uniqueness of the model companion grants that a theory $S$ admitting a
model companion $T$ with $T_{\forall\vee\exists}\neq S_{\forall\vee\exists}$
cannot have an AMC.

The following result is what brought our attention on the notion of AMC:

######  Theorem 1.4.

Let $\tau$ be a signature and $T,S$ be $\tau$-theories. TFAE:

  1. (1)

$T$ is the AMC of $S$;

  2. (2)

$T$ is the model companion of $S$ and is axiomatized by the
$\Pi_{2}$-sentences $\psi$ such that $\psi+R_{\forall\vee\exists}$ is
consistent for all $\tau$-theories $R\supseteq S$.

In case $S$ is complete, $T$ is the AMC of $S$ if and only if it is its model
companion, and the second item of the above equivalence states that the model
companion of $S$ is axiomatized by the $\Pi_{2}$-sentences consistent with
$S_{\forall\vee\exists}$.

The following motivates our terminology for this strengthening of model
companionship:

######  Lemma 1.5.

Assume $S,S^{\prime}$ are $\tau$-theories such that $S^{\prime}$ is the AMC of
$S$. Then for all $T\supseteq S$, $S^{\prime}+T_{\forall}$ is the AMC of $T$.

Note that this characterization does not hold for the weaker notion of model
companionship: for the signature $\tau=\left\\{+,\cdot,0,1\right\\}$
$\mathsf{ACF}$ is the model companion of $\mathsf{Fields}$, but for $T$ the
theory of the rationals in signature $\tau$, $\mathsf{ACF}+T_{\forall}$ is
inconsistent, therefore it cannot be the model companion of $T$.

####  1.1.2. Partial Morleyizations and the AMC-spectrum of a theory

We now introduce the notation we use to relate AMC to set theory.

######  Notation 1.6.

Given a signature $\tau$, let $\phi(x_{0},\dots,x_{n})$ be a $\tau$-formula.

We let:

  * •

$R_{\phi}$ be a new $n+1$-ary relation symbol,

  * •

$f_{\phi}$ be a new $n$-ary function symbol121212As usual we confuse $0$-ary
function symbols with constants.,

  * •

$c_{\tau}$ be a new constant symbol.

We also let:

$\text{{\sf AX}}^{0}_{\phi}\text{ be
}\forall\vec{x}[\phi(\vec{x})\leftrightarrow R_{\phi}(\vec{x})],$
$\displaystyle\text{{\sf AX}}^{1}_{\phi}\text{ be }$ $\displaystyle\forall
x_{1},\dots,x_{n}\,$
$\displaystyle[(\exists!y\phi(y,x_{1},\dots,x_{n})\rightarrow\phi(f_{\phi}(x_{1},\dots,x_{n}),x_{1},\dots,x_{n}))\wedge$
$\displaystyle\wedge(\neg\exists!y\phi(y,x_{1},\dots,x_{n})\rightarrow
f_{\phi}(x_{1},\dots,x_{n})=c_{\tau})]$

for $\phi(x_{0},\dots,x_{n})$ having at least two free variables, and

$\text{{\sf AX}}^{1}_{\phi}\text{ be
}\left[(\exists!y\phi(y))\rightarrow\phi(f_{\phi})\right]\wedge\left[(\neg\exists!y\phi(y))\rightarrow
c_{\tau}=f_{\phi}\right].$

for $\phi(x)$ having exactly one free variable.

Let $\mathsf{Form}_{\tau}$ denotes the set of $\tau$-formulae. For
$A\subseteq\mathsf{Form}_{\tau}\times 2$

  * •

$\tau_{A}$ is the signature obtained by adding to $\tau$ relation symbols
$R_{\phi}$ for the $(\phi,0)\in A$ and function symbols $f_{\phi}$ for the
$(\phi,1)\in A$ (together with the special symbol $c_{\tau}$ if at least one
$(\phi,1)$ is in $A$).

  * •

$T_{\tau,A}$ is the $\tau_{A}$-theory having as axioms the sentences
$\text{{\sf AX}}^{i}_{\phi}$ for $(\phi,i)\in A$.

Note the following:

  * •

$\mathsf{Ax}^{0}_{\phi}$ makes the $\tau$-formula $\phi$ logically equivalent
to an atomic formula. For any $\tau$-theory $T$, if
$C=\mathsf{Form}_{\tau}\times\left\\{0\right\\}$ and $\tau^{*}=\tau_{C}$, then
$T^{*}=T+T_{\tau,A}$ is a $\tau^{*}$-theory admitting quantifier elimination
(the Morleyization of $T$).

  * •

$\mathsf{Ax}^{1}_{\psi}$ introduces a Skolem function for $\psi$ without
appealing to choice: the key point is that the interpretation in some
$\tau$-structure $\mathcal{M}$ of the function symbol $f_{\psi}$ applied to
the parameters $\vec{a}$ is determined by $\psi$ only if there is a unique
solution $b\in\mathcal{M}$ of the formula $\exists y\,\psi(\vec{a},y)$,
otherwise $f^{\mathcal{M}}_{\psi}(\vec{a})=c_{\tau}^{\mathcal{M}}$. This
avoids the possibility of choosing arbitrarily $f_{\psi}(\vec{a})$ when
$\exists y\,\psi(\vec{a},y)$ has more than one solution, and gives that for
each $a\in\mathcal{M}$ any $\tau$-structure admits exactly one expansion to a
$\tau_{A}$-structure which is a model of $T_{\tau,A}$ and interprets
$c_{\tau}$ as $a$.

In the sequel of this paper we are interested to analyze what happens when the
Morleyization process is performed on arbitrary subsets of
$\mathsf{Form}_{\tau}\times 2$.

######  Definition 1.7.

The AMC-spectrum of a $\tau$-theory $T$
($\mathfrak{spec}_{\mathsf{AMC}}\left(T\right)$) is given by those
$A\subseteq\mathsf{Form}_{\tau}\times 2$ such that $T+T_{\tau,A}$ has an AMC
(which we denote by $\mathsf{AMC}(T,A)$).

The MC-spectrum of a $\tau$-theory $T$
($\mathfrak{spec}_{\mathsf{MC}}\left(T\right)$) is given by those
$A\subseteq\mathsf{Form}_{\tau}\times 2$ such that $T+T_{\tau,A}$ has a model
companion (which we denote by $\mathsf{MC}(T,A)$).

Note that $A=\mathsf{Form}_{\tau}\times\left\\{0\right\\}$ is always in the
model companionship spectrum of a theory $T$ (as $T+T_{\tau,A}$ admits
quantifier elimination, hence is model complete and its own AMC in signature
$\tau_{A}$). Note also that $\emptyset$ is in the (A)MC-spectrum of $T$ if and
only if $T$ has a model companion (an AMC).

We now show how we can use AMC and model companionship to extract interesting
information on the continuum problem and on the relation between forcibility
and consistency. Some of the results below holds also if we consider the
_model companionship spectrum_ of set theory, but the most interesting ones
work just in case we focus on the AMC-spectrum131313Moreover as of now we are
not even able to produce an example of an $\in$-theory
$T\supseteq\mathsf{ZFC}$ which has some $A\subseteq\mathsf{Form}\times 2$ in
its model companionship spectrum but not in its AMC-spectrum..

###  1.2. The AMC-spectrum of set theory

From now on for any $A\subseteq\mathsf{Form}_{\left\\{\in\right\\}}\times 2$
we write $\in_{A}$ rather than $\left\\{\in\right\\}_{A}$, and we let
$T_{\in,A}$ be the $\in_{A}$-theory

$T_{\left\\{\in\right\\},A}+\forall x\,\left[(\forall y\,y\notin
x)\leftrightarrow c_{\left\\{\in\right\\}}=x\right],$

where the theory $T_{\left\\{\in\right\\},A}$ (according to Notation 1.6 for
$\left\\{\in\right\\}$ and $A$) is reinforced by an axiom asserting that the
interpretation of the constant symbol $c_{\left\\{\in\right\\}}$ is the empty
set.

We will be interested only in sets
$A\subseteq\mathsf{Form}_{\left\\{\in\right\\}}\times 2$ so that $\in_{A}$
contains a basic signature $\in_{\Delta_{0}}$ where all the basic set
theoretic results can be developed (e.g. those exposed in [14, Chapter IV]
and/or in [12, Chapter 13]). The specific details on $\in_{\Delta_{0}}$ and
the axioms $T_{\Delta_{0}}$ which give the expected interpretation to the
symbols of $\in_{\Delta_{0}}$ are given in Notation 3.1.1, we anticipate here
that:

  * •

$\in_{\Delta_{0}}$ is of the form $\in_{D}$ for a specific recursive set
$D\subseteq\mathsf{Form}_{\left\\{\in\right\\}}\times 2$ so that
$\in_{\Delta_{0}}$ includes constant symbols for $\emptyset,\omega$, relation
symbols for all $\Delta_{0}$-formulae, function symbols for all Goedel
operations as defined in [12, Def. 13.6].

  * •

$T_{\Delta_{0}}$ is a family of $\Pi_{2}$-axioms for $\in_{\Delta_{0}}$ so
that $\mathsf{ZF}^{-}+T_{\Delta_{0}}$ is equivalent to
$\mathsf{ZF}^{-}+T_{\in,D}$.

$\mathsf{ZF}^{-}$ ($\mathsf{ZFC}^{-}$) denotes the $\in$-theory $\mathsf{ZF}$
($\mathsf{ZFC}$) deprived of the powerset axiom,
$\mathsf{ZF}^{-}_{\Delta_{0}}$ (respectively $\mathsf{ZFC}^{-}_{\Delta_{0}}$,
$\mathsf{ZF}_{\Delta_{0}}$, $\mathsf{ZFC}_{\Delta_{0}}$) denotes the
$\in_{\Delta_{0}}$-theory $\mathsf{ZF}^{-}+T_{\Delta_{0}}$ (respectively
$\mathsf{ZFC}^{-}+T_{\Delta_{0}}$, $\mathsf{ZF}+T_{\Delta_{0}}$,
$\mathsf{ZFC}+T_{\Delta_{0}}$).

######  Definition 1.8.

Let $T\supseteq\mathsf{ZFC}^{-}$ be an $\in$-theory. $\kappa$ is a
$T$-definable cardinal if for some $\in$-formula $\phi_{\kappa}(x)$, $T$
proves:

  * •

$\exists!x\,\phi_{\kappa}(x)$ and

$\forall x\,[\phi_{\kappa}(x)\rightarrow(x\emph{ is a cardinal})].$

  * •

$\kappa$ is the constant $f_{\phi_{\kappa}}$ existing in the signature
$\in_{\left\\{(\phi_{\kappa},1)\right\\}}$.

The first result shows that the AMC spectrum of set theory isolates a rich set
of theories which produce models of $\mathsf{ZFC}^{-}$, e.g. structures which
behave like an $H_{\lambda}$ for some regular $\lambda$.

######  Theorem 1.9.

Let $R$ be an $\in$-theory extending $\mathsf{ZFC}$.

  1. (i)

Assume $A\in\mathfrak{spec}_{\mathsf{MC}}\left(R\right)$ and
$\in_{A}\supseteq\in_{\Delta_{0}}$. Then any model of $\mathsf{MC}(R,A)$ is
closed under Goedel operations and satisfies Extensionality, Foundation,
Infinity, and Choice141414It is also a model of a large fragment of the
Collection Principle, however it requires a bit of technicalities to sort out
this fragment.

  2. (ii)

Assume $\kappa$ is an $R$-definable cardinal. Then there exists
$A_{\kappa}\in\mathfrak{spec}_{\mathsf{AMC}}\left(R\right)$ with
$\in_{A}\supseteq\in_{\Delta_{0}}$ and such that $\mathsf{AMC}(R,A_{\kappa})$
is given by the $\in_{A_{\kappa}}$-theory common to the
structures151515$H_{\kappa^{+}}^{\mathcal{M}}$ denotes the substructure of
$\mathcal{M}$ whose extension is given by the formula defining
$H_{\kappa^{+}}$ in the model (using the parameter $\kappa$).
$H_{\kappa^{+}}^{\mathcal{M}}$ as $\mathcal{M}$ ranges among the
$\in_{A_{\kappa}}$-models of $R+T_{\in,A_{\kappa}}$.

###  1.3. Forcibility versus absolute model companionship

The following is the major result of the paper161616The reader unaware of what
is $\text{{\sf MM}}^{++}$ or a stationary set preserving forcing can skip the
second and third items of the theorem.:

######  Theorem 1.10.

Let $S$ be the $\in$-theory

$\mathsf{ZFC}+\emph{there exists class many supercompact cardinals}.$

Then there is a set $B\in\mathfrak{spec}_{\mathsf{AMC}}\left(S\right)$ with
$\in_{B}\supseteq\in_{\Delta_{0}}$ and such that for any $\Pi_{2}$-sentence
$\psi$ for $\in_{B}$ and any $\in$-theory $R\supseteq S$ the following are
equivalent:

  1. (a)

$\psi\in\mathsf{AMC}(R,B)$;

  2. (b)

$(R+T_{\in,B})_{\forall\vee\exists}+S+\text{{\sf MM}}^{++}+T_{\in,B}$
proves171717Here and elsewhere we write $\psi^{N}$ to denote the
relativization of $\psi$ to a definable class (or set) $N$; see [14, Def.
IV.2.1] for details. $\psi^{H_{\omega_{2}}}$;

  3. (c)

$R$ proves that $\psi^{H_{\omega_{2}}}$ is forcible181818Here and in the next
item we mean that the $\in$-formula $\theta$ which is $T_{\in,B}$-equivalent
to $\psi$ is such that $\theta^{H_{\omega_{2}}}$ is forcible by the
appropriate forcing. by a stationary set preserving forcing;

  4. (d)

$R$ proves that $\psi^{H_{\omega_{2}}}$ is forcible by some forcing;

  5. (e)

For any $R^{\prime}\supseteq R$,
$\psi+(R^{\prime}+T_{\in,B})_{\forall\vee\exists}$ is consistent.

Furthermore for any $\theta$ which is a boolean combination of
$\Pi_{1}$-sentences for $\in_{B}$ and any $(V,\in)$ model of $S$, TFAE:

  1. (A)

$(V,\in_{B}^{V})$ models $\theta$;

  2. (B)

$(V,\in_{B}^{V})$ models that some forcing notion $P$ forces $\theta$;

  3. (C)

$(V,\in_{B}^{V})$ models that all forcing notions $P$ force $\theta$.

The second part of the theorem shows that forcing cannot change the
$\Pi_{1}$-fragment of the theory of $V$ in signature
$\in_{B}\supseteq\in_{\Delta_{0}}$. Note also that if $(V,\in)$ is a model of
$S$ and $R$ is the $\in_{B}$-theory of its unique extension to a model of
$T_{\in,B}$, we get that a $\Pi_{2}$-sentence $\psi$ for $\in_{B}$ is
consistent with the universal and existential fragments of $R$ if and only if
$\psi^{H_{\omega_{2}}}$ is forcible over $V$.

We will give a detailed definition of $\in_{B}$ at the proper stage (see
Notation 5.3); we anticipate here that it is a recursive set extending
$\in_{\Delta_{0}}$ with a predicate symbol for the non-stationary ideal on
$\omega_{1}$, a constant symbol for $\omega_{1}$, a finite list of function
and relation symbols for provably $\Delta_{1}$-definable operations and
relations, and predicate symbols for all sets of reals definable by
$\in$-formulae without parameters in the Chang model
$L(\mathrm{Ord}^{\omega})$ (which by a result of Woodin form an interesting
subclass of the universally Baire sets, assuming the large cardinal hypothesis
of the Theorem).

###  1.4. The AMC-spectrum of set theory and the continuum problem

We show that large cardinals place $2^{\aleph_{0}}=\aleph_{2}$ in a very
special position of the AMC spectrum of set theory.

We formalize CH and $2^{\aleph_{0}}>\aleph_{2}$ in signature
$\in_{\Delta_{0}}$ as follows:

  * •

$(x\text{ is a cardinal})$ is the $\Pi_{1}$-formula

$(x\text{ is an ordinal})\wedge\forall f\,\left[(f\text{ is a
function}\wedge\operatorname{dom}(f)\in x)\rightarrow\operatorname{ran}(f)\neq
x\right].$

  * •

$(x\text{ is }\aleph_{1})$ is the boolean combination of $\Sigma_{1}$-formulae

$\displaystyle(x\text{ is a cardinal})\wedge(\omega\in x)\wedge$
$\displaystyle\wedge\exists F\,\left[(F:\omega\times x\to
x)\wedge\forall\alpha\in
x\,(F\restriction\omega\times\left\\{\alpha\right\\}\text{ is a surjection on
}\alpha)\right].$

  * •

CH is the $\Sigma_{2}$-sentence

$\exists f\,\left[(f\text{ is a function}\wedge\operatorname{dom}(f)\text{ is
}\aleph_{1})\wedge\forall
r\subseteq\omega\,(r\in\operatorname{ran}(f))\right].$

and $\neg\text{{\sf CH}}$ is the boolean combination of
$\Pi_{2}$-sentences191919We let $\neg\text{{\sf CH}}$ include the
$\Sigma_{2}$-sentence $\exists x\,(x\text{ is }\aleph_{1})$, for otherwise its
failure could be witnessed by the assertion that there is no uncountable
cardinal, a statement which holds true in $H_{\omega_{1}}$, regardless of
whether CH or its negation is true in the corresponding universe of sets.

$\exists x\,(x\text{ is }\aleph_{1})\wedge\forall
f\,\left[(\operatorname{dom}(f)\text{ is }\aleph_{1}\wedge f\text{ is a
function})\rightarrow\exists
r\subseteq\omega\,(r\not\in\operatorname{ran}(f))\right].$

  * •

$(x\text{ is }\aleph_{2})$ is the $\Sigma_{2}$-formula

$\displaystyle(x\text{ is a cardinal})\wedge$ $\displaystyle\wedge\exists
F\exists y\,\left[(y\text{ is }\aleph_{1})\wedge(y\in x)\wedge(F:y\times x\to
x)\wedge\forall\alpha\in x\,(F\restriction
y\times\left\\{\alpha\right\\}\text{ is a surjection on }\alpha)\right].$

  * •

$2^{\aleph_{0}}>\aleph_{2}$ is the boolean combination of $\Pi_{2}$-sentences

$\exists x\,(x\text{ is }\aleph_{2})\wedge\forall f\,\left[(f\text{ is a
function}\wedge\operatorname{dom}(f)\text{ is }\aleph_{2})\rightarrow\exists
r\,(r\subseteq\omega\wedge r\not\in\operatorname{ran}(f))\right].$

  * •

$2^{\aleph_{0}}\leq\aleph_{2}$ is the $\Sigma_{2}$-sentence

$\exists f\,\left[(f\text{ is a function})\wedge\operatorname{dom}(f)\text{ is
}\aleph_{2}\wedge\forall r\,(r\subseteq\omega\rightarrow
r\in\operatorname{ran}(f))\right].$

######  Theorem 1.11.

Let $S$ be the $\in$-theory of Thm. 1.10. The following holds:

  1. (1)

Let $R\supseteq S$ be an $\in$-theory. Assume
$A\in\mathfrak{spec}_{\mathsf{AMC}}\left(R\right)$ with
$\in_{A}\supseteq\in_{\Delta_{0}}$ and $\neg\text{{\sf
CH}}+(R+T_{\in,A})_{\forall\vee\exists}$ is consistent. Then $\text{{\sf
CH}}\not\in\mathsf{AMC}(R,A)$.

  2. (2)

For the signature $\in_{B}$ of Thm. 1.10 $\neg\text{{\sf CH}}$ is in
$\mathsf{AMC}(R,B)$ for any $\in$-theory $R\supseteq S$.

We can prove exactly the same type of result replacing CH by
$2^{\aleph_{0}}>\aleph_{2}$. Specifically Moore introduced in [18] a
$\Pi_{2}$-sentence $\theta_{\mathrm{Moore}}$ for $\in_{\Delta_{0}}$ to show
the existence of a definable well order of the reals in type $\omega_{2}$ in
models of the bounded proper forcing axiom202020We use here the definable
well-ordering of the reals in type $\omega_{2}$ existing in models of bounded
forcing axioms isolated by Moore, but Thm. 1.12 could be proved replacing
$\theta_{\mathrm{Moore}}$ with any other coding device which produce the same
effects, for example those introduced in [7, 21], or the sentence
$\psi_{\mathrm{AC}}$ of Woodin as in [15, Section 6].. We can use
$\theta_{\mathrm{Moore}}$ as follows:

######  Theorem 1.12.

There is a $\Pi_{2}$-sentence $\theta_{\mathrm{Moore}}$ for $\in_{\Delta_{0}}$
such that the following holds:

  1. (1)

$\theta_{\mathrm{Moore}}$ is independent of $S+T_{\Delta_{0}}$, where $S$ is
the $\in$-theory of Thm. 1.10.

  2. (2)

$\mathsf{ZFC}^{-}_{\Delta_{0}}+\exists x\,(x\text{ is
}\aleph_{1})+\theta_{\mathrm{Moore}}$ proves that there exists a well-ordering
of212121More precisely: there is a a $\mathsf{ZFC}^{-}_{\Delta_{0}}$-provably
$\Delta_{1}$-property $\psi(x,y,z)$ such that in any model $\mathcal{M}$ of
the mentioned theory there is a parameter $d\in\mathcal{M}$ such that
$\psi(x,y,d)$ defines an injection of $\mathcal{P}\left(\omega\right)$ of the
model with the class of ordinals of size at most $\omega_{1}$ of the model.
$\mathcal{P}\left(\omega\right)$ in type at most $\omega_{2}$.

  3. (3)

$\mathsf{ZFC}^{-}_{\Delta_{0}}+\exists x\,(x\text{ is
}\aleph_{2})+\theta_{\mathrm{Moore}}$ proves that
$2^{\aleph_{0}}\leq\omega_{2}$.

  4. (4)

For $S$ and $\in_{B}$ the theory and signature considered in Thm. 1.10,
$\exists x\,(x\text{ is }\aleph_{1}),\theta_{\mathrm{Moore}}$ are both in
$\mathsf{AMC}(R,B)$ for any $\in$-theory $R$ extending $S$.

  5. (5)

If $R$ extends $S$, $A\in\mathfrak{spec}_{\mathsf{AMC}}\left(R\right)$ is such
that $\in_{A}\supseteq\in_{\Delta_{0}}$, $\exists x\,(x\text{ is
}\aleph_{2})\in\mathsf{AMC}(R,A)$, $\mathsf{AMC}(R,A)$ implies the Replacement
schema, and

$\theta_{\mathrm{Moore}}+(R+T_{\in,A})_{\forall\vee\exists}+\mathsf{ZFC}$

is consistent, then $2^{\aleph_{0}}>\aleph_{2}$ is not in $\mathsf{AMC}(R,A)$.

The two theorems show an asimmetry between $2^{\aleph_{0}}=\aleph_{2}$ and all
other solutions of the continuum problem assuming large cardinals: for any $R$
extending $\mathsf{ZFC}+$_large cardinals_ there is at least one
$B\in\mathfrak{spec}_{\mathsf{AMC}}\left(R\right)$ with $\neg\text{{\sf CH}}$
(and a definable version of $2^{\aleph_{0}}\leq\aleph_{2}$) in
$\mathsf{AMC}(R,B)$, and this occurs even if $R\models\text{{\sf CH}}$ or
$R\models 2^{\aleph_{0}}>\aleph_{2}$. On the other hand if CH is independent
of $R$, CH is never in $\mathsf{AMC}(R,A)$ for any
$A\in\mathfrak{spec}_{\mathsf{AMC}}\left(R\right)$ (with
$\in_{A}\supseteq\in_{\Delta_{0}}$) and similarly if $\theta_{\mathrm{Moore}}$
is independent of $R$, $2^{\aleph_{0}}>\aleph_{2}$ is never in
$\mathsf{AMC}(R,A)$ for any $A\in\mathfrak{spec}_{\mathsf{AMC}}\left(R\right)$
(with $\in_{A}\supseteq\in_{\Delta_{0}}$ and $\mathsf{AMC}(R,A)$ satisfying
the Replacement schema).

Furthermore the last part of Thm. 1.10 outlines that CH,
$2^{\aleph_{0}}=\aleph_{2}$, $2^{\aleph_{0}}>\aleph_{2}$,
$\theta_{\mathrm{Moore}}$ are all boolean combination of $\Pi_{2}$-sentences
in the signature $\in_{\Delta_{0}}$ which cannot be expressed by boolean
combination of $\Pi_{1}$-sentences for the signature
$\in_{B}\supseteq\in_{\Delta_{0}}$ in models of $S$ (with $S$ and $B$ as in
Thm. 1.10), as their truth value can be changed by forcing.

######  Remark 1.4.1.

The above results are in the same spirit of [1] where it is shown that one can
find two $\in$-sentences $\psi_{0},\psi_{1}$ which are $\Pi_{2}$ for the
signature $\in_{\Delta_{0}}\cup\left\\{\omega_{1}\right\\}$ and such that
$\mathsf{ZFC}+\mathsf{LC}+\text{{\sf CH}}+\psi_{i}$ (where $\mathsf{LC}$ is a
shorthand for sufficiently strong large cardinal axioms) are separately
consistent (and both forcible by a semiproper forcing), but
$\mathsf{ZFC}+\psi_{0}+\psi_{1}\vdash\neg\text{{\sf CH}}$. This latter result
shows that the theory given by the $\Pi_{2}$-sentences for $\in_{B}$ which are
individually consistent with CH cannot hold in the $H_{\aleph_{2}}$ of a model
of $\mathsf{ZFC}+T_{\in,B}+\mathsf{LC}+\text{{\sf CH}}$, while our results
show that the theory of $\Pi_{2}$-sentences for $\in_{B}$ which are strongly
consistent with the $\Pi_{1}$-fragment for $\in_{B}$ of
$\mathsf{ZFC}+\mathsf{LC}+T_{\in,B}$ is not only consistent, but is its AMC
(note also that $\psi_{0},\psi_{1}$ are both axioms of this AMC).

###  1.5. Model theoretic formulation of a strong form of Woodin’s axiom $(*)$

Woodin’s axiom $(*)$ is a technical statement which provides a model of set
theory in which any “reasonable” $\Pi_{2}$-sentence for $H_{\aleph_{2}}$ which
is forcible holds true in the $H_{\aleph_{2}}$ of the model. In view of our
previous considerations the following result is not a suprise and provides a
formulation of a strong form of $(*)$ accessible to anyone able to grasp the
meaning of the following concepts: Chang model, stationarity on $\omega_{1}$,
saturation of the non-stationary ideal222222$\mathbf{NS}\emph{ is saturated}$
asserts that the boolean algebra
$\mathcal{P}\left(\omega_{1}\right)/\mathbf{NS}$ has only antichains of size
at most $\aleph_{1}$. $\mathbf{NS}$ on $\omega_{1}$, model completeness:

######  Theorem 1.13.

Let $S$ and $B$ be as in Thm. 1.10. Assume $(V,\in)$ models
$S+\mathbf{NS}\emph{ is saturated}.$

TFAE:

  1. (1)

$(V,\in)$ models232323See Def. 5.5 for details, roughly it states that there
exists a generic filter for $\mathbb{P}_{\mathrm{max}}$ meeting all dense
subsets of this forcing which belong to the Chang model
$L(\mathrm{Ord}^{\omega})$. $(*)\text{-}\mathcal{A}$ for $\mathcal{A}$ being
the powerset of $\mathbb{R}$ as computed in its Chang model;

  2. (2)

The
$\in_{B}\cup\mathcal{P}\left(\mathbb{R}\right)^{L(\mathrm{Ord}^{\omega})}$-theory242424E.g.
we regard $A\subseteq\mathbb{R}^{k}$ as a $k$-ary predicate symbol for any
$A\subseteq\mathbb{R}^{k}$ which belongs to $L(\mathrm{Ord}^{\omega})$. For
more details on $\in_{B}$ see Notation 5.3 or the comments after Thm. 1.10. of
$H_{\omega_{2}}$ is model complete.

## Structure of the paper

The remainder of this paper is organized as follows:

  * •

Section 2 gives an account on the main properties of AMC.

  * •

Section 3 proves Theorem 1.9, part 1 of Thm. 1.11, and Thm. 1.12 (with the
exception of part 4).

  * •

Section 4 proves the generic invariance of the theory of $V$ in signature
$\in_{\Delta_{0}}\cup\left\\{\omega_{1},\mathbf{NS}_{\omega_{1}}\right\\}$
(more precisely a stronger version of the second part of Thm. 1.10).

  * •

Section 5 completes the proof of Thm. 1.10 (and of the missing parts of
Theorems 1.11, 1.12). It also provides the proof of Thm. 1.13.

  * •

We conclude the paper with some comments and open questions.

Any reader familiar enough with model companionship to follow Section 1.1 can
easily grasp the content of Section 2. Section 3 needs Section 2 and
familiarity with set theory at the level of [14, Chapters I, III, IV]. (no
knowledge of forcing is required). The proofs in Section 4 assume familiarity
with Woodin’s stationary tower forcing, and (in its second part, cfr. Section
4.3) also with some bits of Woodin’s $\mathbb{P}_{\mathrm{max}}$-technology.
Section 5 can be fully appreciated only by readers familiar with forcing
axioms, Woodin’s stationary tower forcing, Woodin’s
$\mathbb{P}_{\mathrm{max}}$-technology and take advantage of Asperó and
Schindler’s proof that $\text{{\sf MM}}^{++}$ implies a strong form of
Woodin’s axiom $(*)$ [3].

##  2\. Existentially closed structures and absolute model companionship

This section proves the model theoretic results on model companionship stated
in Section 1.1. Absolute model companionship (AMC) isolates those (possibly
non-complete) theories $T$ whose model companion is axiomatized by the
$\Pi_{2}$-sentences which are consistent with the universal and existential
fragments of any model of $T$. We also show that AMC is strictly stronger than
model companionship and does not imply (nor is implied by) model
completion252525A detailed account of model companionship, Kaiser Hulls,
existentially closed structures in line with our treatment of this topic can
be found in the notes [23]..

We introduce the following terminology:

######  Notation 2.0.1.

__

  * •

$\sqsubseteq$ denotes the substructure relation between structures.

  * •

$\mathcal{M}\prec_{n}\mathcal{N}$ indicates that $\mathcal{M}$ is a
$\Sigma_{n}$-elementary substructure of $\mathcal{N}$, we omit the $n$ to
denote full-elementarity.

  * •

Similarly for $\tau$-structures $\mathcal{M},\mathcal{N}$,
$\mathcal{M}\equiv_{n}\mathcal{N}$ if both satisfy exactly the same
$\Sigma_{n}$-sentences, $\mathcal{M}\equiv\mathcal{N}$ if they are
elementarily equivalent.

  * •

Given a first order signature $\tau$, $\tau_{\forall}$ denotes the universal
$\tau$-sentences; likewise we interpret
$\tau_{\exists},\tau_{\forall\exists},\dots$. $\tau_{\forall\vee\exists}$
denotes the boolean combinations of universal $\tau$-sentences; likewise we
interpret $\tau_{\forall\exists\vee\exists\forall},\dots$.

  * •

Given a first order theory $T$, $T_{\forall}$ denotes the sentences in
$\tau_{\forall}$ which are consequences of $T$, likewise we interpret
$T_{\exists},T_{\forall\exists},T_{\forall\vee\exists},\dots$.

  * •

We often denote a $\tau$-structure $(M,R^{M}:R\in\tau)$ by $(M,\tau^{M})$.

  * •

We often identify a $\tau$-structure $\mathcal{M}=(M,\tau^{M})$ with its
domain $M$ and an ordered tuple $\vec{a}\in M^{<\omega}$ with its set of
elements.

  * •

We often write $\mathcal{M}\models\phi(\vec{a})$ rather than
$\mathcal{M}\models\phi(\vec{x})[\vec{x}/\vec{a}]$ when $\mathcal{M}$ is
$\tau$-structure $\vec{a}\in\mathcal{M}^{<\omega}$, $\phi$ is a
$\tau$-formula.

  * •

We let the atomic diagram $\Delta_{0}(\mathcal{M})$ of a $\tau$-model
$\mathcal{M}=(M,\tau^{M})$ be the family of quantifier free sentences
$\phi(\vec{a})$ in signature $\tau\cup M$ such that
$\mathcal{M}\models\phi(\vec{a})$.

###  2.1. Byembeddability versus absolute byembeddability

Let us give a proof of the following well known fact, since it will be helpful
to outline the subtle difference between model companionship and absolute
model companionship.

######  Lemma 2.1.1.

Let $\tau$ be a signature and $T$, $S$ be $\tau$-theories. TFAE:

  1. (1)

$T_{\forall}\supseteq S_{\forall}$.

  2. (2)

For any $\mathcal{M}$ model of $T$ there is $\mathcal{N}$ model of $S$
superstructure of $\mathcal{M}$.

###### Proof.

__

1 implies 2:

Assume $\mathcal{M}$ models $T$ and is such that no $\mathcal{N}$ model of $S$
is a superstructure of $\mathcal{M}$. Then $S\cup\Delta_{0}(\mathcal{M})$ is
not consistent (where $\Delta_{0}(\mathcal{M})$ is the atomic diagram of
$\mathcal{M}$). By compactness find $\psi(\vec{a})\in\Delta_{0}(\mathcal{M})$
quantifier-free sentence such that $S+\psi(\vec{a})$ is inconsistent. This
gives that

$S\models\forall\vec{x}\,\neg\psi(\vec{x})$

since $\vec{a}$ is a string of constant symbols all outside of $\tau$.
Therefore $\forall\vec{x}\,\neg\psi(\vec{x})\in S_{\forall}\subseteq
T_{\forall}$. Hence

$\mathcal{M}\models\forall\vec{x}\,\neg\psi(\vec{x})\wedge\psi(\vec{a}),$

a contradiction.

2 implies 1:

Left to the reader.

∎

The following is a natural question: assume $S$ and $T$ are $\tau$-theories
such that $T_{\forall}=S_{\forall}$, can we extend a model $\mathcal{M}$ of
$T$ to a superstructure $\mathcal{M}$ of $S$ so that $\mathcal{M}$ and
$\mathcal{N}$ satisfy exactly the same universal sentences? The answer is no
as shown by $\tau=\left\\{\cdot,+,0,1\right\\}$, $T$ the $\tau$-theory of
fields, $S$ the $\tau$-theory of algebraically closed fields: it is easy to
see that $T_{\forall}=S_{\forall}$ in view of Lemma 2, but $\mathbb{Q}$ cannot
be extended to an algebraically closed field without killing the universal
$\tau$-sentence stating the non existence of the square root of $-1$.

The clarification of this issue is what has brought our attention to
$T_{\forall\vee\exists}$.

Note that any sentence in $T_{\forall\vee\exists}$ is either logically
equivalent to $\theta\vee\psi$ or equivalent to $\theta\wedge\psi$ with
$\theta$ universal and $\psi$ existential.

Note also that $T_{\forall\vee\exists}$ may contain more information than
$T_{\forall}\cup T_{\exists}$ as there could be a universal $\theta\not\in
T_{\forall}$ and an existential $\psi\not\in T_{\exists}$ with
$\theta\vee\psi\in T_{\forall\vee\exists}$.

######  Lemma 2.1.2.

Let $\tau$ be a signature and $T$, $S$ be $\tau$-theories. TFAE:

  1. (1)

$T_{\forall\vee\exists}\supseteq S_{\forall\vee\exists}$.

  2. (2)

For any $\mathcal{M}$ model of $T$ there is $\mathcal{N}$ model of $S$
superstructure of $\mathcal{M}$ realizing exactly the same universal
sentences.

  3. (3)

For every boolean combination of universal sentences $\theta$, $T+\theta$ is
consistent only if so is $S+\theta$.

###### Proof.

__

1 implies 2:

Assume $\mathcal{M}$ models $T$ and is such that no $\mathcal{N}$ model of $S$
which is a superstructure of $\mathcal{M}$ realizes exactly the same universal
sentences.

For any such $\mathcal{N}$ with $\mathcal{M}\sqsubseteq\mathcal{N}\models S$
we get that some universal $\tau$-sentence $\theta_{\mathcal{N}}$ true in
$\mathcal{M}$ fails in $\mathcal{N}$. We claim that the
$\tau\cup\mathcal{M}$-theory

$S^{*}=\Delta_{0}(\mathcal{M})\cup
S\cup\left\\{\theta_{\mathcal{N}}:\mathcal{M}\sqsubseteq\mathcal{N},\,\mathcal{N}\models
S\right\\}$

is inconsistent. If not let $\mathcal{P}^{*}$ be a model of $S^{*}$. Then
$\mathcal{P}=(\mathcal{P}^{*}\restriction\tau)\sqsupseteq\mathcal{M}$ is a
model of

$S\cup\left\\{\theta_{\mathcal{N}}:\mathcal{M}\sqsubseteq\mathcal{N},\,\mathcal{N}\models
S\right\\}.$

Hence it models $\theta_{\mathcal{P}}$ and $\neg\theta_{\mathcal{P}}$ at the
same time.

By compactness we can find a universal sentence $\phi_{\mathcal{M}}$ given by
the conjunction of a finite set

$\left\\{\theta_{\mathcal{P}_{i}}:\,i=1,\dots,n,\mathcal{M}\sqsubseteq\mathcal{P}_{i}\models
S\right\\}$

and a quantifier free sentence $\psi_{\mathcal{M}}(\vec{a})$ of
$\Delta_{0}(\mathcal{M})$ such that

$S+\psi_{\mathcal{M}}(\vec{a})+\phi_{\mathcal{M}}$

is inconsistent. Hence

$S\models\neg\phi_{\mathcal{M}}\vee\neg\exists\vec{x}\,\psi_{\mathcal{M}}(\vec{x}).$

Now observe that:

  * •:

$\neg\phi_{\mathcal{M}}\vee\neg\exists\vec{x}\,\psi_{\mathcal{M}}(\vec{x})$ is
a boolean combination of universal sentences,

  * •:

$\mathcal{M}\models
T+\exists\vec{x}\,\psi_{\mathcal{M}}(\vec{x})\wedge\phi_{\mathcal{M}}$.

Therefore we get that
$\neg\phi_{\mathcal{M}}\vee\neg\exists\vec{x}\,\psi_{\mathcal{M}}(\vec{x})$ is
in $S_{\forall\vee\exists}\setminus T_{\forall\vee\exists}$.

2 implies 3:

Left to the reader.

3 implies 1:

If $T_{\forall\vee\exists}\not\supseteq S_{\forall\vee\exists}$ there is
$\theta\in S_{\forall\vee\exists}\setminus T_{\forall\vee\exists}$. Then
$\neg\theta$ is inconsistent with $S$ and consistent with $T$.

∎

######  Definition 2.1.3.

Let $\tau$ be a signature and $T,S$ be $\tau$-theories.

  * •

$T$ and $S$ are _cotheories_ if $T_{\forall}=S_{\forall}$.

  * •

$T$ and $S$ are _absolute cotheories_ if
$T_{\forall\vee\exists}=S_{\forall\vee\exists}$.

######  Remark 2.1.4.

Say that a $\tau$-theory $T$ is _$\Pi_{1}$ -complete_ if $T\vdash\phi$ or
$T\vdash\neg\phi$ for any universal $\tau$-sentence $\phi$.

Now consider the $\left\\{+,\cdot,0,1\right\\}$-theories $\mathsf{ACF}_{0}$
and $\mathsf{Fields}_{0}$ expanding $\mathsf{ACF}$ and $\mathsf{Fields}$ with
the axioms fixing the characteristic of their models to be $0$. Note that
$\mathsf{ACF}_{0}$ is $\Pi_{1}$-complete (it is actually complete) while
$\mathsf{Fields}_{0}$ is not, even if
$(\mathsf{ACF}_{0})_{\forall}=(\mathsf{Fields}_{0})_{\forall}$. In particular
$T_{\forall}=S_{\forall}$ is well possible with $T$ $\Pi_{1}$-complete and $S$
not $\Pi_{1}$-complete. Absolute cotheories rule out this confusing
discrepancy. In particular we will use the following trivial fact crucially in
the proof of Lemma 2.3.4: if $S$ is a complete theory $S_{\forall\vee\exists}$
is $\Pi_{1}$-complete, while $S_{\forall}$ may not.

###  2.2. Existentially closed models, Kaiser hulls, strong consistency

######  Definition 2.2.1.

Given a $\tau$-theory $T$, $\mathcal{M}$ is $T$-existentially closed ($T$-ec)
if:

  * •

There is some $\mathcal{N}\sqsupseteq\mathcal{M}$ which models $T$.

  * •

$\mathcal{M}\prec_{1}\mathcal{N}$ for all superstructures
$\mathcal{N}\sqsupseteq\mathcal{M}$ which model $T$.

######  Remark 2.2.2.

Among the many nice properties of $T$-ec structures, note that
$\Pi_{2}$-sentences (with parameters in $\mathcal{M}$) which hold in some
$T$-model $\mathcal{N}$ superstructure of $\mathcal{M}$ reflect to
$\mathcal{M}$.

In view of the above Lemmas it is not hard to check the following:

######  Fact 2.2.3.

TFAE for a $\tau$-theory $T$ and a $\tau$-structure $\mathcal{M}$:

  1. (1)

$\mathcal{M}$ is $T$-ec.

  2. (2)

$\mathcal{M}$ is $T_{\forall\vee\exists}$-ec.

  3. (3)

$\mathcal{M}$ is $T_{\forall}$-ec.

######  Definition 2.2.4.

Let $T$ be a $\tau$-theory.

  * •

A $\tau$-sentence $\psi$ is _strongly $T_{\forall\vee\exists}$-consistent_ if
$\psi+R_{\forall\vee\exists}$ is consistent for all $R\supseteq T$.

  * •

The _Kaiser hull of $T$_ ($\mathsf{KH}(T)$) consists of the
$\Pi_{2}$-sentences for $\tau$ which hold in all $T$-ec models.

  * •

The _strong consistency hull of $T$_ ($\mathsf{SCH}(T)$) consists of the
$\Pi_{2}$-sentences for $\tau$ which are strongly
$T_{\forall\vee\exists}$-consistent.

The Kaiser hull of a theory is a well known notion describing an equivalent of
model companionship which can be defined also for non-companionable theories
(see for example [20, Lemma 3.2.12, Lemma 3.2.13, Thm. 3.2.14]); the strong
consistency hull is a slight weakening of the Kaiser hull not considered till
now (at least to my knowledge) and which does the same with respect to the
notion of absolute model companionship (defined below in Def. 2.3.1).

######  Fact 2.2.5.

For any $\tau$-theory $T$:

  1. (i)

$\mathsf{KH}(T)_{\forall}=T_{\forall}$;

  2. (ii)

$\mathsf{SCH}(T)_{\forall\vee\exists}=T_{\forall\vee\exists}$;

  3. (iii)

$\mathsf{SCH}(T)\subseteq\mathsf{KH}(T)$;

  4. (iv)

$\mathsf{SCH}(T)=\mathsf{KH}(T)$ if
$T_{\forall\vee\exists}=\mathsf{KH}(T)_{\forall\vee\exists}$, which is the
case if $T$ is complete.

  5. (v)

For any $\Pi_{2}$-sentence $\psi$ such that $T_{\forall\vee\exists}+\psi$ is
consistent, there is a model of $\mathsf{SCH}(T)+\psi$.

###### Proof.

__

  1. (i)

By definition any model of $\mathsf{KH}(T)$ is a model of $T_{\forall}$;
conversely any model of $T$ can be extended to a $T$-ec model (see for example
[20, Lemma 3.2.11]). We conclude by Lemma 2.1.1.

  2. (ii)

Trivial.

  3. (iii)

Assume a $\Pi_{2}$-sentence $\psi$ is strongly
$T_{\forall\vee\exists}$-consistent. Let $\mathcal{M}$ be a $T$-ec model. Then
$\mathcal{M}$ is $T_{\forall\vee\exists}$-ec. Let $R$ be the $\tau$-theory of
$\mathcal{M}$. Since $\mathcal{M}$ is $T$-ec, any superstructure of
$\mathcal{M}$ which models $T$ is also a model of $R_{\forall\vee\exists}$ (by
Fact 2.2.3). Since $\psi$ is strongly $T_{\forall\vee\exists}$-consistent,
$\psi+R_{\forall\vee\exists}$ is consistent. By Lemma 2.1.2, $\psi$ holds in
some $\mathcal{N}\sqsupseteq\mathcal{M}$ which models
$R_{\forall\vee\exists}$. Since $\mathcal{M}$ is $T$-ec and
$R_{\forall\vee\exists}\supseteq T_{\forall}$, we get that $\psi$ reflects to
$\mathcal{M}$ (being a $\Pi_{2}$-sentence which holds in $\mathcal{N}$ which
is a $\Sigma_{1}$-superstructure of $\mathcal{M}$).

  4. (iv)

Assume a $\Pi_{2}$-sentence $\psi$ is in $\mathsf{KH}(T)$. Let $R$ be any
complete extension of $T$ and $\mathcal{M}$ be a model of $R$. By Lemma 2.1.2
(since $T_{\forall\vee\exists}=\mathsf{KH}(T)_{\forall\vee\exists}$) there is
$\mathcal{N}$ which is a model of $\mathsf{KH}(T)$ and of
$R_{\forall\vee\exists}$. In particular $\mathcal{N}$ models
$\psi+R_{\forall\vee\exists}$. Since $R$ is arbitrary, $\psi$ is strongly
$T_{\forall\vee\exists}$-consistent.

Clearly if $T$ is complete, $T_{\forall\vee\exists}$ is $\Pi_{1}$-complete,
and $\psi$ is strongly $T_{\forall\vee\exists}$-consistent if and only if
$\psi+T_{\forall\vee\exists}$ is consistent. We conclude also in this case
that a $\Pi_{2}$-sentence $\psi$ holds in some $T$-ec model if and only if it
is strongly $T_{\forall\vee\exists}$-consistent.

  5. (v)

Note that $\mathsf{SCH}(T)$ is axiomatized by its $\Pi_{2}$-fragment and
$\mathsf{SCH}(T)_{\forall\vee\exists}=T_{\forall\vee\exists}$. Therefore we
can apply Lemma 2.2.7 below.

∎

######  Remark 2.2.6.

There can be $\tau$-theories $T$ whose Kaiser hull strictly contains its
strong consistency hull.

Consider the $\left\\{0,1,\cdot,+\right\\}$-theories $\mathsf{ACF}_{0}$ (of
algebraically closed fields of characteristic $0$) and $\mathsf{Fields}_{0}$
(of fields of characteristic $0$). Note that $\mathsf{ACF}_{0}$ is complete
while $\mathsf{Fields}_{0}$ is not. Furthermore $\mathsf{ACF}_{0}$ is the
Kaiser hull of $\mathsf{Fields}_{0}$ (note that: $\mathsf{ACF}_{0}$ is
axiomatized by its $\Pi_{2}$-fragment; any $\mathsf{Fields}_{0}$-ec model is
an algebraically closed field; any model of $\mathsf{ACF}_{0}$ is
$\mathsf{Fields}_{0}$-ec).

We get that $\exists x\,(x^{2}+1=0)$ is a $\Pi_{2}$-sentence (in fact
existential) in the Kaiser hull of $\mathsf{Fields}_{0}$ but not in its strong
consistency hull, since it is not consistent with $R_{\forall\vee\exists}$,
where $R$ is the $\left\\{0,1,\cdot,+\right\\}$-theory of the rationals.

######  Lemma 2.2.7.

Let $S,T$ be $\tau$-theories such that $S_{\forall\vee\exists}\subseteq
T_{\forall\vee\exists}$ and $S$ is axiomatized by its $\Pi_{2}$-fragment. Then
for any $\Pi_{2}$-sentence $\psi$ consistent with $T$ there is a model of
$S+\psi$.

###### Proof.

We prove a stronger conclusion which is the following:

> _Let $R$ be a complete theory extending $T+\psi$. Then there is a model of
> $R_{\forall\vee\exists}+S+\psi$._

Let $\left\\{\mathcal{M}_{n}:\,n\in\omega\right\\}$ be a sequence of
$\tau$-structures such that for all $n\in\omega$:

  * •

$\mathcal{M}_{n}$ is a $\tau$-substructure of $\mathcal{M}_{n+1}$;

  * •

$\mathcal{M}_{n}$ models $R_{\forall\vee\exists}$;

  * •

$\mathcal{M}_{2n}$ models $R$;

  * •

$\mathcal{M}_{2n+1}$ models $S$.

Such a sequence can be defined letting $\mathcal{M}_{0}$ be a model of $R$,
$\mathcal{M}_{1}$ be a model of $S$ which satisfies $R_{\forall\vee\exists}$
(which is possible in view of Lemma 2.1.2) and defining $\mathcal{M}_{n}$ as
required for all other $n$ appealing to the fact that
$S+R_{\forall\vee\exists}$ and $T+\psi+R_{\forall\vee\exists}$ are absolute
cotheories with $R_{\forall\vee\exists}$ being the $\Pi_{1}$-complete fragment
shared by both theories. Then
$\mathcal{M}=\bigcup_{n\in\omega}\mathcal{M}_{n}$ is a model of
$R_{\forall\vee\exists}+S+\psi$ since it realizes all $\Pi_{2}$-sentences
which hold in an infinite set of $\mathcal{M}_{n}$ (see for example [20, Lemma
3.1.6]) and satisfies exactly the same $\Pi_{1}$-sentences of each of the
$\mathcal{M}_{n}$. ∎

Note that the above cannot be proved if $S,T$ are just cotheories: performing
the above construction under this weaker assumption, we may not be able to
define $\mathcal{M}_{2}$ as required if $\mathcal{M}_{1}$ does not realize
exactly the same universal sentences of $\mathcal{M}_{0}$.

A useful consequence of the above results is the following:

######  Proposition 2.2.8.

Assume $T$ is a $\tau$-theory and $\psi$ is a $\Pi_{2}$-sentence for $\tau$
such that $T+\psi$ is consistent. Then for all
$A\subseteq\mathsf{Form}_{\tau}\times 2$ we have that
$\neg\psi\not\in\mathsf{SCH}(T+T_{\tau,A})$.

###### Proof.

By assumption for any $A\subseteq\mathsf{Form}_{\tau}\times 2$
$(T+T_{\tau,A})_{\forall\vee\exists}$ is consistent. By Fact 2.2.5v, there is
a model of $\mathsf{SCH}(T+T_{\tau,A})+\psi$, hence
$\neg\psi\not\in\mathsf{SCH}(T+T_{\tau,A})$. ∎

###  2.3. Absolute model companionship

######  Definition 2.3.1.

A $\tau$-theory $T$ is:

  * •

_model complete_ if $\mathcal{M}\models T$ if and only if it is
$T_{\forall}$-ec;

  * •

the _model companion_ of a $\tau$-theory $S$ if $T$ and $S$ are cotheories and
$T$ is model complete;

  * •

the _absolute model companion_ (AMC) of a $\tau$-theory $S$ if $T$ and $S$ are
absolute cotheories and $T$ is model complete.

Our definition of model completeness and model companionship takes advantage
of [8, Prop. 3.5.15].

We will use repeatedly Robinson’s test providing different equivalent
characterizations of model completeness:

######  Lemma 2.3.2.

[20, Lemma 3.2.7] (Robinson’s test) Let $T$ be a $\tau$-theory. The following
are equivalent:

  1. (a)

$T$ is model complete.

  2. (b)

Whenever $\mathcal{M}\sqsubseteq\mathcal{N}$ are models of $T$,
$\mathcal{M}\prec\mathcal{N}$.

  3. (c)

Each _existential_ $\tau$-formula $\phi(\vec{x})$ in free variables $\vec{x}$
is $T$-equivalent to a universal $\tau$-formula $\psi(\vec{x})$ in the same
free variables.

  4. (d)

Each $\tau$-formula $\phi(\vec{x})$ in free variables $\vec{x}$ is
$T$-equivalent to a universal $\tau$-formula $\psi(\vec{x})$ in the same free
variables.

######  Remark 2.3.3.

d (or c) shows that being a model complete $\tau$-theory $T$ is expressible by
a $\Delta_{0}$-property in parameters $\tau,T$ in any model of $\mathsf{ZFC}$,
hence it is absolute with respect to forcing. They also show that quantifier
elimination implies model completeness. c also shows that model complete
theories are axiomatized by their $\Pi_{2}$-fragment.

The following characterization of absolute model companionship has brought our
attention to this notion.

######  Lemma 2.3.4.

Assume $T,T^{\prime}$ are $\tau$-theories and $T^{\prime}$ is model complete.
TFAE:

  1. (i)

$T^{\prime}$ is the absolute model companion of $T$.

  2. (ii)

$T^{\prime}$ is axiomatized by the strong consistency hull of $T$.

###### Proof.

__

i implies ii:

First of all we note that any model complete theory $S$ is axiomatized by its
strong consistency hull in view of Robinson’s test d and Fact 2.2.3.

We also note that for absolute cotheories $T,T^{\prime}$, their strong
consistency hull overlap (in view of Lemma 2.1.2).

Putting everything together we obtain the desired implication.

ii implies i:

Note that for $\theta$ a boolean combination of universal $\tau$-sentences, we
have that $\theta$ is in the strong consistency hull of some $\tau$-theory $S$
if and only if $\theta\in S_{\forall\vee\exists}$. Combined with ii, this
gives that $T^{\prime}_{\forall\vee\exists}=T_{\forall\vee\exists}$.

∎

Finally the following Lemma motivates our terminology for AMC:

######  Lemma 2.3.5.

Assume $T,T^{\prime}$ are $\tau$-structures such that $T^{\prime}$ is the AMC
of $T$. Then any $S$ extending $T$ has as AMC $T^{\prime}+S_{\forall}$.

Note that this fails for the standard notion of model companionship:
$\mathsf{ACF}$ is the model companion of $\mathsf{Fields}$ in signature
$\tau=\left\\{0,1,\cdot,+\right\\}$, but if $S$ is the theory of the rationals
in signature $\tau$, $S_{\forall}+\mathsf{ACF}$ is inconsistent, hence it
cannot be the model companion of $S$.

###### Proof.

Assume $S\supseteq T$ is consistent. If $\mathcal{M}\models S$, $\mathcal{M}$
has a superstructure which models $T^{\prime}+S_{\forall\vee\exists}$, since
$T$ and $T^{\prime}$ are absolute cotheories. This gives that
$S^{\prime}=T^{\prime}+S_{\forall}$ is consistent. Since $T^{\prime}$ is model
complete, so is $S^{\prime}$ by Robinson’s test (cfr. Remark 2.3.3 and Lemma
2.3.2c). Now observe that $S^{\prime}$ and $S$ satisfy item 2 of Lemma 2.1.2
(since $S^{\prime}\supseteq T^{\prime}$ and $S\supseteq T$ with $T$ and
$T^{\prime}$ absolute cotheories), yielding easily that
$S_{\forall\vee\exists}=S_{\forall\vee\exists}^{\prime}$. Therefore
$S^{\prime}$ is the AMC of $S$. ∎

######  Remark 2.3.6.

Absolute model companionship is strictly stronger than model companionship: if
$T$ is model complete, $T$ is the model companion of $T_{\forall}$ and the
absolute model companion of $T_{\forall\vee\exists}$; the two notions do not
coincide whenever $T_{\forall}$ is strictly weaker than
$T_{\forall\vee\exists}$.

If $T^{\prime}$ is the model companion of $T$,
$T_{\forall\vee\exists}^{\prime}\supseteq T_{\forall\vee\exists}$: assume
$\mathcal{M}\models T^{\prime}$, then there is a superstructure $\mathcal{N}$
of $\mathcal{M}$ which models $T$ (since $T^{\prime}$ is the model companion
of $T$). Now $\mathcal{M}\prec_{1}\mathcal{N}$, since $\mathcal{M}$ is $T$-ec.
Hence $\mathcal{N}$ has the same $\Pi_{1}$-theory of $\mathcal{M}$. The
inclusion can be strict as shown by the counterexample given by
$\mathsf{Fields}$ versus $\mathsf{ACF}$.

Recall that $T$ is the model completion of $S$ if it is its model companion
and admits quantifier elimination (See [8, Prop. 3.5.19]).

Absolute model companionship does not imply model completion:

######  Fact 2.3.7.

Let $\in_{B}$ and $S$ be the signature and theory appearing in Theorem 1.10.
Then $\mathsf{AMC}(S,B)$ does not admit quantifier elimination, hence
$S+T_{\in,B}$ does not have the amalgamation property and has no model
completion.

###### Proof.

We show that $S+T_{\in,B}$ does not have the amalgamation property. This
suffices to prove the Fact by [8, Prop. 3.5.19]. Given $(V,\in)$ model of
$S+\text{{\sf MM}}^{++}$, let $G$ be $V$-generic for Namba forcing at
$\aleph_{2}$ and $H$ be $V$-generic for
$\operatorname{Coll}(\omega_{1},\omega_{2})$. Note that (by Thm. 4.2.1, since
all predicate symbols of $\in_{B}$ are either universally Baire sets or
$\Delta_{0}$-definable formulae or the non-stationary ideal on $\omega_{1}$)

$(V,\in_{B}^{V})\sqsubseteq(V[G],\in_{B}^{V[G]}),(V[H],\in_{B}^{V[H]}).$

If $W$ is an $\in_{B}$-amalgamation of $V[G]$ and $V[H]$ over $V$, and $W$
models $S+T_{\in,B}$, we get that in $W$ $\omega_{2}^{V}$ has cofinality
$\omega_{1}^{V}$ and $\omega$ at the same time (both properties are
expressible by $\Delta_{0}$-formulae in parameters
$\omega,\omega_{1}^{V},\omega_{2}^{V},f,g$ stating that
$f:\omega\to\omega_{2}^{V}$ is cofinal and $g:\omega_{1}\to\omega_{2}^{V}$ is
cofinal), hence $\omega_{1}^{V}$ is countable in $W$. This is impossible since
$(H_{\omega_{2}}^{V},\in_{B}^{V})\prec_{1}(W,\in_{A}^{W})$ and
$\omega_{1}^{V}$ is uncountable in $(H_{\omega_{2}}^{V},\in_{B}^{V})$. ∎

###  2.4. Preservation of the substructure relation and of
$\Sigma_{1}$-elementarity by expansions via definable Skolem functions

These technical results will be needed for the enhanced version of Levy
absoluteness given by Lemma 3.1. This is crucial for the proofs of Thm. 1.9i
and in Sections 4 and 5 to define the signature $\in_{B}$ of Thm. 1.10. Recall
Notation 1.6.

######  Definition 2.4.1.

Let $S$ be a $\tau$-theory. A $\tau$-formula $\phi(\vec{x})$ is
$\Delta_{1}(S)$ if there are quantifier free $\tau$-formulae
$\psi_{\phi}(\vec{x},\vec{y})$ and $\theta_{\phi}(\vec{x},\vec{z})$ such that
$S$ proves

$\forall\vec{x}\,[\phi(\vec{x})\leftrightarrow\forall\vec{y}\,\psi_{\phi}(\vec{x},\vec{y})\leftrightarrow\exists\vec{z}\,\theta_{\phi}(\vec{x},\vec{z})].$

######  Fact 2.4.2.

Assume $T$ is a $\tau$-theory, $\mathcal{M},\mathcal{N}$ are $\tau$-models of
$T$ and $A\subseteq\mathsf{Form}_{\tau}\times 2$ is such that:

  * •

for each $\langle\phi,i\rangle$ in $A$, $\phi$ is a provably
$\Delta_{1}(T)$-formula;

  * •

for each $(\phi,1)\in A$

$T\models\forall\vec{x}\exists!y\phi(\vec{x},y).$

The following holds for $\mathcal{M}_{A},\mathcal{N}_{A}$ the unique
expansions of $\mathcal{M},\mathcal{N}$ to $\tau_{A}$-models of $T_{\tau,A}$
both interpreting as some $a\in\mathcal{M}$ the constant $c_{\tau}$:

  1. (A)

If $\mathcal{M}\sqsubseteq\mathcal{N}$ (for $\tau$), then
$\mathcal{M}_{A}\sqsubseteq\mathcal{N}_{A}$ (for $\tau_{A}$).

  2. (B)

if $\mathcal{M}\equiv_{1}\mathcal{N}$ (for $\tau$), then
$\mathcal{M}_{A}\equiv_{1}\mathcal{N}_{A}$ (for $\tau_{A}$).

  3. (C)

if $\mathcal{M}\prec_{1}\mathcal{N}$ (for $\tau$), then
$\mathcal{M}_{A}\prec_{1}\mathcal{N}_{A}$ (for $\tau_{A}$).

###### Proof.

The proof is an elementary application of basic observations. First one shows
that whenever $\phi(\vec{x},y)$ is a provably $\Delta_{1}(T)$ $\tau$-formula
and $A_{0}=\left\\{\langle\phi,0\rangle\right\\}$, then:

  1. (i)

for every quantifier free formula $\gamma(\vec{v})$ of $\tau_{A_{0}}$ there
are a $\psi_{\gamma}(\vec{v},\vec{u}),\theta_{\gamma}(\vec{v},\vec{z})$
quantifier free $\tau$-formulae such that $T+T_{\tau,A_{0}}$ models

$\forall\vec{v}\,[\gamma(\vec{v})\leftrightarrow\forall\vec{u}\,\psi_{\gamma}(\vec{v},\vec{u})\leftrightarrow\exists\vec{z}\,\theta_{\gamma}(\vec{v},\vec{z})].$

Then one shows that if furthermore

$T\models\forall\vec{x}\exists!y\phi(\vec{x},y),$

and let $A_{1}=\left\\{\langle\phi,1\rangle\right\\}$. Then:

  1. (ii)

i holds replacing $A_{0}$ with $A_{1}$ all over.

Using the above inductively one proves that every quantifier free
$\tau_{A}$-formula is provably equivalent over $T+T_{\tau,A}$ both to a
$\Sigma_{1}$-formula for $\tau$ and to a $\Pi_{1}$-formula for $\tau$.

The three conclusions for $\mathcal{M}_{A},\mathcal{N}_{A}$ follow then
easily. ∎

###  2.5. Summing up

  * •

We see model completeness, model companionship, AMC as tameness properties of
elementary classes $\mathcal{E}$ defined by a theory $T$ rather than of the
theory $T$ itself: these model-theoretic notions outline certain regularity
patterns for the substructure relation on models of $\mathcal{E}$, patterns
which may be unfolded only when passing to a signature distinct from the one
in which $\mathcal{E}$ is first axiomatized (much the same way as it occurs
for Birkhoff’s characterization of algebraic varieties in terms of universal
theories).

  * •

We will see in the next sections that set theory together with large cardinal
axioms has (until now unexpected) tameness properties when formalized in
certain natural signatures (already implicitly considered in most of the
prominent set theoretic results of the last decades). These tameness
properties couple perfectly with well known (or at least published) generic
absoluteness results. The notion of AMC-spectrum gives an additional model
theoretic criterium for selecting these “natural” signatures out of the
continuum many signatures which produce definable extensions of
$\mathsf{ZFC}$.

##  3\. The AMC spectrum of set theory

In this section we prove the basic properties of the AMC spectrum of set
theory. Specifically we prove Thm. 1.9, Thm. 1.11, Thm. 1.12 (items 2 of Thm.
1.11 and 4 of Thm. 1.12 are proved conditionally to the proof of Thm. 1.10).

In 3.1 we define precisely the signatures in which we need to formalize set
theory in order to prove all our main results.

3.2 formulates in precise mathematical terms the role Levy absoluteness plays
in the proofs of the above theorems.

In 3.3 we show that for any $A\subseteq\mathsf{Form}_{\in}\times 2$ with
$\in_{A}\supseteq\in_{\Delta_{0}}$ in the model companionship spectrum of set
theory and $R\supseteq\mathsf{ZFC}$, $\mathsf{MC}(R,A)$ is a model of a large
chun of the replacement schema, choice, and is closed under Goedel operations.
Typically these axioms characterize the structures of type $H_{\kappa}$ for
$\kappa$ a regular cardinal.

In 3.4 we show how to produce sets $A$ such that
$\in_{A}\supseteq\in_{\Delta_{0}}$ and the AMC of set theory with respect to
$\in_{A}$ exists and is the theory of $H_{\kappa^{+}}$ for some infinite
cardinal $\kappa$.

In 3.5 we show why it is artificial to put CH and $2^{\aleph_{0}}>\aleph_{2}$
in some AMC of set theory.

3.3, 3.4, 3.5 can be read independently of one another.

All proofs in this section are elementary (e.g. knowledge of [14, Chapters
$I$, $III$, $IV$] suffices to follow the arguments); however —especially in
3.4— the notation is heavy. We haven’t been able to simplify it.

###  3.1. Basic notation and terminology

We introduce the basic signatures and fragments of set theory we will always
include in any signature of interest to us.

######  Notation 3.1.1.

We let $\in_{\Delta_{0}}$ be $\in_{D}$ for
$D\subseteq\mathsf{Form}_{\in}\times 2$ extending the set
$\Delta_{0}\times\left\\{0\right\\}$ with the pairs $(\phi,1)$ as $\phi$
ranges over the following $\Delta_{0}$-formulae:

  * •

The $\Delta_{0}$-formulae $\phi_{\omega}(x)$, $\phi_{\emptyset}(x)$ defining
$\emptyset$ and $\omega$ in any model of $\mathsf{ZF}^{-}$, where the latter
includes all axioms of $\mathsf{ZF}$ with the exception of power-set axiom
(also we denote by $\omega$ and $\emptyset$ the constants
$f_{\phi_{\emptyset}}$, $f_{\phi_{\omega}}$).

  * •

The $\Delta_{0}$-formulae $\phi_{i}(\vec{x},y)$ as $G_{i}$ ranges over the
Goedel operations $G_{1},\dots,G_{10}$ as defined in [12, Def. 13.6] and
$\phi_{i}(\vec{x},y)$ is the $\Delta_{0}$-formula defining the graph of
$G_{i}$ in any $\in$-model of 262626In models of $\mathsf{ZF}^{-}$ the Goedel
operations $G_{1},\dots,G_{10}$ as listed and defined in [12, Def. 13.6] and
their compositions have as graph the extension of a $\Delta_{0}$-formula (by
[12, Lemma 13.7]). $\mathsf{ZF}^{-}$.

We let $T_{\Delta_{0}}$ be given by the axioms:

(1) $\forall\vec{x}\,(R_{\forall z\in
y\phi}(y,z,\vec{x})\leftrightarrow\forall z(z\in y\rightarrow
R_{\phi}(y,z,\vec{x})),$ (2)
$\forall\vec{x}\,[R_{\phi\wedge\psi}(\vec{x})\leftrightarrow(R_{\phi}(\vec{x})\wedge
R_{\psi}(\vec{x}))],$ (3)
$\forall\vec{x}\,[R_{\neg\phi}(\vec{x})\leftrightarrow\neg R_{\phi}(\vec{x})]$
(4) $\forall x\,(x\not\in\emptyset)$ (5) $\omega\emph{ is a non-empty ordinal
all whose elements are successor ordinals or $\emptyset$}.$ (6)
$\forall\vec{x}\,\exists!y\,R_{\phi_{i}}(\vec{x},y))$ (7)
$\forall\vec{x}\,\forall y\,[y=G_{i}(\vec{x})\leftrightarrow
R_{\phi_{i}}(\vec{x},y)]$

for the Goedel operations $G_{1},\dots,G_{10}$.

We axiomatize suitable fragments of the $\in$-theory
$\mathsf{ZFC}+T_{\Delta_{0}}$ as follows:

  * •

$\mathsf{Z}^{-}_{\Delta_{0}}$ stands for the $\in_{\Delta_{0}}$-theory given
by:

    1. (a)

the Extensionality Axiom

$\forall x,y,z\,\left[(z\in x\leftrightarrow z\in y)\rightarrow x=y\right],$

    2. (b)

the Foundation Axiom

$\forall x\left[x=\emptyset\vee\exists y\in x\,\forall z\in x\,(z\not\in
y)\right],$

    3. (c)

$T_{\Delta_{0}}$.

  * •

$\mathsf{Z}_{\Delta_{0}}$ enriches $\mathsf{Z}^{-}_{\Delta_{0}}$ adding the
power-set axiom

$\forall x\exists y\,\left[\forall z\,(z\subseteq x\leftrightarrow z\in
y\right].$

  * •

$\mathsf{ZC}^{-}_{\Delta_{0}}$ enriches $\mathsf{Z}^{-}_{\Delta_{0}}$ adding
the axiom of choice $\mathsf{AC}$

$\forall x\exists f\,\left[(f\text{ is a
bijection})\wedge\operatorname{dom}(f)=x\wedge(\operatorname{ran}(f)\text{ is
an ordinal})\right].$

  * •

$\mathsf{ZF}^{-}_{\Delta_{0}}$ enriches $\mathsf{Z}^{-}_{\Delta_{0}}$ adding
the Strong Replacement Axiom 272727Which is a strong form of the Collection
Principle and is defined in (8) below. for all $\in_{\Delta_{0}}$-formulae.

  * •

$\mathsf{ZFC}^{-}_{\Delta_{0}}$, $\mathsf{ZF}_{\Delta_{0}}$,
$\mathsf{ZFC}_{\Delta_{0}}$ are defined as expected.

######  Remark 3.1.2.

We took the pain of giving an explicit axiomatization of
$\mathsf{Z}^{-}_{\Delta_{0}}$ using Extensionality, Foundation, and axioms
1,…,7 because this axiomatizion is given by $\Pi_{2}$-sentences of
$\in_{\Delta_{0}}$, hence it is preserved by $\Sigma_{1}$-substructures. Note
that $\mathsf{AC}$ is a $\Pi_{2}$-axiom of $\in_{\Delta_{0}}$ while the power-
set axiom and the (strong) replacement schema for a quantifier free
$\in_{\Delta_{0}}$-formula are both $\Pi_{3}$.

A simple inductive argument shows that $\mathsf{ZF}^{-}+T_{\in,D}$ (where $D$
is the subset of $\mathsf{Form}_{\in}\times 2$ used in Not. 3.1.1 to define
$\in_{\Delta_{0}}$) is logically equivalent to $\mathsf{ZF}^{-}$ enriched with
axioms 1,…,7 (with $\emptyset$ taking the place of $c_{\in}$ and $\omega$
being the constant of $\in_{\Delta_{0}}$ associated to the
$\Delta_{0}$-formula defining it). We skip the details.

We now introduce the terminology to handle set theory formalized in signatures
richer than $\in_{\Delta_{0}}$.

######  Notation 3.1.3.

Let $\tau\supseteq\in_{\Delta_{0}}$. For a $\tau$-formula
$\phi(\vec{x},\vec{y},\vec{z})$:

  * •

The _Replacement Axiom_ for $\phi$ ($\mathsf{Rep}(\phi)$) states:

$\forall\vec{z}\left[(\forall
x\,\exists!y\,\phi(x,y,\vec{z}))\rightarrow\forall X\exists F\,(F\text{ is a
function}\wedge\operatorname{dom}(F)=X\wedge\forall x\in
X\,\phi(x,F(x),\vec{z}))\right];$

$\mathsf{Rep}_{\tau}$ holds if $\mathsf{Rep}(\phi)$ holds for all
$\tau$-formulae $\phi$.

  * •

The _Strong Replacement Axiom_ for282828This is a strong form of the
Collection Principle. Note that Replacement, the Collection Principle and
Strong Replacement are equivalent over the other axioms of $\mathsf{ZFC}$
(when including powerset and choice and excluding replacement), but are not
equivalent if one drops either one among choice and powerset and retains the
other axioms of $\mathsf{ZFC}$ (but not replacement). $\phi$
($\mathsf{StrRep}(\phi)$) states:

(8) $\forall\vec{z}\left[(\forall x\,\exists
y\,\phi(x,y,\vec{z}))\rightarrow\forall X\exists F\,(F\text{ is a
function}\wedge\operatorname{dom}(F)=X\wedge\forall x\in
X\,\phi(x,F(x),\vec{z}))\right];$

$\mathsf{StrRep}_{\tau}$ holds if $\mathsf{StrRep}(\phi)$ holds for all
$\tau$-formulae $\phi$.

  * •

$\mathsf{ZF}^{-}_{\tau}$ is
$\mathsf{Z}^{-}_{\Delta_{0}}+\mathsf{StrRep}_{\tau}$.

  * •

Accordingly we define $\mathsf{ZFC}_{\tau}$, $\mathsf{ZFC}^{-}_{\tau}$,
$\mathsf{ZF}_{\tau}$, $\mathsf{ZFC}_{\tau}$,…

  * •

We write $\mathsf{ZFC}_{\Delta_{0}}$ rather than $\mathsf{ZFC}_{\tau}$ when
$\tau=\in_{\Delta_{0}}$, etc.

  * •

If $A\subseteq\mathsf{Form}_{\in}\times 2$ is such that
$\in_{\Delta_{0}}\subseteq\in_{A}$, we write $\mathsf{ZFC}^{-}_{A}$ rather
than $\mathsf{ZFC}^{-}+T_{\in,A}$,…

Clearly (the suitable fragment of) $\mathsf{ZFC}+T_{\in,A}$ is logically
equivalent to (the suitable fragment of) $\mathsf{ZFC}_{A}$.

####  3.1.1. Further notational conventions

Let us introduce notation we will use to handle the substructure relation over
expanded signatures. The following supplements Notations 1.6, 2.0.1.

######  Notation 3.1.4.

Given some signature
$\tau\supseteq\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$ and a
$\tau$-structure $(M,\tau^{M})$ and some $B\subseteq\mathsf{Form}_{\tau}\times
2$, $(M,\tau_{B}^{M})$ is the unique extension of $(M,\tau)$ defined in
accordance with Notations 1.6, 2.0.1 which satisfies $T_{\tau,B}$. In
particular $(M,\tau_{B}^{M})$ is a shorthand for $(M,S^{M}:S\in\tau_{B})$. If
$(N,\tau^{N})$ is a substructure of $(M,\tau^{M})$ we also write
$(N,\tau_{B}^{M})$ as a shorthand for $(N,S^{M}\restriction N:S\in\tau_{B})$.

######  Remark 3.1.5.

Note that even if $(N,\tau^{N})$ is a substructure of $(M,\tau^{M})$,
$(N,\tau_{B}^{M})$ and $(N,\tau_{B}^{N})$ could be differents structures and
$(N,\tau_{B}^{N})$ may not be a substructure of $(M,\tau_{B}^{M})$:

  * •

$(N,\tau_{B}^{M})$ is obtained by restricting to $N$ the intepretation of the
new symbols of $\tau_{B}$ in $M$ according to how $M$ realizes $T_{\tau,B}$;

  * •

$(N,\tau_{B}^{N})$ is obtained by interpreting the new symbols of $\tau_{B}$
in $N$ according to $T_{\tau,B}$ as realized in $N$.

We are spending a great deal of attention to isolate those set theoretic
concepts which grant that $(N,\tau_{B}^{M})$ and $(N,\tau_{B}^{N})$ are the
same (at least when $(N,\tau^{N})\prec_{1}(M,\tau^{M})$).

######  Notation 3.1.6.

Let $(M,E)$ be an $\in$-structure. For $a\in M$,

$\text{Ext}^{M}_{\in}(a)=\left\\{b\in M:\,M\models b\in a\right\\}.$

$N\subseteq M$ is a _transitive subset of $M$_ if
$\text{Ext}^{M}_{\in}(a)=\text{Ext}^{N}_{\in}(a)$ for all $a\in N$.

There are many basic set theoretic properties which are established in
standard textbooks (such as [12, 14]) only for transitive models of fragments
of $\mathsf{ZFC}$ and can be established for arbitrary models of these
fragments:

######  Fact 3.1.7.

Let $(M,E)$ be an $\in$-structure with $N\subseteq M$ a transitive subset.
Then for all $\Delta_{0}$-formula $\phi$ and $\vec{a}\in N^{<\omega}$,
$N\models\phi(\vec{a})$ if and only if $M\models\phi(\vec{a})$.

###### Proof.

By the results of [14, Section IV.3] (which are there established under the
further assumptions that $M,N$ are transitive, but can be proved just assuming
the weaker assumption that $(M,E)$, $(N,E)$ are models of $\mathsf{ZFC}^{-}$
with $N$ a transitive subset of $M$). We leave the details to the reader. ∎

######  Notation 3.1.8.

Let $\in_{\Delta_{1}}$ be $\in_{D}$ for $D\subseteq\mathsf{Form}_{\in}\times
2$ given by the $\phi_{i}$ such that:

  * •

$\phi$ is $\Delta_{1}(\mathsf{ZFC}^{-}_{\Delta_{0}})$

  * •

if $(\phi,1)\in E$, then $\mathsf{ZFC}^{-}$ proves
$\forall\vec{x}\,\exists!y\phi(\vec{x},y)$.

$T_{\Delta_{1}}$ is $T_{\Delta_{0}}$ enriched with $\mathsf{Ax}^{i}_{\phi}$
for $(\phi,i)\in D$; $\mathsf{ZFC}^{-}_{\Delta_{1}}$ is
$\mathsf{ZFC}^{-}_{\Delta_{0}}+T_{\Delta_{1}}$; accordingly we define
$\mathsf{ZFC}_{\Delta_{1}}$, etc.

Note that $\in_{\Delta_{0}}$ and $\mathsf{ZFC}_{\Delta_{0}}$ are recursive
sets while $\in_{\Delta_{1}}$ and $\mathsf{ZFC}_{\Delta_{1}}$ are just semi-
recursive.

######  Fact 3.1.9.

Assume $\tau_{0}\supseteq\in_{\Delta_{0}}$ and
$\mathcal{M}_{0},\mathcal{N}_{0}$ are $\tau_{0}$-structures which model
$\mathsf{ZFC}^{-}_{\Delta_{0}}$. Let $\tau_{1}$ be
$\tau_{0}\cup\in_{\Delta_{1}}$ and $\mathcal{M}_{1},\mathcal{N}_{1}$ be the
unique extensions to $\tau_{1}$-structures which model
$\mathsf{ZFC}^{-}_{\Delta_{1}}$. Then

$\mathcal{M}_{0}\mathrel{R}\mathcal{N}_{0}\text{ if and only if
}\mathcal{M}_{1}\mathrel{R}\mathcal{N}_{1}$

holds for any binary relation
$R\in\left\\{\equiv_{1},\sqsubseteq,\prec_{1}\right\\}$.

###### Proof.

Use Fact 2.4.2. ∎

In view of the above facts if $(N,E)$ models $\mathsf{ZFC}^{-}$ and $M$ is a
transitive subset of $N$ such that $(M,E)$ also models $\mathsf{ZFC}^{-}$ the
unique expansions of $M,N$ to $\in_{\Delta_{1}}$-models of
$\mathsf{ZFC}_{\Delta_{1}}$ are still substructures also according to
$\in_{\Delta_{1}}$.

###  3.2. Levy absoluteness

We need a generalization of Levy’s absoluteness in most proofs of the
remainder of this paper. We state and prove the Lemma under the assumption
that the model of $\mathsf{ZFC}$ we work in is transitive; but this assumption
is unnecessary. Here and in other places of this paper we just need that the
models in question satisfy $\mathsf{ZFC}^{-}$ or slightly more.

Recall that $(H_{\lambda},\in)$ is a model of $\mathsf{ZFC}^{-}$ for all
regular cardinal $\lambda$.

######  Lemma 3.1.

Let $(V,\in_{\Delta_{0}})$ be a model of $\mathsf{ZFC}_{\Delta_{i}}$ and
$\lambda>\kappa$ be infinite cardinals for $V$ with $\lambda$ regular. Then
for both $i=0,1$ the structure

$(H_{\lambda},\in_{\Delta_{i}}^{H_{\lambda}},\,A:A\subseteq\mathcal{P}\left(\kappa\right)^{k},\,k\in\mathbb{N})$

is $\Sigma_{1}$-elementary in

$(V,\in_{\Delta_{i}}^{V},\,A:A\subseteq\mathcal{P}\left(\kappa\right)^{k},\,k\in\mathbb{N}).$

Its proof is a variant of the classical result of Levy (which is the above
theorem stated just for the signature $\in_{\Delta_{0}}$); it is a slight
expansion of [22, Lemma 5.3]; we include it here since it is not literally the
same:

###### Proof.

Let $\tau$ be either one of the signature $\in_{\Delta_{0}},\in_{\Delta_{1}}$
and $\phi(\vec{x},y)$ be a quantifier free formula for
$\tau\cup\left\\{A_{1},\dots,A_{k}\right\\}$ with each of the $A_{i}$ being a
subset of some292929Note that $\exists x\in yA(y)$ is not a quantifier free
formula, and is actually equivalent to the $\Sigma_{1}$-formula $\exists
x(x\in y)\wedge A(y)$. $\mathcal{P}\left(\kappa\right)^{n_{i}}$, and
$\vec{a}\in H_{\lambda}$ be such that

$(V,\,\tau^{V},\,A_{1},\dots,A_{k})\models\exists y\phi(\vec{a},y).$

Let $\alpha>\kappa$ be large enough so that for some $b\in V_{\alpha}$

$(V,\,\tau^{V},\,A_{1},\dots,A_{k})\models\phi(\vec{a},b).$

Then

$(V_{\alpha},\,\tau^{V_{\alpha}},\,A_{1},\dots,A_{k})\models\phi(\vec{a},b)$

(since
$(V_{\alpha},\tau^{V_{\alpha}},A_{1},\dots,A_{k})\sqsubseteq(V,\,\tau^{V},\,A_{1},\dots,A_{k})$
by Fact 3.1.9). By the downward Lowenheim-Skolem theorem, we can find
$X\subseteq V_{\alpha}$ which is the domain of a
$\tau\cup\left\\{A_{1},\dots,A_{k}\right\\}$-elementary substructure of

$(V_{\alpha},\,\tau^{V_{\alpha}},\,A_{1},\dots,A_{k})$

such that $X$ is a set of size $\kappa$ containing $\kappa$ and such that
$A_{1},\dots,A_{k},\kappa,b,\vec{a}\in X$. Since $|X|=\kappa\subseteq X$, a
standard argument shows that $H_{\lambda}\cap X$ is a transitive set, and that
$\kappa^{+}$ is the least ordinal in $X$ which is not contained in $X$. Let
$M$ be the transitive collapse of $X$ via the Mostowski collapsing map
$\pi_{X}$.

We have that the first ordinal moved by $\pi_{X}$ is $\kappa^{+}$ and
$\pi_{X}$ is the identity on $H_{\kappa^{+}}\cap X$. Therefore $\pi_{X}(a)=a$
for all $a\in H_{\kappa^{+}}\cap X$. Moreover for
$A\subseteq\mathcal{P}\left(\kappa\right)^{n}$ in $X$

(9) $\pi_{X}(A)=A\cap M.$

We prove equation (9):

###### Proof.

Since $X\cap V_{\kappa+1}\subseteq X\cap H_{\kappa^{+}}$, $\pi_{X}$ is the
identity on $X\cap H_{\kappa^{+}}$, and
$A\subseteq\mathcal{P}\left(\kappa\right)\subseteq V_{\kappa+1}$, we get that

$\pi_{X}(A)=\pi_{X}[A\cap X]=\pi_{X}[A\cap X\cap V_{\kappa+1}]=A\cap M\cap
V_{\kappa+1}=A\cap M.$

∎

It suffices now to show that

(10)
$(M,\tau^{M},\pi_{X}(A_{1}),\dots,\pi_{X}(A_{k}))\sqsubseteq(H_{\lambda},\tau^{H_{\lambda}},A_{1},\dots,A_{k}).$

Assume 10 holds; since $\pi_{X}$ is an isomorphism and
$\pi_{X}(A_{j})=\pi_{X}[A_{j}\cap X]$, we get that

$(M,\tau^{M},\pi_{X}(A_{1}),\dots,\pi_{X}(A_{k}))\models\phi(\pi_{X}(b),\vec{a})$

since

$(X,\tau^{V},A_{1}\cap X,\dots,A_{k}\cap X)\models\phi(b,\vec{a}).$

By (10) we get that

$(H_{\lambda},\tau^{H_{\lambda}},A_{1},\dots,A_{k})\models\phi(\pi_{X}(b),\vec{a})$

and we are done.

We prove (10):

###### Proof.

since $(M,\in)$ is a transitive model of $\mathsf{ZFC}^{-}$ with $M\subseteq
H_{\lambda}$, any atomic $\tau$-formula holds true in $(M,\tau^{M})$ if and
only if it holds in $(H_{\lambda},\tau^{H_{\lambda}})$ (again by Fact 3.1.9).
It remains to argue that the same occurs for the formulae of type $A_{j}(x)$,
i.e. that $A_{j}\cap M=\pi_{X}(A_{j})$ for all $j=1,\dots,n$; which is the
case by (9). ∎

∎

###  3.3. What holds in a $T$-ec structure for
$T\supseteq\mathsf{ZFC}_{\Delta_{0}}$

######  Lemma 3.3.1.

Assume $\mathcal{N}$ is a $\tau$-structure for some
$\tau\supseteq\in_{\Delta_{i}}$ (for some $i=0,1$) such that
$\mathcal{N}\models\mathsf{ZF}^{-}_{\Delta_{i}}$. Assume further that
$\mathcal{M}\prec_{1}\mathcal{N}$. Then:

  1. (1)

$\mathcal{M}$ models $\mathsf{Z}^{-}_{\Delta_{0}}$.

  2. (2)

$\mathcal{M}\models\mathsf{StrRep}(\phi)$ for any existential
$\in_{\Delta_{i}}$-formula $\phi(x,y,\vec{z})$ such that for all
$X\in\mathcal{M}$ and parameters $\vec{a}\in\mathcal{M}^{<\omega}$

$\mathcal{N}\models\forall x\in X\exists y\phi(x,y,\vec{a}).$

  3. (3)

Furthermore if $\mathcal{N}\models\mathsf{AC}$, so does $\mathcal{M}$.

###### Proof.

__

  1. (1)

All axioms of $\mathsf{Z}^{-}_{\Delta_{0}}$ hold in $\mathcal{N}$ and are
reflected to $\mathcal{M}$, since $\mathcal{M}\prec_{1}\mathcal{N}$, and those
axioms are at most $\Pi_{2}$-sentences of $\in_{\Delta_{0}}$.

  2. (2)

In view of Fact 3.1.9 it suffices to prove the Lemma just for a quantifier
free $\in_{\Delta_{0}}$-formula $\phi(x,y,\vec{z})$. By
$\mathsf{StrRep}(\phi)$ applied in $\mathcal{N}$ and the assumptions, we get
that for any $X\in\mathcal{M}$ and parameters
$\vec{a}\in\mathcal{M}^{<\omega}$

$\mathcal{N}\models\exists F\,[F\text{ is a
function}\wedge\operatorname{dom}(F)=X\wedge\forall
y\in\operatorname{ran}(F)\phi(x,y,\vec{a})].$

Now observe that

$F\text{ is a function}\wedge\operatorname{dom}(F)=X\wedge\forall
y\in\operatorname{ran}(F)\phi(x,y,\vec{a})$

is a $\Delta_{0}$-formula in parameters $F,X,\vec{a}$, hence logically
equivalent to an atomic $\in_{\Delta_{0}}$-formula. By
$\Sigma_{1}$-elementarity

$\mathcal{M}\models\exists F\,[F\text{ is a
function}\wedge\operatorname{dom}(F)=X\wedge\forall
y\in\operatorname{ran}(F)\phi(x,y,\vec{a})]$

and we are done.

  3. (3)

It is immediately checked that if $\mathcal{N}\models\mathsf{AC}$,
$\mathcal{M}$ satisfies also the axiom of choice, as this is a
$\Pi_{2}$-sentence for $\in_{\Delta_{0}}$ which holds in $\mathcal{N}$ and
thus reflects to $\mathcal{M}$.

∎

This proves Theorem 1.9i.

###  3.4. The theory of $H_{\kappa^{+}}$ as the model companion of set theory

In this section we prove Thm. 1.9ii. To motivate the result we first show the
following:

######  Lemma 3.4.1.

Assume $\tau\supseteq\in_{\Delta_{i}}\cup\left\\{\kappa\right\\}$ (for some
$i=0,1$) is such that
$(H_{\kappa^{+}}^{\mathcal{N}},\tau^{\mathcal{N}})\prec_{1}(\mathcal{N},\tau^{\mathcal{N}})$
whenever $\mathcal{N}$ models $S\supseteq\mathsf{ZFC}_{\tau}+\kappa$_is a
cardinal_. Then every $S$-ec structure satisfies

$\forall x\exists f\,(f:\kappa\to x\text{ is a surjection}).$

In particular for any $\tau,S$ as in the Lemma, the $S$-ec models satisfy
$\mathsf{ZC}_{\Delta_{0}}$, Strong Replacement for most existential
$\tau$-formulae, and the $\Pi_{2}$-sentence stating that all sets have size at
most $\kappa$, e.g. they provide a reasonable class of models describing a
theory of $H_{\kappa^{+}}$.

###### Proof.

Let $\mathcal{M}$ be $S$-ec and $\mathcal{N}\sqsupseteq\mathcal{M}$ be a model
of $S$. Since
$(H_{\kappa^{+}}^{\mathcal{N}},\tau^{\mathcal{N}})\prec_{1}(\mathcal{N},\tau^{\mathcal{N}}),$
the two structures share the same $\Pi_{1}$-theory (which is clearly
$\Pi_{1}$-complete); hence by Lemma 2.1.2, there is
$\mathcal{P}\sqsupseteq\mathcal{N}$ satisfying the theory of
$(H_{\kappa^{+}}^{\mathcal{N}},\tau^{\mathcal{N}})$, and in particular the
universal theory of $S$ and the $\Pi_{2}$-sentence $\forall x\exists
f\,(f:\kappa\to x\emph{ is a surjection}).$ Since $\mathcal{M}$ is $S$-ec, the
latter sentence reflects to $\mathcal{M}$ and we are done. ∎

We now give existence results stating that there are many signatures
$\tau\supseteq\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$ so that
$\mathsf{ZFC}_{\tau}+\kappa$_is a cardinal_ admits as AMC the theory

$\mathsf{ZFC}^{-}_{\tau}+(\kappa\text{ is a cardinal})+\forall x\exists
f\,(f:\kappa\to x\text{ is a surjection}).$

####  3.4.1. By-interpretability of the first order theory of $H_{\kappa^{+}}$
with the first order theory of $\mathcal{P}\left(\kappa\right)$

Let’s compare the first order theory of the structure

$(\mathcal{P}\left(\kappa\right),\in_{\Delta_{0}}^{V})$

with that of the $\in$-theory of $H_{\kappa^{+}}$ in models $(V,\in)$ of
$\mathsf{ZFC}$. We show that they are $\mathsf{ZFC}_{\Delta_{0}}$-provably by-
interpretable with a by-interpetation translating $H_{\kappa^{+}}$ in a
$\Pi_{1}$-definable subset of $\mathcal{P}\left(\kappa^{2}\right)$ (in
signature $\in_{\Delta_{0}}$) and the $\in$-relation into a
$\Sigma_{1}$-relation over this set (in signature $\in_{\Delta_{0}}$). This
result is the key to the proof of Thm. 1.9i and is just outlining the model
theoretic consequences of the well-known fact that sets can be coded by well-
founded extensional graphs.

######  Definition 3.4.2.

Given $a\in H_{\kappa^{+}}$, $R\in\mathcal{P}\left(\kappa^{2}\right)$ codes
$a$, if $R$ codes a well-founded extensional relation on some
$\alpha\leq\kappa$ with top element $0$ so that the transitive collapse
mapping of $(\alpha,R)$ maps $0$ to $a$.

  * •

$\mathsf{WFE}_{\kappa}$ is the set of $R\in\mathcal{P}\left(\kappa^{2}\right)$
which are a well founded extensional relation with domain $\alpha\leq\kappa$
and top element $0$.

  * •

$\text{{\rm Cod}}_{\kappa}:\mathsf{WFE}_{\kappa}\to H_{\kappa^{+}}$ is the map
assigning $a$ to $R$ if and only if $R$ codes $a$.

The following theorem shows that the structure $(H_{\kappa^{+}},\in)$ is
interpreted by means of “imaginaries” in the structure
$(\mathcal{P}\left(\kappa\right),\in_{\Delta_{0}}^{V})$ by means of:

  * •

a universal $\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-formula (with
quantifiers ranging over subsets of $\kappa^{<\omega}$) defining a set
$\mathsf{WFE}_{\kappa}\subseteq\mathcal{P}\left(\kappa^{2}\right)$.

  * •

an equivalence relation $\cong_{\kappa}$ on $\mathsf{WFE}_{\kappa}$ defined by
an existential $\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-formula (with
quantifiers ranging over subsets of $\kappa^{<\omega}$)

  * •

A binary relation $E_{\kappa}$ on $\mathsf{WFE}_{\kappa}$ invariant under
$\cong_{\kappa}$ representing the $\in$-relation as the extension of an
existential $\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-formula (with
quantifiers ranging over subsets of $\kappa^{<\omega}$)303030See [12, Section
25] for proofs of the case $\kappa=\omega$; in particular the statement and
proof of Lemma 25.25 and the proof of [12, Thm. 13.28] contain all ideas on
which one can elaborate to draw the conclusions of Thm. 3.4.4. Note that the
map $x\mapsto x^{<\omega}$ has a $\Delta_{0}$-graph in models of
$\mathsf{ZF}^{-}$\- Therefore quantification over $\kappa$ or over
$\kappa^{<\omega}$ are the same modulo an $\in_{\Delta_{0}}$-term. In the
sequel we might be sloppy and identify at our convenience $\kappa$ with
$\kappa^{<\omega}$..

######  Notation 3.4.3.

Recall Notation 3.1.1.

$\phi_{\mathrm{Card}}(x)$

is the $\Pi_{1}$-formula for $\in_{\Delta_{0}}$

$(x\emph{ is a cardinal})\wedge\omega\subseteq x$

  * •

$\mathsf{Z}^{-}_{\kappa}$ is the
$\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-theory
$\mathsf{Z}^{-}_{\Delta_{0}}\cup\left\\{\phi_{\mathrm{Card}}(\kappa)\right\\}$.

  * •

Accordingly we define the
$\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-theories
$\mathsf{ZF}^{-}_{\kappa}$, $\mathsf{ZFC}^{-}_{\kappa}$,
$\mathsf{ZF}_{\kappa}$, $\mathsf{ZFC}_{\kappa}$.

######  Theorem 3.4.4.

Assume $\mathsf{ZFC}^{-}_{\kappa}$. The following holds313131Many transitive
supersets of $H_{\kappa^{+}}$ are
$\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-models of
$\mathsf{ZFC}^{-}_{\kappa}$ for $\kappa$ an infinite cardinal (see [14,
Section IV.6]). To simplify notation we assume to have fixed a transitive
$\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-model $\mathcal{N}$ of
$\mathsf{ZFC}^{-}_{\kappa}$ with domain $N\supseteq H_{\kappa^{+}}$. The
reader can easily realize that all these statements holds for an arbitrary
model $\mathcal{N}$ of $\mathsf{ZFC}^{-}_{\kappa}$ replacing $H_{\kappa^{+}}$
with its version according to $\mathcal{N}$.:

  1. (1)

The map $\mathrm{Cod}_{\kappa}$ and $\mathsf{WFE}_{\kappa}$ are defined by
$\left\\{\in,\kappa\right\\}$-formulae which are
$\Delta_{1}(\mathsf{ZFC}^{-}_{\kappa})$. Moreover $\text{{\rm
Cod}}_{\kappa}:\mathsf{WFE}_{\kappa}\to H_{\kappa^{+}}$ is surjective
(provably in $\mathsf{ZFC}^{-}_{\kappa}$), and $\mathsf{WFE}_{\kappa}$ is
defined by a universal $\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-formula
with quantifiers ranging over subsets of $\kappa^{<\omega}$.

  2. (2)

There are existential $\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-formulae
(with quantifiers ranging over subsets of $\kappa^{<\omega}$),
$\phi_{\in},\phi_{=}$ such that for all $R,S\in\mathsf{WFE}_{\kappa}$,
$\phi_{=}(R,S)$ if and only if $\text{{\rm Cod}}_{\kappa}(R)=\text{{\rm
Cod}}_{\kappa}(S)$ and $\phi_{\in}(R,S)$ if and only if $\text{{\rm
Cod}}_{\kappa}(R)\in\text{{\rm Cod}}_{\kappa}(S)$. In particular letting

$E_{\kappa}=\left\\{(R,S)\in\mathsf{WFE}_{\kappa}:\phi_{\in}(R,S)\right\\},$
$\cong_{\kappa}=\left\\{(R,S)\in\mathsf{WFE}_{\kappa}:\phi_{=}(R,S)\right\\},$

$\cong_{\kappa}$ is a $\mathsf{ZFC}^{-}_{\kappa}$-provably definable
equivalence relation, $E_{\kappa}$ respects it, and

$(\mathsf{WFE}_{\kappa}/_{\cong_{\kappa}},E_{\kappa}/_{\cong_{\kappa}})$

is isomorphic to $(H_{\kappa^{+}},\in)$ via the map $[R]\mapsto\text{{\rm
Cod}}_{\kappa}(R)$.

###### Proof.

A detailed proof requires a careful examination of the syntactic properties of
$\Delta_{0}$-formulae, in line with the one carried in Kunen’s [14, Chapter
IV]. We outline the main ideas, following (as we already did) Kunen’s book
terminology for certain set theoretic operations on sets, functions and
relations (such as $\operatorname{dom}(f),\operatorname{ran}(f)$,
$\text{Ext}(R)$, etc). To simplify the notation, we prove the results for a
transitive $\mathsf{ZFC}^{-}$-model $(N,\in)$ which is then extended to a
structure $(N,\in_{\Delta_{0}}^{N},\kappa^{N})$ which models
$\mathsf{ZFC}^{-}_{\kappa}$, and whose domain contains $H_{\kappa^{+}}$. The
reader can verify by itself that the argument is modular and works for any
other model of $\mathsf{ZFC}^{-}_{\kappa}$ (transitive or ill-founded,
containing the “true” $H_{\kappa^{+}}$ or not).

  1. (1)

This is proved in details in [14, Chapter IV]. To define
$\mathsf{WFE}_{\kappa}$ by a universal
$\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-property over subsets of
$\kappa$ and $\text{{\rm Cod}}_{\kappa}$ by a $\Delta_{1}$-property for
$\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$ over $H_{\kappa^{+}}$, we
proceed as follows:

     * •

_$R$ is an extensional relation with top element $0$_ is defined by the
$\in_{\Delta_{0}}$-atomic formula $\psi_{\mathrm{EXT}}(R)$
$\mathsf{ZF}^{-}_{\Delta_{0}}$-provably equivalent to the
$\Delta_{0}$-formula:

(11) $\displaystyle(R\emph{ is a binary
relation})\wedge(0\in\text{Ext}(R))\wedge$ $\displaystyle\wedge\forall
z,w\in\text{Ext}(R)\,[\forall u\in\text{Ext}(R)\,(u\mathrel{R}z\leftrightarrow
u\mathrel{R}w)\rightarrow(z=w)]\wedge$ $\displaystyle\wedge\forall
z\in\text{Ext}(R)\,(0\neq z\rightarrow\exists
y\in\text{Ext}(R)\,z\mathrel{R}y).$

     * •

$\mathsf{WFE}_{\kappa}$ is defined by the universal
$\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-formula
$\phi_{\mathsf{WFE}_{\kappa}}(R)$ (quantifying only over subsets of
$\kappa^{<\omega}$)

$\displaystyle\psi_{\mathrm{EXT}}(R)\wedge$
$\displaystyle(\text{Ext}(R)\in\kappa\vee\text{Ext}(R)=\kappa)\wedge$
$\displaystyle\wedge[\forall f\,(f:\omega\to\text{Ext}(R)\text{ is a function
}\rightarrow\exists n\in\omega\,\neg(\langle f(n+1),f(n)\rangle\in R))].$

     * •

To define $\text{{\rm Cod}}_{\kappa}$, consider the $\in_{\Delta_{0}}$-atomic
formula $\psi_{\text{{\rm Cod}}}(G,R)$ provably equivalent to the
$\in_{\Delta_{0}}$-formula:

$\displaystyle\psi_{\mathrm{EXT}}(R)\wedge$ $\displaystyle\wedge(G\text{ is a
function})\wedge$
$\displaystyle\wedge(\operatorname{dom}(G)=\text{Ext}(R))\wedge(\operatorname{ran}(G)\text{
is transitive})\wedge$
$\displaystyle\wedge\forall\alpha,\beta\in\text{Ext}(R)\,[\alpha\mathrel{R}\beta\leftrightarrow
G(\alpha)\in G(\beta)].$

Then $\text{{\rm Cod}}_{\kappa}(R)=a$ can323232Note that $\text{{\rm
Cod}}_{\kappa}$ can be defined without any reference to $\kappa$. This
reference appears once we decide to restrict the domain of $\text{{\rm
Cod}}_{\kappa}$ to $\mathsf{WFE}_{\kappa}$. be defined either by the
existential $\in_{\Delta_{0}}$-formula333333Given an $R$ such that
$\psi_{\mathrm{EXT}}(R)$ holds, _$R$ is a well founded relation_ holds in a
model of $\mathsf{ZFC}^{-}_{\kappa}$ if and only if $\text{{\rm
Cod}}_{\kappa}$ is defined on $R$. In the theory $\mathsf{ZFC}^{-}_{\kappa}$,
$\mathsf{WFE}_{\kappa}$ can be defined using a universal property by a
$\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-formula quantifying only over
subsets of $\kappa$. On the other hand if we allow arbitrary quantification
over elements of $H_{\kappa^{+}}$, we can express the well-foundedness of $R$
also using the existential formula $\exists G\,\psi_{\text{{\rm
Cod}}_{\kappa}}(G,R)$. This is why $\mathsf{WFE}_{\kappa}$ is defined by a
universal $\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$-property in the
structure $(\mathcal{P}\left(\kappa\right),\in_{\Delta_{0}}^{V},\kappa)$,
while the graph of $\text{{\rm Cod}}_{\kappa}$ can be defined by a
$\Delta_{1}$-property for $\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$ in the
structure $(H_{\kappa^{+}},\in_{\Delta_{0}}^{V},\kappa^{V})$.

$\exists G\,(\psi_{\text{{\rm Cod}}}(G,R)\wedge G(0)=a)$

or by the universal $\in_{\Delta_{0}}$-formula

$\forall G\,(\psi_{\text{{\rm Cod}}}(G,R)\rightarrow G(0)=a).$

  2. (2)

The equality relation in $H_{\kappa^{+}}$ is transferred to the isomorphism
relation between elements of $\mathsf{WFE}_{\kappa}$: if $X,Y$ are well-
founded extensional on $\kappa$ with a top-element, the Mostowski collapsing
theorem entails that $\text{{\rm Cod}}_{\kappa}(X)=\text{{\rm
Cod}}_{\kappa}(Y)$ if and only if
$(\mathrm{Ext}(R),R)\cong(\mathrm{Ext}(S),S)$. Isomorphism of the two
structures $(\mathrm{Ext}(X),X)\cong(\mathrm{Ext}(Y),Y)$ is expressed by the
$\Sigma_{1}$-formula for $\tau_{\kappa}$:

$\phi_{=}(X,Y)\equiv\exists f\,(f\text{ is a bijection of $\text{Ext}(X)$ onto
$\text{Ext}(Y)$ and $\alpha\mathrel{X}\beta$ if and only if
$f(\alpha)\mathrel{Y}f(\beta)$}).$

In particular we get that $\phi_{=}(X,Y)$ holds in $H_{\kappa^{+}}$ for
$X,Y\in\mathsf{WFE}_{\kappa}$ if and only if $\text{{\rm
Cod}}_{\kappa}(X)=\text{{\rm Cod}}_{\kappa}(Y)$.

Similarly one can express $\text{{\rm Cod}}_{\kappa}(X)\in\text{{\rm
Cod}}_{\kappa}(Y)$ by the $\Sigma_{1}$-property $\phi_{\in}$ in
$\tau_{\kappa}$ stating that $(\mathrm{Ext}(X),X)$ is isomorphic to
$(\mathrm{pred}_{Y}(\alpha),Y)$ for some $\alpha\in\kappa$ with
$\alpha\mathrel{Y}0$, where $\mathrm{pred}_{Y}(\alpha)$ is given by the
elements of $\mathrm{Ext}(Y)$ which are $Y$-connected by a finite path to
$\alpha$.

Moreover letting $\cong_{\kappa}\subseteq\mathsf{WFE}_{\kappa}^{2}$ denote the
isomorphism relation between elements of $\mathsf{WFE}_{\kappa}$ and
$E_{\kappa}\subseteq\mathsf{WFE}_{\kappa}^{2}$ denote the relation which
translates into the $\in$-relation via $\text{{\rm Cod}}_{\kappa}$, it is
clear that $\cong_{\kappa}$ is a congruence relation over $E_{\kappa}$, i.e.:
if $X_{0}\cong_{\kappa}X_{1}$ and $Y_{0}\cong_{\kappa}Y_{1}$,
$X_{0}\mathrel{E_{\kappa}}Y_{0}$ if and only if
$X_{1}\mathrel{E_{\kappa}}Y_{1}$.

This gives that the structure
$(\mathsf{WFE}_{\kappa}/_{\cong_{\kappa}},E_{\kappa}/_{\cong_{\kappa}})$ is
isomorphic to $(H_{\kappa^{+}},\in)$ via the map $[X]\mapsto\text{{\rm
Cod}}_{\kappa}(X)$ (where $\mathsf{WFE}_{\kappa}/_{\cong_{\kappa}}$ is the set
of equivalence classes of $\cong_{\kappa}$ and the quotient relation
$[X]\mathrel{E_{\kappa}/_{\cong_{\kappa}}}[Y]$ holds if and only if
$X\mathrel{E_{\kappa}}Y$).

This isomorphism is defined via the map $\text{{\rm Cod}}_{\kappa}$, which is
by itself defined by a $\mathsf{ZFC}^{-}_{\kappa}$-provably
$\Delta_{1}$-property for $\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$.

The very definition of $\mathsf{WFE}_{\kappa},\cong_{\kappa},E_{\kappa}$ show
that

$\mathsf{WFE}_{\kappa}=\phi_{\mathsf{WFE}_{\kappa}}^{N},$
$\cong_{\kappa}=(\phi_{\mathsf{WFE}_{\kappa}}(x)\wedge\phi_{\mathsf{WFE}_{\kappa}}(y)\wedge\phi_{=}(x,y))^{N},$
$E_{\kappa}=(\phi_{\mathsf{WFE}_{\kappa}}(x)\wedge\phi_{\mathsf{WFE}_{\kappa}}(y)\wedge\phi_{\in}(x,y))^{N}.$

Note that we crucially use the axiom of choice to prove the surjectivity of
$\text{{\rm Cod}}_{\kappa}$ on $H_{\kappa^{+}}$. ∎

####  3.4.2. Model completeness for the theory of $H_{\kappa^{+}}$

The following definition isolates those signatures extending
$\in_{\Delta_{0}}$ with predicates defining a family of subsets of
$\bigcup_{n\in\omega}\mathcal{P}\left(\kappa\right)^{n}$ closed under finite
unions, complementations, and projections.

######  Definition 3.4.5.

Let $\tau$ be a signature extending
$\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$.

$\tau$ is $\kappa$-projectively closed for some $\tau$-theory343434Recall
Notation 3.1.1 for $\mathsf{ZFC}_{\tau}$, $\mathsf{ZF}^{-}_{\tau}$,…
$T\supseteq\mathsf{ZFC}^{-}_{\tau}$ if the following holds for a certain
family of predicate symbols in $\tau_{0}\subseteq\tau$:

  1. (1)

$T\models(\kappa\emph{ is a cardinal})$

  2. (2)

$T\models\forall
x_{0},\dots,x_{n}\,(R(x_{0},\dots,x_{n})\rightarrow\bigwedge_{i=0}^{n}x_{i}\subseteq\kappa)$

for all predicate symbols $R\in\tau_{0}$.

  3. (3)

For all quantifier free $\tau$-formulae $\phi$ there exists
$R_{\phi}\in\tau_{0}$ such that

$T\models\forall
x_{0},\dots,x_{n},y_{0},\dots,y_{n}\,\left[(\bigwedge_{i=0}^{n}x_{i}=\text{{\rm
Cod}}_{\kappa}(y_{i})\wedge\phi(x_{0},\dots,x_{n}))\leftrightarrow
R_{\phi}(y_{0},\dots,y_{n})\right]$

  4. (4)

For all $R,S\in\tau_{0}$ there is some $U$ also in $\tau_{0}$ such that

$T\models\forall\vec{x},\vec{y}\,\left[(R(\vec{x})\vee
S(\vec{y}))\leftrightarrow U(\vec{x},\vec{y})\right].$

  5. (5)

For all $R,S\in\tau_{0}$ there is some $U$ also in $\tau_{0}$ such that

$T\models\forall\vec{x},\vec{y}\,\left[(R(\vec{x})\wedge
S(\vec{y}))\leftrightarrow U(\vec{x},\vec{y})\right].$

  6. (6)

For all $R\in\tau_{0}$ there is some $U$ also in $\tau_{0}$ such that

$T\models\forall
x_{0},\dots,x_{n}\,\left[\bigwedge_{i=0}^{n}x_{i}\subseteq\kappa\rightarrow(R(x_{0},\dots,x_{n})\leftrightarrow\neg
U(x_{0},\dots,x_{n})\right]$

  7. (7)

For all $R\in\tau_{0}$ and $i=0,\dots,n$ there is some $U$ also in $\tau_{0}$
such that

$T\models\forall
x_{0},\dots,x_{i-1},x_{i+1},\dots,x_{n}\,\left[U(x_{0},\dots,x_{i-1},x_{i+1},\dots,x_{n})\leftrightarrow\exists
x_{i}\,(x_{i}\subseteq\kappa\wedge R(x_{0},\dots,x_{n}))\right].$

We just write _projectively closed_ if $\kappa=\omega$ and
$\tau_{0}=\tau\setminus\in_{\Delta_{0}}$.

The following is a trivial but fundamental remark:

######  Fact 3.4.6.

Let $\tau$ be a $\kappa$-projectively closed family. Axioms 1 to 7 are all
$\Pi_{2}$-sentences for $\tau$.

######  Lemma 3.4.7.

Let $\tau\supseteq\in_{\Delta_{0}}$, $T\supseteq\mathsf{ZFC}_{\tau}$ be a
$\tau$-theory, and $\kappa$ be a $T$-definable cardinal.

Assume $\phi$ is a $\tau$-formula which defines in any model $(V,\tau^{V})$ of
$T$ a transitive model $M_{\phi}\subseteq V$ of $\mathsf{ZF}^{-}_{\tau}$ which
contains $H_{\kappa^{+}}^{V}$. Let $A=\bar{A}\times\left\\{0\right\\}$, where

$\bar{A}=\left\\{\theta^{M_{\phi}}(x_{0},\dots,x_{n})\wedge\bigwedge_{i=0}^{n}x_{i}\subseteq\kappa:\,\theta\emph{
a $\tau$-formula}\right\\}.$

Then $\tau_{A}$ is $\kappa$-projectively closed for $T+T_{\tau,A}$.

###### Proof.

This is an almost immediate consequence of the fact that whenever
$(V,\tau^{V})$ models $T$, $M_{\phi}\supseteq H_{\kappa^{+}}^{V}$ is a
transitive model of $\mathsf{ZF}^{-}_{\tau}$, hence such that $\text{{\rm
Cod}}_{\kappa}^{V}=\text{{\rm Cod}}_{\kappa}^{M}$ and
$\mathcal{P}\left(\kappa\right)^{V}=\mathcal{P}\left(\kappa\right)^{M}$. Using
this fact together with the fact that $(M_{\phi},\tau^{V})$ is a model of
$\mathsf{ZF}^{-}_{\tau}$, we get that the collection of subsets of
$\mathcal{P}\left(\kappa\right)^{V}$ definable in $M_{\phi}$ without (or even
with) parameters defines a family which is closed under projections (e.g.
Axiom 7), finite unions and complementations (e.g. Axioms 4, 5, 6), and
satisfies also axiom 3.

The conclusion follows. ∎

######  Theorem 3.4.8.

Let $\tau$ be a $\kappa$-projectively closed signature for some $\tau$-theory
$T\supseteq\mathsf{ZFC}_{\tau}$.

Let $S$ be the $\tau$-theory extending $\mathsf{ZFC}^{-}_{\tau}$ with the
axioms 1,…,7 which follow from $T$ and the axiom

$\forall x\exists f\,\left[f\text{ is a
surjection}\wedge\operatorname{dom}(f)=\kappa\wedge\operatorname{ran}(f)=x\right].$

Then $S$ is model complete.

###### Proof.

To simplify notation, we conform to the assumption of Thm. 3.4.4, i.e. we
assume that the model $(N,\tau^{N})$ of $S$ on which we work is a transitive
superstructure of $H_{\kappa^{+}}$.

The statement _every set has size at most $\kappa$_ is satisified by a
$\mathsf{ZF}^{-}$-model $(N,\in^{N},\kappa^{N})$ with $N\supseteq
H_{\kappa}^{+}$ if and only if $N=H_{\kappa^{+}}$. From now on we proceed
assuming this equality.

It suffices to show that for all $\tau$-formulae $\phi(\vec{x})$

$S\models\forall\vec{x}\,(\phi(\vec{x})\leftrightarrow\psi_{\phi}(\vec{x})),$

for some universal $\tau$-formula $\psi_{\phi}$.

By Axiom 2 applied to the atomic $\tau$-formulae $x=x$, $x\in y$ and $x=y$, we
obtain predicate symbols $R_{\mathsf{WFE}_{\kappa}}$, $R_{=}$, $R_{\in}$ in
$\tau_{0}$ such that:

$R_{\mathsf{WFE}_{\kappa}}^{H_{\kappa^{+}}}=\mathsf{WFE}_{\kappa},$
$R_{x=y}^{H_{\kappa^{+}}}=\cong_{\kappa}=\left\\{(X,Y)\in(\mathsf{WFE}_{\kappa})^{2}:\,\text{{\rm
Cod}}_{\kappa}(X)=\text{{\rm Cod}}_{\kappa}(Y)\right\\},$ $R_{x\in
y}^{H_{\kappa^{+}}}=E_{\kappa}=\left\\{(X,Y)\in(\mathsf{WFE}_{\kappa})^{2}:\,\text{{\rm
Cod}}_{\kappa}(X)\in\text{{\rm Cod}}_{\kappa}(Y)\right\\}.$

Now by assumptions on $\tau$, we get that for any quantifier free
$\tau$-formula $\phi(\vec{x})$, there is some predicate $R_{\phi}\in\tau_{0}$
such that:

$S\models R_{\phi}(x_{0},\dots,x_{n})\leftrightarrow\exists
y_{0},\dots,y_{n}\,\left[\phi(y_{0},\dots,y_{n})\wedge\bigwedge_{i=0}^{n}y_{i}=\text{{\rm
Cod}}_{\kappa}(x_{i})\right].$

Now we proceed to define $R_{\psi}(\vec{x})$ for any $\tau$-formula $\phi$
letting:

  * •

$R_{\psi\wedge\psi}(\vec{x})$ be $U(\vec{x})$ for the $U$ given by Axiom 5
applied to $R_{\psi},R_{\phi}$,

  * •

$R_{\neg\psi}(\vec{x})$ be $U(\vec{x})$ for the $U$ given by Axiom 6 applied
to $R_{\psi},R_{\phi}$,

  * •

$R_{\exists y\psi(y,\vec{x})}(\vec{x})$ be $U(\vec{x})$ for the $U$ given by
Axiom 7 applied to $R_{\psi}$.

An easy induction on the complexity of the $\tau_{0}$-formulae
$R_{\phi}(\vec{x})$ gives that for any $\tau$-definable subset $X$ of
$(H_{\kappa^{+}})^{n}$ which is the extension of some $\tau$-formula
$\phi(x_{1},\dots,x_{n})$

$\left\\{(Y_{1},\dots,Y_{n})\in(\mathsf{WFE}_{\kappa})^{n}:\,(\text{{\rm
Cod}}_{\kappa}(Y_{1}),\dots,\text{{\rm Cod}}_{\kappa}(Y_{n}))\in
X\right\\}=R_{\phi}^{H_{\kappa^{+}}},$

with the further property that
$R_{\phi}^{H_{\kappa^{+}}}\subseteq(\mathsf{WFE}_{\kappa})^{n}$ respects the
$\cong_{\kappa}$-relation.

Then for any $\tau$-formula $\phi(x_{1},\dots,x_{n})$
$(H_{\kappa^{+}},\tau^{H_{\kappa^{+}}})\models\phi(a_{1},\dots,a_{n})$ if and
only if

$(\mathsf{WFE}_{\kappa}/_{\cong_{\kappa}},E_{\kappa}/_{\cong_{\kappa}},R_{\phi}^{H_{\kappa^{+}}}:\,\phi\emph{
a $\tau$-formula})\models\phi([X_{1}],\dots,[X_{n}])$

whenever $\text{{\rm Cod}}_{\kappa}(X_{i})=a_{i}$ for $i=1,\dots,n$, if and
only if

$(H_{\kappa^{+}},\tau^{H_{\kappa^{+}}})\models\forall
X_{1},\dots,X_{n}\,[(\bigwedge_{i=1}^{n}R_{\mathsf{WFE}_{\kappa}}(X_{i})\wedge\text{{\rm
Cod}}_{\kappa}(X_{i})=a_{i})\rightarrow R_{\phi}(X_{1},\dots,X_{n})].$

Since this argument can be repeated verbatim for any model of $S$, we have
proved the following:

######  Claim 1.

For any $\tau$-formula $\phi(x_{1},\dots,x_{n})$, $S$ proves

$\forall x_{1},\dots,x_{n}\,[\phi(x_{1},\dots,x_{n})\leftrightarrow\forall
y_{1},\dots,y_{n}\,[(\bigwedge_{i=1}^{n}R_{\mathsf{WFE}_{\kappa}}(y_{i})\wedge\text{{\rm
Cod}}_{\kappa}(y_{i})=x_{i})\rightarrow R_{\phi}(y_{1},\dots,y_{n})]].$

But $\text{{\rm Cod}}_{\kappa}(y)=x$ is expressible provably in $S$ by an
existential $\tau$-formula (since $\tau\supseteq\in_{\Delta_{0}}$). Therefore

$\forall
y_{1},\dots,y_{n}\,[(\bigwedge_{i=1}^{n}R_{\mathsf{WFE}_{\kappa}}(y)\wedge\text{{\rm
Cod}}_{\kappa}(y_{i})=x_{i})\rightarrow R_{\phi}(y_{1},\dots,y_{n})]$

is a universal $\tau$-formula, and we are done. ∎

####  3.4.3. Proof of Thm. 1.9i

Conforming to the notation of Thm. 1.9i, it is clear that if $T$ extends
$\mathsf{ZFC}$ and $\kappa$ is a $T$-definable cardinal

$\bar{A}_{\kappa}=\left\\{\phi^{\mathcal{P}\left(\kappa\right)}(x_{1},\dots,x_{n})\wedge\bigwedge_{i=1}^{n}x_{i}\subseteq\kappa:\,\phi\emph{
an $\in$-formula}\right\\},$

we get that

$A_{\kappa}=\left\\{(\phi,0):\,\phi\in\bar{A}_{\kappa}\right\\}\cup D$

(where $D$ is the set used in Notation 3.1.1) is such that $\in_{A_{\kappa}}$
is $\kappa$-projectively closed for $\mathsf{ZFC}$.

Therefore the following result completes the proof of Thm. 1.9i.

######  Theorem 3.4.9.

Let $\tau$ be $\kappa$-projectively closed for some $\tau$-theory
$T\supseteq\mathsf{ZFC}_{\tau}$ such that

$H_{\kappa^{+}}^{\mathcal{M}}\prec_{1}\mathcal{M}$

whenever $\mathcal{M}$ models $T$.

Then $T$ has an AMC $T^{\prime}$ in signature $\tau$.

Observe that the assumptions of the theorem apply whenever
$\tau\setminus\tau_{0}$ is $\in_{\Delta_{0}}\cup\left\\{\kappa\right\\}$ or
any signature satisfying the assumptions of Lemma 3.1.

###### Proof.

By Thm. 3.4.8, any $\tau$-theory extending

$\mathsf{ZFC}^{-}_{\tau}+\emph{every set has size at most $\kappa$}$

with the axioms 1,…,7 which follow from $T$ is model complete.

Therefore:

  * •

$H_{\kappa^{+}}^{\mathcal{M}}$ models $\mathsf{ZFC}^{-}_{\tau}$, by the
standard properties of $H_{\kappa^{+}}$ and the regularity of $\kappa^{+}$
(any function $F$ with domain in $H_{\kappa^{+}}$ and range contained in
$H_{\kappa^{+}}$ is in $H_{\kappa^{+}}$).

  * •

Clearly $H_{\kappa^{+}}^{\mathcal{M}}$ models _every set has size at most
$\kappa$_.

  * •

$H_{\kappa^{+}}^{\mathcal{M}}$ models the axioms 1,…,7 which follow from $T$,
as those are all expressible by $\Pi_{2}$-sentences for $\tau$ holding in
$\mathcal{M}$.

Therefore

$T^{\prime}=\left\\{\phi:H_{\kappa^{+}}^{\mathcal{M}}\models\phi,\,\mathcal{M}\models
T\right\\}$

is model complete.

Clearly $T_{\forall\vee\exists}^{\prime}=T_{\forall\vee\exists}$, since
$H_{\kappa^{+}}^{\mathcal{M}}\prec_{1}\mathcal{M}$ whenever
$\mathcal{M}\models T$.

The theorem is proved. ∎

######  Remark 3.4.10.

Note that there is no reason to expect that the family of models
$\left\\{H_{\kappa^{+}}^{\mathcal{M}}:\,\mathcal{M}\models T\right\\}$ we used
to define $T^{\prime}$ is an elementary class for $\tau$.

By Lemma 3.4.7 for any $\in$-theory $S\supseteq\mathsf{ZFC}$ and any
$S$-definable cardinal $\kappa$ there is plenty of
$A\supseteq\mathsf{Form}_{\in}\times 2$ such that $\in_{A}$ is
$\kappa$-projectively closed for $\mathsf{ZFC}^{-}_{\tau_{A}}$.

However linking model companionship results for set theory to forcibility as
we do in Theorem 1.10 requires much more care in the definition of the
signature. We will pursue this matter in more details in the next sections.

###  3.5. Absolute codings of the continuum in type $\omega_{2}$

We prove variants of Theorems 1.11 and 1.12. Here we focus on the strong
consistency hull of set theory as formalized in a given signature (which -when
model complete- is also its AMC) rather than on its possible Kaiser hull
(which -when model model complete- is only its MC) (recall Def. 2.2.4, Fact
2.2.5, Lemma 2.3.4, [20, Lemma 3.2.12, Lemma 3.2.13, Thm. 3.2.14]).

We start with Thm. 1.11. We need the following:

######  Proposition 3.5.1.

Assume $\tau\supseteq\in_{\Delta_{0}}$ and $S\supseteq\mathsf{ZFC}_{\tau}$ is
a $\tau$-theory such that $S+\neg\text{{\sf CH}}$ is consistent. Then CH is
not in $\mathsf{SCH}(S)$.

###### Proof.

Note that $\neg\text{{\sf CH}}$ as formalized in 1.4 is expressible in
signature $\tau$ as the conjunction of the purely $\Pi_{2}$-sentence for
$\in_{\Delta_{0}}$ _“every function with domain $\aleph_{1}$ is not surjective
on the reals”_ with the the purely $\Sigma_{2}$-sentence for
$\in_{\Delta_{0}}$ _“ $\aleph_{1}$ exists”_.

Either $\mathsf{SCH}(S)$ does not model _“ $\aleph_{1}$ exists”_ in which case
there is a model of $\mathsf{SCH}(S)+\neg\text{{\sf CH}}$ (e.g. a model of
$\mathsf{SCH}(S)+$_“all sets are countable”_), or $\mathsf{SCH}(S)$ models _“
$\aleph_{1}$ exists”_. In this case we note that CH holds in $\mathsf{SCH}(S)$
if and only if it $\mathsf{SCH}(S)$ models the $\Sigma_{2}$-sentence $\psi$
for $\in_{\Delta_{0}}$ stating that _“there is a bijection $f$ with domain
$\omega_{1}$ and whose range contains all subsets of $\omega$”_, and also that
$S+\neg\text{{\sf CH}}$ is consistent and logically equivalent to
$S+\neg\psi$. By Fact 2.2.5v we can find a model of
$\mathsf{SCH}(S)+\neg\psi$, hence also in this case $\mathsf{SCH}(S)$ does not
model CH. ∎

######  Remark 3.5.2.

The above Proposition is the instantiation to $\neg\text{{\sf CH}}$ of a
result which can be obtained for any $\tau\supseteq\in_{\Delta_{0}}$, any
$\psi$ which is a $\Pi_{2}$-sentences for $\tau$, any
$S\supseteq\mathsf{ZFC}_{\tau}$ such that $\psi+S$ is consistent. We can use
Proposition 2.2.8 to infer that $\neg\psi\not\in\mathsf{SCH}(S)$.

######  Proposition 3.5.3.

Let $C\subseteq\mathsf{Form}\in\times 2$ be such that $\langle(x\text{ is
}\aleph_{1}),1\rangle\in C$ and $\in_{C}\supseteq\in_{\Delta_{0}}$; let
$S\supseteq\mathsf{ZFC}$. Assume that $(S+{T}_{\in,C})_{\forall\vee\exists}$
is invariant across forcing extensions353535E.g. for any universal
$\in_{C}$-sentence $\psi$ $(V,\in_{C}^{V})\models\psi$ if and only if so does
$(V[G],\in_{C}^{V[G]})$ in any forcing extension of $V$. of any model of $S$.
Then $\neg\text{{\sf CH}}\in\mathsf{SCH}(S+T_{\in,C})$.

###### Proof.

We have that the $\Pi_{2}$-sentence for $\in_{C}$

$\forall f(\operatorname{dom}(f)\text{ is }\aleph_{1}\wedge f\text{ is a
function})\rightarrow\exists r\,(r\subseteq\omega\wedge
r\not\in\operatorname{ran}(f))]$

is strongly consistent with $S_{\forall\vee\exists}$ as it can be forced by
Cohen’s original forcing over any model of $S$, and (by assumption) such
forcings do not change the universal $\in_{C}$-theory of the model. ∎

The two Propositions combine with Thm. 4.1 of the next Section to give the
following variant of Thm. 1.11:

######  Corollary 3.5.4.

Assume $S_{0}$ is any extension of $\mathsf{ZFC}+$_there are class many Woodin
cardinals_. Then:

  * •

$\text{{\sf CH}}\in\mathsf{SCH}(S_{0}+T_{\in,A})$ for no
$A\subseteq\mathsf{Form}\in\times 2$ such that
$\in_{A}\supseteq\in_{\Delta_{0}}$;

  * •

$\neg\text{{\sf CH}}\in\mathsf{SCH}(S_{0}+T_{\in,C})$ for
$C\subseteq\mathsf{Form}\in\times 2$ such that
$\in_{C}=\in_{\Delta_{0}}\cup\left\\{\omega_{1}\right\\}$ and
$\left\\{\langle(x\text{ is }\aleph_{1}),1\rangle\in C\right\\}$.

###### Proof.

By Thm. 4.1 the truth value of universal $\in_{C}$-sentences is invariant
across forcing extensions of any model of $S_{0}$. Hence we can apply the
previous propositions to get the desired conclusions. ∎

The first item of Theorem 1.11 follows from the above Corollary. To get the
second item it suffices to find $B\subseteq\mathsf{Form}\in\times 2$ and
$S\supseteq S_{0}$ such that $\in_{B}$ is invariant across forcing extensions
of models of $S$ and $B\in\mathfrak{spec}_{\mathsf{AMC}}\left(S\right)$. Then
again the second Proposition above yields the second part of Theorem 1.11. We
will define such a $B$ and $S$ in Section 5.

######  Remark 3.5.5.

Corollary 3.5.4 is the instantiation for $\neg\text{{\sf CH}}$ of a results
which can be obtained for any $\Pi_{2}$-sentence $\psi$ formalized in a
signature $\in_{C}$ and a set theory $T$ formalized in $\in$ so that:

  * •

The universal $\in_{C}$-theory of $(V,\in)$ is invariant across forcing
extensions of $V$ in any model $(V,\in)$ of $T$;

  * •