# Boolean valued semantics for infinitary logics

Juan M. Santiago Suárez & Matteo Viale

###### Abstract.

It is well known that the completeness theorem for
$\mathrm{L}_{\omega_{1}\omega}$ fails with respect to Tarski semantics.
Mansfield showed that it holds for $\mathrm{L}_{\infty\infty}$ if one replaces
Tarski semantics with Boolean valued semantics. We use forcing to improve his
result in order to obtain a stronger form of Boolean completeness (but only
for $\mathrm{L}_{\infty\omega}$). Leveraging on our completeness result, we
establish the Craig interpolation property and a strong version of the
omitting types theorem for $\mathrm{L}_{\infty\omega}$ with respect to Boolean
valued semantics. We also show that a weak version of these results holds for
$\mathrm{L}_{\infty\infty}$ (if one leverages instead on Mansfield’s
completeness theorem). Furthermore we bring to light (or in some cases just
revive) several connections between the infinitary logic
$\mathrm{L}_{\infty\omega}$ and the forcing method in set theory.

The first author acknowledges support from INDAM through GNSAGA and from the
project: _PRIN 2017-2017NWTM8R Mathematical Logic: models, sets,
computability._ MSC: _03C75, 03E40._ Keywords: _Infinitary Logics, Forcing,
Consistency Properties._ We thank Boban Velickovic (who outlined us the
relevance of consistency properties in the analysis of forcing), and Ben de
Bondt (for many useful comments).

###### Contents

  1. 1 Introduction
  2. 2 The infinitary logics $\mathrm{L}_{\kappa\lambda}$
    1. 2.1 Syntax
    2. 2.2 Boolean valued semantics
    3. 2.3 Boolean satisfiability
    4. 2.4 Proof systems for $\mathrm{L}_{\infty\infty}$
  3. 3 Main model theoretic results
    1. 3.1 Results for $\mathrm{L}_{\infty\omega}$
    2. 3.2 Results for $\mathrm{L}_{\infty\infty}$
  4. 4 Consistency properties for relational $\omega$-signatures
  5. 5 Forcing with consistency properties
  6. 6 Mansfield’s Model Existence Theorem
  7. 7 Proofs of model theoretic results
    1. 7.1 Proof of Thm. 3.1
    2. 7.2 Proof of Thm. 3.2
    3. 7.3 Proof of Thm. 3.5
    4. 7.4 Proof of Thm. 3.7
  8. 8 Forcing notions as consistency properties
  9. 9 Appendix
    1. 9.1 Separating Tarski satisfiability from boolean satisfiability
    2. 9.2 Proof of Fact 2.6 and Proposition 2.9

##  1\. Introduction

This paper revives and brings to light several connections existing between
infinitary logics and forcing. The main objective of the paper is to show that
boolean valued semantics is a right semantics for infinitary logics, more
precisely: the class of boolean valued models with the mixing property (e.g.
sheaves on compact extremally disconnected spaces by the results of [11])
provides a complete semantics for $\mathrm{L}_{\infty\omega}$ with respect to
the natural sequent calculus for infinitary logics (obtained by trivially
adapting to this logic the inference rules of Gentzen’s sequent calculus for
first order logic, see Section 2.4 below)111We note that Mansfield [10] proves
that the larger class of boolean valued models (e.g. presheaves on compact
extremally disconnected spaces by the results of [11]) gives a complete
semantics for the logic $\mathrm{L}_{\infty\infty}$. Our completeness result
is weaker than Mansfield’s (as it applies only to $\mathrm{L}_{\infty\omega}$)
but also stronger than his (as it provides completeness with respect to a much
better behaved class of models, e.g. sheaves instead of presheaves on compact
extremally disconnected topological spaces).. Leveraging on our completeness
result we are able to prove the natural form of Craig’s interpolation theorem
for our deductive system for $\mathrm{L}_{\infty\omega}$, as well as a natural
generalization to $\mathrm{L}_{\infty\omega}$ with respect to boolean valued
semantics of the standard omitting types theorems which can be proved for
first order logic with respect to Tarski semantics. We are also able to prove
weaker forms of these results for $\mathrm{L}_{\infty\infty}$, in this latter
case appealing to a completeness result of Mansfield.

A central role in our analysis of $\mathrm{L}_{\infty\omega}$ is played by the
notion of consistency property. Roughly a consistency property for a signature
$\tau$ is a partial order whose elements are consistent families of infinitary
$\tau$-formulae ordered by reverse inclusion. The clauses for being a
consistency property in signature $\tau$ grant that a generic filter for such
a forcing notion produces a maximal set of consistent $\tau$-formulae, which
then can be turned into a Tarski $\tau$-structure (a term model) realizing
each of them. However generic filters do not exist in the standard universe of
set theory $V$, hence such Tarski $\tau$-structures do not exist in $V$ as
well, but just in a generic extension of $V$; on the other hand their
semantics can be instead described in $V$ by means of boolean valued models,
e.g. forcing.

Keeping in mind this idea we can show that:

  * •

any forcing notion is forcing equivalent to a consistency property for
$\mathrm{L}_{\infty\omega}$;

  * •

every consistency property defines an “elementary class” of boolean valued
models for $\mathrm{L}_{\infty\omega}$ and conversely;

  * •

most of the standard results for first order logic transfer to infinitary
logic if we replace Tarski semantics with boolean valued semantics; e.g. in
this paper we show that this is the case for the completeness theorem, Craig’s
interpolation, Beth definability, the omitting types theorem (on the other
hand we can show that compactness fails for boolean valued semantics also for
$\mathrm{L}_{\infty\omega}$).

Some caveats and further comments are in order.

  * •

Sections 5 and 8 require a basic familiarity with the forcing method (at the
level of Kunen’s book [7]). The rest of the paper can be read by people with a
loose or null knowledge of the forcing method.

  * •

Most of our results generalize to $\mathrm{L}_{\infty\omega}$ (and in some
cases also to $\mathrm{L}_{\infty\infty}$) with respect to boolean valued
semantics, results and proofs that Keisler obtains for
$\mathrm{L}_{\omega_{1}\omega}$ with respect to Tarski semantics [6]. Roughly
Keisler’s proofs are divided in two parts: the first designs a suitable
countable consistency property associated to a given countable
$\mathrm{L}_{\omega_{1}\omega}$-theory $T$ of interest; the second appeals to
Baire’s category theorem taking advantage of the considerations to follow.

Consistency properties are designed in order that the Tarski structure induced
by a maximal filter $F$ on them (seen as partial orders) realizes a certain
formula $\phi$ if and only if $F$ meets a dense set $D_{\phi}$ associated to
$\phi$. If one focuses on countable theories $T$ for
$\mathrm{L}_{\omega_{1}\omega}$, one can appeal to Baire’s category theorem to
find a maximal filter $F$ for the associated consistency property: $F$ meets
the countable family of dense sets associated to the formulae in $T$. This is
what Keisler’s proofs usually do.

However, if one considers an arbitrary $\mathrm{L}_{\infty\omega}$-theory $T$,
one could drop the use of Baire’s category theorem and replace it by
describing (using forcing) as a boolean valued model the Tarski structure that
Keisler’s method would produce in a forcing extension where $T$ becomes a
countable $\mathrm{L}_{\omega_{1}\omega}$-theory. This is what we will do
here.

  * •

One has to pay attention to our formulation of Craig’s interpolation property
(e.g. Thm. 3.2). We prove our result with respect to the natural deduction
calculus for $\mathrm{L}_{\infty\omega}$; for this calculus it is known that
the completeness theorem with respect to Tarski semantics fails (we give an
explicit counterexample in Fact 9.1). It is no surprise hence that the
semantic version of Craig’s interpolation fails as well with respect to Tarski
semantics (see [9, Thm. 3.2.4]). On the other hand Malitz has proved an
interpolation theorem for $\mathrm{L}_{\infty\infty}$ with respect to Tarski
semantics using another deductive system for $\mathrm{L}_{\infty\omega}$
(introduced by Karp) which is complete for Tarski semantics [9]. However we
believe that our deductive system is better than Malitz’s, since the notion of
proof for our system is independent of the model of set theory we work with;
for example our deductive system when restricted to
$\mathrm{L}_{\infty\omega}$ is forcing invariant. Even more, for sets
$\Gamma$, $\Delta$ of $\mathrm{L}_{\infty\omega}$-formulae in $V$, $\Gamma$
proves $\Delta$ is a provably $\Delta_{1}$-property in the parameters
$\Gamma,\Delta$ in any model of $\mathsf{ZFC}$ to which $\Gamma$ and $\Delta$
belong: note that the existence of a proof is expressible by a
$\Sigma_{1}$-statement while being true in any boolean valued model is
expressible by a $\Pi_{1}$-statement (according to the Levy hierarchy as in
[3, Pag. 183]). In particular, $\Gamma$ proves $\Delta$ holds in $V$ according
to our deductive system if and only if it holds in any (equivalently some)
forcing extension of $V$. This fails badly for Malitz’s deductive system, e.g.
there is a sentence $\phi$ such that “$\phi$ is valid according to Malitz’s
deductive system” holds in some generic extension of $V$, but fails in $V$ and
conversely.

  * •

The fact that forcing and consistency properties are closely related concepts
is implicit in the work of many; for example we believe this is behind
Jensen’s development of $\mathrm{L}$-forcing [4] and the spectacular proof by
Asperó and Schindler that $\text{{\sf MM}}^{++}$ implies Woodin’s axiom $(*)$
[1] (see also [14] -which gives a presentation of their proof more in line
with the spirit of this paper); it also seems clear that Keisler is to a large
extent aware of this equivalence in his book on infinitary logics [6], as well
as Mansfield in his paper proving the completeness theorem for
$\mathrm{L}_{\infty\infty}$ using boolean valued semantics [10]. On the other
hand we have not been able to find anywhere an explicit statement that every
complete boolean algebra is the boolean completion of a consistency property
(e.g. Thm. 8.2) even if the proof of this theorem is rather trivial once the
right definitions are given.

  * •

While some of the results we present in this paper were known at least to some
extent (e.g. the completeness theorem via boolean valued semantics for
$\mathrm{L}_{\infty\infty}$ — see Mansfield [10] and independently Karp [5]),
we believe that this paper gives a unified presentation of the sparse number
of theorems connecting infinitary logics to boolean valued semantics we have
been able to trace in the literature. Furthermore, we add to the known results
some original contributions, e.g. Craig’s interpolation property, Beth’s
definability property, the omitting types theorem, the equivalence of forcing
with consistency properties, the completeness theorem for
$\mathrm{L}_{\infty\omega}$ with respect to to the semantics produced by
sheaves on compact extremally disconnected spaces.

The paper is organized as follows:

  * •

2 introduces the basic definitions for the infinitary logics
$\mathrm{L}_{\kappa\lambda}$, including their boolean valued semantics and a
Gentzen’s style proof system for them.

  * •

3 states the main model theoretic results we obtain for
$\mathrm{L}_{\infty\omega}$ and $\mathrm{L}_{\infty\infty}$.

  * •

4 introduces the key notion of consistency property on which we leverage to
prove all the main results of the paper.

  * •

5 shows that we can use consistency properties to produce boolean valued
models with the mixing property (e.g. sheaves on extremally disconnected
compact spaces) for any consistent $\mathrm{L}_{\infty\omega}$ theory.

  * •

6 gives a proof rephrased in our terminology of the main technical result of
Mansfield on this topic, e.g. that any consistency property for
$\mathrm{L}_{\infty\infty}$ gives rise to a corresponding boolean valued model
(which however may not satisfy the mixing property).

  * •

7 leverages on 5 and 6 to prove the theorems stated in 3.

  * •

8 shows that any forcing notion can be presented as the boolean completion of
a consistency property for $\mathrm{L}_{\infty\omega}$.

  * •

The Appendix 9 collects some counterexamples to properties which do not
transfer from first order logic to infinitary logics (for example the failure
of boolean compactness), as well as the proof of some basic facts regarding
boolean valued models.

  * •

We close the paper with a brief list of open problems and comments.

##  2\. The infinitary logics $\mathrm{L}_{\kappa\lambda}$

The set of formulae for a language in first order logic is constructed by
induction from atomic formulae by taking negations, finite conjunctions and
finite quantifications. $\mathrm{L}_{\kappa\lambda}$ generalizes both
“finites” to cardinals $\kappa$ and $\lambda$ allowing disjunctions and
conjunctions of size less than $\kappa$ and simultaneous universal
quantification of a string of variables of size less than $\lambda$. Our basic
references on this topic is Väänänen’s book [12]. To simplify slightly our
notation we confine our attention to relational languages, i.e. languages that
do not have function symbols222With some notational efforts which we do not
spell out all our results transfer easily to arbitrary signatures. Also, when
interested in logics with quantification of infinite strings we consider
natural to include signatures containing relation symbols of infinite arity.

###  2.1. Syntax

######  Definition 2.1.

$\mathrm{L}$ is a relational $\lambda$-signature if it contains only relation
symbols of arity less than $\lambda$ and eventually constant symbols;
relational $\omega$-signatures are first order signatures without function
symbols.

Fix two cardinals $\lambda,\kappa$, a set of $\kappa$ variables,
$\\{v_{\alpha}:\alpha<\kappa\\}$, and consider a relational
$\lambda$-signature $\mathrm{L}$. The set of terms and atomic formulae for
$\mathrm{L}_{\kappa\lambda}$ is constructed in analogy to first order logic
using the symbols of $\mathrm{L}\cup\\{v_{\alpha}:\alpha<\kappa\\}$. The other
$\mathrm{L}_{\kappa\lambda}$-formulae are defined by induction as follows:

  * •

if $\phi$ is a $\mathrm{L}_{\kappa\lambda}$-formula, then so is $\neg\phi$;

  * •

if $\Phi$ is a set of $\mathrm{L}_{\kappa\lambda}$-formulae of size $<\kappa$
with free variables in the set $V=\left\\{v_{i}:i\in I\right\\}$ for some
$I\in[\kappa]^{<\lambda}$, then so are $\bigwedge\Phi$ and $\bigvee\Phi$;

  * •

if $V=\left\\{v_{i}:i\in I\right\\}$ for some $I\in[\kappa]^{<\lambda}$ and
$\phi$ is a $\mathrm{L}_{\kappa\lambda}$-formula, then so are $\forall V\phi$
and $\exists V\phi$.

We let $\mathrm{L}_{\infty\lambda}$ be the family of
$\mathrm{L}_{\kappa\lambda}$-formulae for some $\kappa$, and
$\mathrm{L}_{\infty\infty}$ be the family of
$\mathrm{L}_{\kappa\lambda}$-formulae for some $\kappa,\lambda$.

The restriction on the number of free variables for the clauses $\bigwedge$
and $\bigvee$ is intended to avoid formulae for which there is no quantifier
closure. Another common possibility is to call pre-formula any “formula”, and
formula the ones that verify this property.

###  2.2. Boolean valued semantics

Let us recall the following basic facts about partial orders and their Boolean
completions:

######  Definition 2.2.

Given a Boolean algebra $\mathsf{B}$ and a partial order
$\mathbb{P}=(P,\leq)$:

  * •

$\mathsf{B}^{+}$ denotes the partial order given by its positive elements and
ordered by $a\leq_{\mathsf{B}}b$ if $a\wedge b=a$.

  * •

$\mathsf{B}$ is $<\lambda$-complete if any subset of $\mathsf{B}$ of size less
than $\lambda$ has an infimum and a supremum according to $\leq_{\mathsf{B}}$.

  * •

A set $G\subset P$ is a prefilter if for any $a_{1},\ldots,a_{n}\in G$ we can
find $b\in G$, $b\leq a_{1},\ldots,a_{n}$.

  * •

A set $F\subset P$ is a filter if it is a prefilter and is upward close:

$(a\in F\wedge a\leq b)\Rightarrow b\in F.$

######  Remark 2.3.

Given a partial order $\mathbb{P}=(P,\leq)$:

  * •

The order topology on $P$ is the one whose open sets are given by the downward
closed subsets of $P$; the sets $N_{p}=\left\\{q\in P:q\leq p\right\\}$ form a
basis for this topology.

  * •

$\operatorname{RO}(P)$ is the complete Boolean algebra given by the regular
open sets of the order topology on $P$.

  * •

The map $p\mapsto\text{Reg}\left(N_{p}\right)$ defines an order and
incompatibility preserving map of $P$ into a dense subset of
$(\operatorname{RO}(P)^{+},\subseteq)$; hence $(P,\leq)$ and
$(\operatorname{RO}(P)^{+},\subseteq)$ are equivalent forcing notions.

If $\mathsf{B}$ is a Boolean algebra, $\mathsf{B}^{+}$ sits inside its Boolean
completion $\operatorname{RO}(\mathsf{B}^{+})$ as a dense subset via the map
$b\mapsto N_{b}$ (e.g. for all $A\in\operatorname{RO}(\mathsf{B}^{+})$ there
is $b\in\mathsf{B}$ such that $N_{b}\subseteq A$).

From now on we identify $\mathsf{B}$ with its image in
$\operatorname{RO}(\mathsf{B}^{+})$ via the above map.

######  Definition 2.4.

Let $\mathrm{L}$ be a relational $\lambda$-signature and $\mathsf{B}$ a
$<\lambda$-complete Boolean algebra. A $\mathsf{B}$-valued model $\mathcal{M}$
for $\mathrm{L}$ is given by:

  1. (1)

a non-empty set $M$;

  2. (2)

the Boolean value of equality,

$\displaystyle M^{2}$ $\displaystyle\rightarrow\mathsf{B}$
$\displaystyle(\tau,\sigma)$
$\displaystyle\mapsto\left\llbracket\tau=\sigma\right\rrbracket^{\mathcal{M}}_{\mathsf{B}};$

  3. (3)

the interpretation of relation symbols $R\in\mathrm{L}$ of arity
$\alpha<\lambda$ by maps

$\displaystyle M^{\alpha}$ $\displaystyle\rightarrow\mathsf{B}$
$\displaystyle(\tau_{i}:i\in\alpha)$ $\displaystyle\mapsto\left\llbracket
R(\tau_{i}:i\in\alpha)\right\rrbracket^{\mathcal{M}}_{\mathsf{B}};$

  4. (4)

the interpretation $c^{\mathcal{M}}\in M$ of constant symbols $c$ in
$\mathrm{L}$.

We require that the following conditions hold:

  1. (A)

For all $\tau,\sigma,\pi\in M$,

$\displaystyle\left\llbracket\tau=\tau\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}=1_{\mathsf{B}},$
$\displaystyle\left\llbracket\tau=\sigma\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}=\left\llbracket\sigma=\tau\right\rrbracket^{\mathcal{M}}_{\mathsf{B}},$
$\displaystyle\left\llbracket\tau=\sigma\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\wedge\left\llbracket\sigma=\pi\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\leq\left\llbracket\tau=\pi\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}.$

  2. (B)

If $R\in\mathrm{L}$ is an $\alpha$-ary relation symbol, for all
$(\tau_{i}:\,i<\alpha),(\sigma_{i}:\,i<\alpha)\in M^{\alpha}$,

$\bigg{(}\bigwedge_{i\in\alpha}\left\llbracket\tau_{i}=\sigma_{i}\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\bigg{)}\wedge\left\llbracket
R(\tau_{i}:\,i<\alpha)\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\leq\left\llbracket
R(\sigma_{i}:\,i<\alpha)\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}.$

######  Definition 2.5.

Fix $\mathsf{B}$ a $<\lambda$-complete Boolean algebra and $\mathcal{M}$ a
$\mathsf{B}$-valued structure for a relational $\lambda$-signature
$\mathrm{L}$. We define the $\operatorname{RO}(\mathsf{B}^{+})$-value of an
$\mathrm{L}_{\infty\infty}$-formula $\phi(\overline{v})$ with assignment
$\overline{v}\mapsto\overline{m}$ by induction as follows:

$\displaystyle\left\llbracket
R(t_{i}:i\in\alpha)[\overline{v}\mapsto\overline{m}]\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B}^{+})}=\left\llbracket
R(t_{i}[\overline{v}\mapsto\overline{m}]:i\in\alpha)\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\text{
for $R\in\mathrm{L}$ of arity $\alpha<\lambda$},$
$\displaystyle\left\llbracket(\neg\phi)[\overline{v}\mapsto\overline{m}]\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B}^{+})}=\neg\left\llbracket\phi[\overline{v}\mapsto\overline{m}]\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B}^{+})},$
$\displaystyle\left\llbracket(\bigwedge\Phi)[\overline{v}\mapsto\overline{m}]\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B}^{+})}=\bigwedge_{\phi\in\Phi}\left\llbracket\phi[\overline{v}\mapsto\overline{m}]\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B}^{+})},$
$\displaystyle\left\llbracket(\bigvee\Phi)[\overline{v}\mapsto\overline{m}]\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B}^{+})}=\bigvee_{\phi\in\Phi}\left\llbracket\phi[\overline{v}\mapsto\overline{m}]\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B}^{+})},$
$\displaystyle\left\llbracket(\forall
V\phi)[\overline{v}\mapsto\overline{m}]\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B}^{+})}=\bigwedge_{\overline{a}\in
M^{V}}\left\llbracket\phi[\overline{v}\mapsto\overline{m},V\mapsto\overline{a}]\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B}^{+})},$
$\displaystyle\left\llbracket(\exists
V\phi)[\overline{v}\mapsto\overline{m}]\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B}^{+})}=\bigvee_{\overline{a}\in
M^{V}}\left\llbracket\phi[\overline{v}\mapsto\overline{m},V\mapsto\overline{a}]\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B}^{+})}.$

A $\mathsf{B}$-valued model is well behaved333We believe this is the right
generalization that should become standard in future papers. for
$\mathrm{L}_{\kappa\lambda}$ if
$\left\llbracket\phi(t_{i}:i\in\alpha)[\overline{v}\mapsto\overline{m}]\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B}^{+})}\in\mathsf{B}$
for any $\mathrm{L}_{\kappa\lambda}$ formula $\phi(\overline{v})$.

Let $T$ be an $\mathrm{L}_{\infty\infty}$ theory and $\mathcal{M}$ be a well
behaved $\mathsf{B}$-valued $\mathrm{L}$-structure. The relation

$\mathcal{M}\vDash T$

holds if

$\left\llbracket\bigwedge
T\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}=1_{\mathsf{B}}.$

Note that if $\mathsf{B}$ is complete any $\mathsf{B}$-valued model is well
behaved. We feel free to write just
$\left\llbracket\phi(\tau_{i}:\,i<\alpha)\right\rrbracket$ or
$\left\llbracket\phi(\tau_{i}:\,i<\alpha)\right\rrbracket^{\mathcal{M}}$ or
$\left\llbracket\phi(\tau_{i}:\,i<\alpha)\right\rrbracket_{\mathsf{B}}$ when
no confusion arises on which structure we are considering or in which Boolean
algebra we are evaluating the predicate $R$.

A key (but not immediately transparent) observation is that for any
$\lambda$-signature $\mathrm{L}$, any well behaved $\mathsf{B}$-valued model
$\mathcal{M}$ for $\mathrm{L}$ satisfies B with $R$ replaced by any
$\mathrm{L}_{\infty\infty}$-formula. More precisely the following holds:

######  Fact 2.6.

Let $\mathrm{L}$ be a $\lambda$-relational signature and $\mathsf{B}$ a
$<\lambda$-complete Boolean algebra. Then for any $\mathsf{B}$-valued model
$\mathcal{M}$ for $\mathrm{L}$, any $\mathrm{L}_{\infty\infty}$-formula
$\phi(x_{i}:i<\alpha)$ in displayed free variables, and any sequence
$(\sigma_{i}:i<\alpha)$, $(\tau_{i}:i<\alpha)$ in $\mathcal{M}^{\alpha}$

(1)
$\bigg{(}\bigwedge_{i\in\alpha}\left\llbracket\tau_{i}=\sigma_{i}\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B})^{+}}\bigg{)}\wedge\left\llbracket\phi(\tau_{i}:\,i<\alpha)\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B})^{+}}\leq\left\llbracket\phi(\sigma_{i}:\,i<\alpha)\right\rrbracket^{\mathcal{M}}_{\operatorname{RO}(\mathsf{B})^{+}}.$

We prove this in Section 9.2.

######  Definition 2.7.

Let $\mathsf{B}$ be a complete Boolean algebra and $\mathcal{M}$ a well
behaved $\mathsf{B}$-valued model for some $\lambda$-signature $\mathrm{L}$.
$\mathcal{M}$ has the mixing property if for any antichain
$A\subset\mathsf{B}$ and $\\{\tau_{a}:a\in A\\}\subset M$ there is some
$\tau\in M$ such that
$a\leq\left\llbracket\tau=\tau_{a}\right\rrbracket_{\mathsf{B}}$ for all $a\in
A$.

######  Definition 2.8.

Let $\lambda\leq\kappa$ be infinite cardinals, $\mathsf{B}$ be a
$<\lambda$-complete Boolean algebra, and $\mathcal{M}$ be a well behaved
$\mathsf{B}$-valued model for $\mathrm{L}_{\kappa\lambda}$.

$\mathcal{M}$ is full for the logic $\mathrm{L}_{\kappa\lambda}$ if for every
$\mathrm{L}_{\kappa,\lambda}$-formula $\phi(\overline{v},\overline{w})$ and
$\overline{m}\in M^{\overline{w}}$ there exists $\overline{n}\in
M^{\overline{v}}$ such that

$\left\llbracket\exists\overline{v}\phi(\overline{v},\overline{m})\right\rrbracket_{\mathsf{B}}=\left\llbracket\phi(\overline{n},\overline{m})\right\rrbracket_{\mathsf{B}}.$

######  Proposition 2.9.

Let $\mathrm{L}$ be a $\lambda$-relational signature and $\mathsf{B}$ a
complete Boolean algebra. Any $\mathsf{B}$-valued model for $\mathrm{L}$ with
the mixing property is full for $\mathrm{L}_{\infty\infty}$.

The proof of this proposition is deferred to Section 9.2.

######  Definition 2.10.

Let $\mathsf{B}$ be a $<\lambda$-complete Boolean algebra, $\mathcal{M}$ a
full $\mathsf{B}$-valued model for $\mathrm{L}_{\kappa\lambda}$ where
$\mathrm{L}$ is a relational $\lambda$-signature, and $F\subset\mathsf{B}$ a
$<\lambda$-complete filter. The quotient of $\mathcal{M}$ by $F$ is the
$\mathrm{L}$-structure $\mathcal{M}/_{F}$ defined as follows:

  1. (1)

its domain $M/_{F}$ is the quotient of $M$ by the equivalence

$\tau\equiv_{F}\sigma\leftrightarrow\left\llbracket\tau=\sigma\right\rrbracket\in
F,$

  2. (2)

if $R\in\mathrm{L}$ is an $\alpha$-ary relation symbol,

$R^{\mathcal{M}/_{F}}=\\{({[\tau_{i}]}_{F}:i<\alpha)\in(M/_{F})^{\alpha}:\left\llbracket
R(\tau_{i}:i<\alpha)\right\rrbracket\in F\\},$

  3. (3)

if $c\in\mathrm{L}$ is a constant symbol,

$c^{\mathcal{M}/_{F}}=\bigl{[}c^{\mathcal{M}}\bigr{]}_{F}\in M/F.$

######  Remark 2.11.

If $\mathcal{M}$ a $\mathsf{B}$-valued model for $\mathrm{L}_{\kappa\lambda}$
so is $\mathcal{M}/_{F}$ is for $\mathsf{B}/_{F}$: condition B of Def. 2.10 is
satisfied by the quotient structure $\mathcal{M}/_{F}$ appealing to the
$<\lambda$-completeness of $F$. All other conditions of Def. 2.10 holds for
$\mathcal{M}/_{F}$ just assuming $F$ being a filter. Furthermore if
$\mathcal{M}$ is full for $\mathrm{L}_{\kappa\lambda}$ and $F$ is also
$<\kappa$-complete, so is $\mathcal{M}/_{F}$ (appealing to the
$<\kappa$-completeness of $F$ to handle infinitary disjunctions and
conjunctions and to the $<\lambda$-completeness of $F$ to handle infinitary
quantifiers).

######  Theorem 2.12 (Łoś).

Let $\lambda\leq\kappa$ be infinite cardinals, $\mathsf{B}$ be a
$<\lambda$-complete Boolean algebra, $\mathcal{M}$ an
$\mathrm{L}_{\kappa\lambda}$-full $\mathsf{B}$-valued model for
$\mathrm{L}_{\kappa\lambda}$, and $U\subset\mathsf{B}$ a
$<\max\left\\{\kappa,\lambda\right\\}$-complete ultrafilter. Then, for every
$\mathrm{L}_{\kappa\lambda}$-formula $\phi(\overline{v})$ and
$\overline{\tau}\in M^{|\overline{v}|}$,

$\mathcal{M}/_{U}\vDash\phi(\overline{{[\tau]}_{U}})\iff\left\llbracket\phi(\overline{\tau})\right\rrbracket_{\mathsf{B}}\in
U.$

###### Proof.

A proof of the Theorem for $\mathrm{L}_{\omega\omega}$ for $\omega$-relational
signatures is given in [13, Thm. 5.3.7]. The general case uses the
$<\kappa$-completeness of the ultrafilter to handle $<\kappa$-sized
disjunctions and conjunctions, and its $<\lambda$-completeness and the
fullness of $\mathcal{M}$ to handle quantifiers on infinite strings. ∎

From now on we will work only with complete Boolean algebras $\mathsf{B}$,
hence $\mathsf{B}$-valued models are automatically well behaved for
$\mathrm{L}_{\infty\infty}$.

###  2.3. Boolean satisfiability

######  Definition 2.13.

$\mathrm{BVM}$ denotes the class of Boolean valued models with values on a
complete Boolean algebra and $\mathrm{Sh}$ the subclass of Boolean valued
models with values on a complete Boolean algebra which have the mixing
property. Let $\Gamma$ and $\Delta$ be sets of
$\mathrm{L}_{\infty\infty}$-formulae. In case $\Gamma=\emptyset$ we let

$\left\llbracket\bigwedge\Gamma\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}=1_{\mathsf{B}},$

and if $\Delta=\emptyset$ we let

$\left\llbracket\bigvee\Delta\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}=0_{\mathsf{B}}.$

  * •

$\Gamma$ is _weakly Boolean satisfiable_ if there is a complete Boolean
algebra $\mathsf{B}$ and a $\mathsf{B}$-valued model $\mathcal{M}$ such that
$\left\llbracket\phi\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}>0_{\mathsf{B}}$
for each $\phi\in\Gamma$.

  * •

$\Gamma$ is _Boolean satisfiable_ if there is a complete Boolean algebra
$\mathsf{B}$ and a $\mathsf{B}$-valued model $\mathcal{M}$ such that
$\left\llbracket\phi\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}=1_{\mathsf{B}}$
for each $\phi\in\Gamma$.

  * •

$\Gamma\vDash_{\mathrm{BVM}}\Delta$ if

$\left\llbracket\bigwedge\Gamma\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\leq\left\llbracket\bigvee\Delta\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}$

for any complete Boolean algebra $\mathsf{B}$ and $\mathsf{B}$-valued model
$\mathcal{M}$.

  * •

$\Gamma\vDash_{\mathrm{Sh}}\Delta$ if

$\left\llbracket\bigwedge\Gamma\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}\leq\left\llbracket\bigvee\Delta\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}$

for any complete Boolean algebra $\mathsf{B}$ and $\mathsf{B}$-valued model
$\mathcal{M}$ with the mixing property.

  * •

$\Gamma\equiv_{\mathrm{BVM}}\Delta$ if $\Gamma\vDash_{\mathrm{BVM}}\Delta$ and
$\Delta\vDash_{\mathrm{BVM}}\Gamma$.

  * •

$\Gamma\equiv_{\mathrm{Sh}}\Delta$ if $\Gamma\vDash_{\mathrm{Sh}}\Delta$ and
$\Delta\vDash_{\mathrm{Sh}}\Gamma$.

###  2.4. Proof systems for $\mathrm{L}_{\infty\infty}$

We present a proof system for $\mathrm{L}_{\infty\infty}$ that is a direct
generalization of the Sequent Calculus from first order logic.
$\Gamma$,$\Gamma^{\prime}$,$\Delta$ and $\Delta^{\prime}$ denote sets of
$\mathrm{L}_{\infty\infty}$-formulae of any cardinality,
$\overline{v},\overline{w}$ denote set-sized sequences of variables,
$\overline{t},\overline{u}$ denote set-sized sequences of terms, and $I$
denotes an index set. When dealing with sequents, and in order to make proofs
shorter, we will assume that formulae only contain $\neg,\bigwedge$ and
$\forall$ as logical symbols; this is not restrictive as all reasonable
semantics for these logics (among which all those we consider in this paper)
should validate the natural logical equivalences
$\neg\forall\vec{v}\neg\phi\equiv\exists\vec{v}\phi$, $\neg\bigwedge_{i\in
I}\neg\phi_{i}\equiv\bigvee_{i\in I}\phi_{i}$.

######  Definition 2.14.

Given $\Gamma,\Delta$ arbitrary sets of $\mathrm{L}_{\infty\infty}$-formulae,
a proof of $\Gamma\vdash\Delta$ in $\mathrm{L}_{\infty\infty}$ is a sequence
$(s_{\alpha})_{\alpha\leq\beta}$ of sequents, where $s_{\beta}$ is
$\Gamma\vdash\Delta$ and each element $s_{\alpha}$ is either an axiom or comes
from an application of the following rules to $(s_{i})_{i<\alpha}$.

$\begin{array}[]{lcc@{\qquad}l}\mbox{Axiom
rule}&\prftree{}{\Gamma,\phi\vdash\phi,\Delta}&\prftree{\Gamma,\phi\vdash\Delta}{\Gamma^{\prime}\vdash\phi,\Delta^{\prime}}{\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}}&\mbox{Cut
Rule}\\\ \\\
\mbox{Substitution}&\prftree{\Gamma\vdash\Delta}{\Gamma(\overline{w}\diagup\overline{v})\vdash\Delta(\overline{w}\diagup\overline{v})}&\prftree{\Gamma\vdash\Delta}{\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}}&\mbox{Weakening}\\\
\\\ \mbox{Left
Negation}&\prftree{\Gamma\vdash\phi,\Delta}{\Gamma,\neg\phi\vdash\Delta}&\prftree{\Gamma,\phi\vdash\Delta}{\Gamma\vdash\neg\phi,\Delta}&\mbox{Right
Negation}\\\ \\\ \mbox{Left
Conjunction}&\prftree{\Gamma,\Gamma^{\prime}\vdash\Delta}{\Gamma,\bigwedge\Gamma^{\prime}\vdash\Delta}&\prftree{\Gamma\vdash\phi_{i},\Delta\
,\ i\in I}{\Gamma\vdash\bigwedge_{i\in I}\\{\phi_{i}:i\in
I\\},\Delta}&\mbox{Right Conjunction}\\\ \\\ \mbox{Left
Quantification}&\prftree{\Gamma,\phi(\overline{t}\diagup\overline{v})\vdash\Delta}{\Gamma,\forall\overline{v}\phi(\overline{v})\vdash\Delta}&\prftree[r]{*}{\Gamma\vdash\phi(\overline{w}\diagup\overline{v}),\Delta}{\Gamma\vdash\forall\overline{v}\phi(\overline{v}),\Delta}&\mbox{Right
Quantification}\\\ \\\ \mbox{Equality 1}&\prftree{}{v_{\alpha}=v_{\beta}\vdash
v_{\beta}=v_{\alpha}}&\prftree[r]{}{}{\overline{u}=\overline{t},\phi(\overline{t})\vdash\phi(\overline{u})}&\mbox{Equality
2}\\\ \end{array}$

* The Right Quantification rule can only be applied in the case that none of the variables from $\overline{w}$ occurs free in formulae of $\Gamma\cup\Delta\cup\left\\{\phi\right\\}$.

######  Remark 2.15.

It needs to be noted that with this deduction system the completeness theorem
for $\mathrm{L}_{\infty\infty}$ (even for $\mathrm{L}_{\omega_{2}\omega}$)
fails for the usual semantics given by Tarski structures.

Remark first that our proof system is forcing invariant: the existence of a
proof for a certain sentence is described by a $\Sigma_{1}$ statement in
parameter the sequent to be proved; if the proof exists in $V$ it exists in
any further extension of $V$.

Consider now a set of $\kappa$ constants $\\{c_{\alpha}:\alpha<\kappa\\}$ for
$\kappa>\omega$ and the sentence

$\psi:=\bigg{(}\bigwedge_{\omega\leq\alpha\neq\beta}c_{\alpha}\neq
c_{\beta}\bigg{)}\Rightarrow\exists v\bigg{(}\bigwedge_{n<\omega}v\neq
c_{n}\bigg{)}.$

The sentence $\psi$ is valid in the usual Tarski semantics but it cannot be
proved (in our deduction system or in any forcing invariant system) since the
sentence is no longer valid when moving to $V[G]$ for $G$ a $V$-generic filter
for $\operatorname{Coll}(\omega,\kappa)$.

Malitz [9, Thm. 3.2.4] showed also that the above formula is a counterexample
to Craig’s interpolation property for Tarski semantics in
$\mathrm{L}_{\infty\omega}$.

Our opinion is that a proof system should not depend on the model of set
theory in which one is working, which is the case for the proof system we
present here at least when restricted to $\mathrm{L}_{\infty\omega}$. In
contrast with our point of view, one finds a complete proof system for Tarski
semantics on $\mathrm{L}_{\infty\infty}$ in Malitz’s thesis [9, Thm. 3.3.1],
however this proof system (which by the way is due to Karp [5, Ch. 11]) is not
forcing invariant e.g. a proof of some sequent in some model of set theory may
not be anymore a proof of that same sequent in some forcing extension.

##  3\. Main model theoretic results

These are the main model theoretic results of the paper.

###  3.1. Results for $\mathrm{L}_{\infty\omega}$

######  Theorem 3.1 (Boolean Completeness for $\mathrm{L}_{\infty\omega}$).

Let $\mathrm{L}$ be an $\omega$-relational signature. The following are
equivalent for $T,S$ sets of $\mathrm{L}_{\infty\omega}$-formulae.

  1. (1)

$T\models_{\mathrm{Sh}}S$,

  2. (2)

$T\models_{\mathrm{BVM}}S$,

  3. (3)

$T\vdash S$.

######  Theorem 3.2 (Boolean Craig Interpolation).

Assume $\vDash_{\mathrm{Sh}}\phi\rightarrow\psi$ with
$\phi,\psi\in\mathrm{L}_{\kappa\omega}$. Then there exists a sentence $\theta$
in $\mathrm{L}_{\kappa\omega}$ such that

  * •

$\vDash_{\mathrm{Sh}}\phi\rightarrow\theta$,

  * •

$\vDash_{\mathrm{Sh}}\theta\rightarrow\psi$,

  * •

all non logical symbols appearing in $\theta$ appear both in $\phi$ and
$\psi$.

Recall the Beth definability property:

######  Definition 3.3.

Let $\mathrm{L}$ be a relational $\lambda$-signature and $R$ be an
$\alpha$-ary relation symbol not in $\mathrm{L}$ for some $\alpha<\lambda$.
Given $\lambda,\kappa\in\mathsf{Card}\cup\left\\{\infty\right\\}$, let $T$ be
a $\mathrm{L}_{\kappa\lambda}^{\prime}$-theory for
$\mathrm{L}^{\prime}=\mathrm{L}\cup\left\\{R\right\\}$.

  * •

$R$ is implicitly Boolean definable from $T$ in a relational
$\lambda$-signature $\mathrm{L}$ if the following holds: whenever
$\mathcal{M}$ and $\mathcal{N}$ are $\mathsf{B}$-valued models of $T$ with
domain $M$ such that
$\mathcal{M}\restriction\mathrm{L}=\mathcal{N}\restriction\mathrm{L}$, we have
that $\left\llbracket
R(\tau_{i}:i\in\alpha)\right\rrbracket^{\mathcal{M}}=\left\llbracket
R(\tau_{i}:i\in\alpha)\right\rrbracket^{\mathcal{N}}$ for all
$(\tau_{i}:i\in\alpha)\in M^{\alpha}$.

  * •

$R$ is explicitly Boolean definable from $T$ in $\mathrm{L}_{\kappa\lambda}$
if

$T\vdash\forall(v_{i}:i\in\alpha)\,(R(v_{i}:i\in\alpha)\leftrightarrow\phi(v_{i}:i\in\alpha))$

for some $\mathrm{L}_{\kappa\lambda}$-formula $\phi(v_{i}:i\in\alpha)$.

The Boolean Beth definability property for $\mathrm{L}_{\kappa\lambda}$ (with
$\lambda,\kappa\in\mathsf{Card}\cup\left\\{\infty\right\\}$) states that for
all relational $\lambda$-signatures $\mathrm{L}$ and
$(\mathrm{L}\cup\left\\{R\right\\})_{\infty\omega}$-theory $T$, $R$ is
implicitly definable from $T$ in $\mathrm{L}\cup\left\\{R\right\\}$ if and
only if it is explicitly definable from $T$ in $\mathrm{L}_{\kappa\lambda}$.

This is a standard consequence of Craig’s interpolation and completeness (see
for example [12, Thm. 6.42]; the same proof applies to our context in view of
the properties of our calculus $\vdash$).

######  Theorem 3.4.

$\mathrm{L}_{\infty\omega}$ has the Boolean Beth definability property.

Another main result we present is the Boolean omitting types theorem. We need
to clarify some notation so to make its statement intelligible. Suppose
$\Sigma(v_{1},\dots,v_{n})$ is a set of $\mathrm{L}_{\infty\infty}$-formulae
in free variables $v_{1},\dots,v_{n}$. We say that a model $\mathcal{M}$
realizes $\Sigma(v_{1},\dots,v_{n})$ if there exists some
$m_{1},\dots,m_{n}\in M$ such that

$\mathcal{M}\vDash\bigwedge\Sigma(m_{1},\dots,m_{n}).$

$\mathcal{M}$ omits the type $\Sigma$ amounts to say that for any
$m_{1},\dots,m_{n}\in M$,

$\mathcal{M}\vDash\bigvee_{\phi\in\Sigma}\neg\phi(m_{1},\dots,m_{n}).$

Thus, a model $\mathcal{M}$ omits the family of types
$\mathcal{F}=\\{\Sigma(v_{1},\ldots,v_{n_{\Sigma}}):\Sigma\in\mathcal{F}\\}$
if it models the sentence

$\bigwedge_{\Sigma\in\mathcal{F}}\forall\overline{v}_{\Sigma}\bigvee\left\\{\neg\phi(\overline{v}_{\Sigma}):\phi\in\Sigma\right\\}.$

In the following proof the sets $\Phi$ will be playing the roles of
$\left\\{\neg\psi:\psi\in\Sigma\right\\}$, where $\Sigma$ is the type we wish
to omit. In this context, the type $\Sigma$ is not isolated by a sentence
$\theta$ if whenever there is a model of $\theta$, there is also a model of
$\theta\wedge\neg\phi$ for some $\phi\in\Sigma$.

######  Theorem 3.5 (Boolean Omitting Types Theorem).

Let $T$ be a set-sized Boolean satisfiable $\mathrm{L}_{\infty\omega}$-theory.
Assume $\mathcal{F}$ is a set-sized family such that each $\Phi\in\mathcal{F}$
is a set of $\mathrm{L}_{\infty\omega}$-formulae with the property that each
$\phi\in\Phi$ has free variables among $v_{0},\dots,v_{n_{\Phi}-1}$. Let
$\mathrm{L}_{T,\mathcal{F}}$ be the smallest fragment of
$\mathrm{L}_{\infty\omega}$ such that
$T,\Phi\subset\mathrm{L}_{T,\mathcal{F}}$ for all $\Phi\in\mathcal{F}$.
Suppose that no $\Phi$ is boolean isolated in $\mathrm{L}_{T,\mathcal{F}}$,
i.e.; for all $\mathrm{L}_{T,\mathcal{F}}$-formula $\theta$ in free variables
$v_{0},\dots,v_{n_{\theta}-1}$,

$T+\exists v_{0}\dots v_{n_{\theta}-1}\,\theta$

is Boolean satisfiable if and only if so is

$T+\exists v_{0}\dots
v_{\max\left\\{n_{\theta}-1,n_{\Phi}-1\right\\}}\,[\theta\wedge\phi]$

for some $\phi\in\Phi$. Then there exists a Boolean valued model $\mathcal{M}$
with the mixing property such that

$\mathcal{M}\vDash T+\bigwedge_{\Phi\in\mathcal{F}}\forall v_{0}\dots
v_{n_{\Phi}-1}\,\bigvee\Phi.$

###  3.2. Results for $\mathrm{L}_{\infty\infty}$

Theorem 3.6 below is due to Mansfield [10]. We do not know whether any of
these results hold for $\lambda$-relational signatures which are not first
order.

######  Theorem 3.6.

[10, Thm. 1] Let $\mathrm{L}$ be an $\omega$-relational signature. The
following are equivalent for $T,S$ sets of
$\mathrm{L}_{\infty\infty}$-formulae.

  1. (1)

$T\models_{\mathrm{BVM}}S$,

  2. (2)

$T\vdash S$.

######  Theorem 3.7 (Boolean Craig Interpolation).

Let $\mathrm{L}$ be an $\omega$-relational signature. Assume
$\vDash_{\mathrm{BVM}}\phi\rightarrow\psi$ with
$\phi,\psi\in\mathrm{L}_{\kappa\lambda}$. Then there exists a sentence
$\theta$ in $\mathrm{L}_{\kappa\lambda}$ such that

  * •

$\vDash_{\mathrm{BVM}}\phi\rightarrow\theta$,

  * •

$\vDash_{\mathrm{BVM}}\theta\rightarrow\psi$,

  * •

all non logical symbols appearing in $\theta$ appear both in $\phi$ and
$\psi$.

######  Theorem 3.8.

$\mathrm{L}_{\infty\infty}$ has the Boolean Beth definability property.

As in the case of interpolation, one can prove a version of the omitting types
theorem in $\mathrm{L}_{\infty\infty}$ where the obtained model is not mixing
in general. Nonetheless, we do not think its proof nor the statement while
introduce anything of relevance other than the information found in theorems
3.5 and 3.7.

##  4\. Consistency properties for relational $\omega$-signatures

Consistency properties are partial approximations to the construction of a
model of an infinitary theory. In first order logic the main tool for
constructing Tarski models of a theory is the compactness theorem. However,
this technique is not suited for the infinitary logics
$\mathrm{L}_{\kappa\lambda}$ since it fails even at the simplest case
$\mathrm{L}_{\omega_{1}\omega}$. Actually, a cardinal $\kappa$ is (weakly)
compact if and only if the (weak) compactness theorem holds for the logic
$\mathrm{L}_{\kappa\kappa}$. Thus a new recipe for constructing models is
needed. This is given by the notion of consistency property. Our aim is to
show that by means of consistency properties one gets a powerful tool to
produce boolean valued models of infinitary logic. We follow (generalizing it)
the approach of Keisler’s book [6] to consistency properties for
$\mathrm{L}_{\omega_{1}\omega}$.

First of all it is convenient to reduce the satisfaction problem to formulae
where negations occur only in atomic formulae. We use the abbreviation
$\vec{v}$ to denote a sequence of variables. Similarly $\vec{c}$ denotes a
string of constants.

######  Definition 4.1.

Let $\phi$ be a $\mathrm{L}_{\infty\infty}$-formula. We define $\phi\neg$
(moving a negation inside) by induction on the complexity of formulae:

  * •

If $\phi$ is an atomic formula $\varphi$, $\phi\neg$ is $\neg\varphi$.

  * •

If $\phi$ is $\neg\varphi$, $\phi\neg$ is $\varphi$.

  * •

If $\phi$ is $\bigwedge\Phi$, $\phi\neg$ is
$\bigvee\\{\neg\varphi:\varphi\in\Phi\\}$.

  * •

If $\phi$ is $\bigvee\Phi$, $\phi\neg$ is
$\bigwedge\\{\neg\varphi:\varphi\in\Phi\\}$.

  * •

If $\phi$ is $\forall\vec{v}\varphi(\vec{v})$, $\phi\neg$ is
$\exists\vec{v}\neg\varphi(\vec{v})$.

  * •

If $\phi$ is $\exists\vec{v}\varphi(\vec{v})$, $\phi\neg$ is
$\forall\vec{v}\neg\varphi(\vec{v})$.

It is easily checked that $\neg\phi$ and $\phi\neg$ are equivalent (under any
reasonable notion of equivalence, e.g. boolean satisfiability or provability).
This operation is used in the proof of Thm. 4.3, Thm. 5.7 and Thm. 6.1.

######  Definition 4.2.

Let $\mathrm{L}=\mathcal{R}\cup\mathcal{D}$ be a relational $\omega$-signature
where the relation symbols are in $\mathcal{R}$ and $\mathcal{D}$ is the set
of constants. Given an infinite set of constants $\mathcal{C}$ disjoint from
$\mathcal{D}$, consider $\mathrm{L}(\mathcal{C})$ the signature obtained by
extending $\mathrm{L}$ with the constants in $\mathcal{C}$. A set $S$ whose
elements are set sized subsets of $\mathrm{L}(\mathcal{C})_{\infty\infty}$ is
a consistency property for $\mathrm{L}(\mathcal{C})_{\infty\infty}$ if for
each $s\in S$ the following properties hold:

  1. (Con)

for any $r\in S$ and any $\mathrm{L}(\mathcal{C})_{\infty\infty}$-sentence
$\phi$ either $\phi\not\in r$ or $\neg\phi\not\in r$,

  2. (Ind.1)

if $\neg\phi\in s$, $s\cup\\{\phi\neg\\}\in S$,

  3. (Ind.2)

if $\bigwedge\Phi\in s$, then for any $\phi\in\Phi$, $s\cup\\{\phi\\}\in S$,

  4. (Ind.3)

if $\forall\vec{v}\phi(\vec{v})\in s$, then for any
$\vec{c}\in(\mathcal{C}\cup\mathcal{D})^{|\vec{v}|}$,
$s\cup\\{\phi(\vec{c})\\}\in S$,

  5. (Ind.4)

if $\bigvee\Phi\in s$, then for some $\phi\in\Phi$, $s\cup\\{\phi\\}\in S$,

  6. (Ind.5)

if $\exists\vec{v}\phi(\vec{v})\in s$, then for some
$\vec{c}\in\mathcal{C}^{|\vec{v}|}$, $s\cup\\{\phi(\vec{c})\\}\in S$,

  7. (Str.1)

if $c,d\in\mathcal{C}\cup\mathcal{D}$ and $c=d\in s$, then $s\cup\\{d=c\\}\in
S$,

  8. (Str.2)

if $c,d\in\mathcal{C}\cup\mathcal{D}$ and $\\{c=d,\phi(d)\\}\subset s$, then
$s\cup\\{\phi(c)\\}\in S$,

  9. (Str.3)

if $d\in\mathcal{C}\cup\mathcal{D}$, then for some $c\in\mathcal{C}$,
$s\cup\\{c=d\\}\in S$.

The following result, due to Makkai [8], shows the value of consistency
properties for $\mathrm{L}_{\omega_{1}\omega}$.

######  Theorem 4.3 (Model Existence Theorem).

Let $\mathrm{L}$ be a countable relational $\omega$-signature, $\mathcal{C}$ a
countable set of constants, and
$S\subset[\mathrm{L}(\mathcal{C})_{\omega_{1}\omega}]^{\leq\omega}$ be a
consistency property of countable size. Then any $s\in S$ is realized in some
Tarski model for $\mathrm{L}$.

Now let us give a few examples of consistency properties for
$\mathrm{L}(\mathcal{C})_{\infty\omega}$ and
$\mathrm{L}(\mathcal{C})_{\infty\infty}$.

  1. (1)

Consider $\mathcal{K}$ a class of Tarski structures for
$\mathrm{L}(\mathcal{C})$. The following families are consistency properties
for $\mathrm{L}(\mathcal{C})_{\infty\infty}$:

     * •

for fixed infinite cardinals $\lambda\geq\kappa,\mu$ and $\mathcal{C}$ a set
of constants of size at least $\lambda$,

$S_{\lambda,\kappa}=\\{s\in[\mathrm{L}(\mathcal{C})_{\lambda\mu}]^{\leq\kappa}:\,\exists\mathcal{A}\in\mathcal{K},\
\mathcal{A}\vDash\bigwedge s\\},$

     * •

$S_{\lambda,<\omega}=\\{s\in[\mathrm{L}(\mathcal{C})_{\lambda\mu}]^{<\omega}:\,\exists\mathcal{A}\in\mathcal{K},\
\mathcal{A}\vDash\bigwedge s\\}$,

     * •

$S_{\lambda,\kappa}$ and $S_{\lambda,<\omega}$ where only a finite number of
constants from $\mathcal{C}$ appear in each $s\in S$.

  2. (2)

Let $\mathcal{M}$ be a $\mathsf{B}$-valued model with domain $M$ for a
signature $\mathrm{L}=\mathcal{R}\cup\mathcal{D}$. We let $\mathcal{C}=M$ and
$S$ be the set of finite (less than $\kappa$-sized,…) sets $r$ of
$\mathrm{L}(M)_{\kappa\lambda}$-sentences such that

$\left\llbracket\bigwedge
r\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}>0_{\mathsf{B}}.$

Then $S$ is a consistency property.

  3. (3)

The following families are consistency properties for
$\mathrm{L}(\mathcal{C})_{\infty\omega}$:

     * •

Any of the previous cases where the Tarski structures in $\mathcal{K}$ may
exist only in some generic extension of $V$: e.g. given a
$\mathrm{L}_{\infty\omega}$-theory $T$, $T$ may not be consistent in $V$ with
respect to the Tarski semantics for $\mathrm{L}_{\infty\omega}$, but $T$ may
become consistent with respect to the Tarski semantics for
$\mathrm{L}_{\infty\omega}$ in some generic extension of $V$; one can then use
the forcible properties of the Tarski models of $T$ existing in some generic
extension of $V$ to define a consistency property in $V$.

The last example is based on the following observation: let $S$ be a
consistency property for $\mathrm{L}_{\kappa^{+}\omega}$ of size $\kappa$
whose elements are all sets of formulae of size at most $\kappa$ existing in
$V$. Let $G$ be a $V$-generic filter for the forcing
$\operatorname{Coll}(\omega,\kappa)$. Then, in the generic extension $V[G]$,
$S$ becomes a consistency property of countable size for
$\mathrm{L}_{\omega_{1}^{V[G]}\omega}$ and the Model Existence Theorem 4.3
applied in $V[G]$ provides the desired Tarski model of any $s\in S$.

######  Definition 4.4.

Suppose $\kappa$ is an infinite cardinal and let $\mathrm{L}$ be a signature.
A fragment $\mathrm{L}_{\mathcal{A}}\subset\mathrm{L}_{\kappa\omega}$ consists
in a set of $\mathrm{L}_{\kappa\omega}$-formulas such that:

  * •

$\mathrm{L}_{\mathcal{A}}$ is closed under $\neg$, $\wedge$ and $\vee$,

  * •

if $\phi\in\mathrm{L}_{\mathcal{A}}$ and $v$ is a variable appearing in some
$\mathrm{L}_{\mathcal{A}}$-formula, $\forall v\phi$ and $\exists v\phi$ belong
to $\mathrm{L}_{\mathcal{A}}$,

  * •

$\mathrm{L}_{\mathcal{A}}$ is closed under subformulas,

  * •

if $\phi\in\mathrm{L}_{\mathcal{A}}$, then
$\phi\neg\in\mathrm{L}_{\mathcal{A}}$,

  * •

if $\phi\in\mathrm{L}_{\mathcal{A}}$, then there is a variable appearing in
$\mathrm{L}_{\mathcal{A}}$ which does not occur in $\phi$,

  * •

if $\phi(v)\in\mathrm{L}_{\mathcal{A}}$ and $t$ is any $\mathrm{L}$-term,
$\phi(t)\in\mathrm{L}_{\mathcal{A}}$,

  * •

if $\phi(v_{1},\ldots,v_{n})\in\mathrm{L}_{\mathcal{A}}$ and
$w_{1},\ldots,w_{n}$ are variable appearing in $\mathrm{L}_{\mathcal{A}}$,
$\phi(w_{1},\ldots,w_{n})\in\mathrm{L}_{\mathcal{A}}$.

######  Remark 4.5.

Suppose $\kappa$ is an infinite cardinal and let $\mathrm{L}$ be a signature.
Let $T$ be a set of $\mathrm{L}_{\kappa\omega}$-formulae. Then there exists a
smallest fragment $\mathrm{L}_{\mathcal{A}}$ such that
$T\subset\mathrm{L}_{\mathcal{A}}$ and

$|\mathrm{L}_{\mathcal{A}}|=|L|+|T|+\kappa.$

##  5\. Forcing with consistency properties

In this section we assume that $\mathrm{L}$ denotes a set-sized
$\omega$-relational signature, $\mathcal{C}$ is a set of fresh constants, and
$S\subset\mathcal{P}(\mathrm{L}(\mathcal{C})_{\infty\omega})$ is a set-sized
consistency property.

We start by noting the following:

######  Remark 5.1.

If $S$ is a consistency property, so is
$\\{s\subset\mathrm{L}(\mathcal{C})_{\infty\omega}:\exists s_{0}\in
S\,s\subseteq s_{0}\\}$.

######  Definition 5.2.

Let $S$ be a consistency property over
$\mathrm{L}(\mathcal{C})_{\infty\omega}$ for a set of constants $\mathcal{C}$
and a relational $\omega$-signature $\mathrm{L}$. The forcing notion
$\mathbb{P}_{S}$ is given by:

  * •

domain: $\\{s\subset\mathrm{L}(\mathcal{C})_{\infty\omega}:\exists s_{0}\in
S\,(s\subseteq s_{0})\\}$;

  * •

order: $p\leq q$ if and only if $q\subseteq p$.

Given a filter $F$ on $\mathbb{P}_{S}$, $\Sigma_{F}=\bigcup F$.

The proof of the Model Existence Theorem for $\mathrm{L}_{\omega_{1}\omega}$
as given in [6], corresponds naturally to the construction for a given
consistency property $S$ of a suitable filter $G$ on $\mathbb{P}_{S}$ generic
over countably many dense sets. The clauses of a consistency property are
naturally attached to dense sets a maximal filter $G$ on $\mathbb{P}_{S}$
needs to meet in order to produce a Tarski model of the formulae
$\phi\in\bigcup G$. For example, suppose $\bigvee\Phi\in s_{0}\in S$. Clause
(Ind.4) together with Remark 5.1 states that the set $\\{s\in S:\,\Phi\cap
s\neq\emptyset\\}$ is dense below $s_{0}$. In Keisler’s case the elements of a
consistency property are countable and each
$\mathrm{L}(\mathcal{C})_{\omega_{1}\omega}$-formula has countably many
subformulae. So, one can take an enumeration of all the dense sets at issue
and diagonalize. In the general case for $\mathrm{L}_{\infty\omega}$ one deals
with many more dense sets, hence a filter meeting all the relevant dense sets
may not exists. However we can translate Keisler’s argument using forcing and
produce a Boolean valued model for the associated consistency property.

For the rest of this section we work with consistency properties made up from
finite sets of sentences. The reader familiar with Keisler’s book [6] will
find this restriction natural.

We split our generalization of Keisler’s result in two pieces. The first piece
shows how far one can go in proving the Model Existence Theorem assuming only
the existence of a maximal filter on a consistency property $S$. The second
one shows how genericity fills the missing gaps.

######  Fact 5.3.

Let $S$ be a consistency property for $\mathrm{L}(\mathcal{C})_{\infty\omega}$
for a set of constants $\mathcal{C}$. Assume $S$ consists only of finite sets
of formulae. Suppose $F\subseteq\mathbb{P}_{S}$ is a filter. Then
$[\Sigma_{F}]^{<\omega}=F$.

###### Proof.

The inclusion $F\subset[\Sigma_{F}]^{<\omega}$ follows by definition of
$\Sigma_{F}$. We now prove $[\Sigma_{F}]^{<\omega}\subseteq F$. Suppose
$p=\left\\{\phi_{1},\dots,\phi_{n}\right\\}\in[\Sigma_{F}]^{<\omega}$. Then
there exist $s_{1},\ldots,s_{n}\in F$ with each $\phi_{i}\in s_{i}$. Hence
$p\subseteq\bigcup_{i\leq n}s_{i}$. Since $F$ is a filter, we have
$\bigcup_{i\leq n}s_{i}\in F\subseteq\mathbb{P}_{S}$. The set $p$ is a
condition in $\mathbb{P}_{S}$ since $\mathbb{P}_{S}$ is closed under subsets.
Finally, $\bigcup_{i\leq n}s_{i}\leq p$ and $\bigcup_{i\leq n}s_{i}\in F$
imply $p\in F$. ∎

######  Definition 5.4.

Given a relational $\omega$-signature $\mathrm{L}=\mathcal{R}\cup\mathcal{D}$,
a set of fresh constants $\mathcal{C}$, and a consistency property $S$ for
$\mathrm{L}(\mathcal{C})_{\infty\omega}$, let $F$ be a maximal filter for
$\mathbb{P}_{S}$.

$\mathcal{A}_{F}=(A_{F},R_{F}:R\in\mathcal{R},d_{F}:d\in\mathcal{D})$ is the
following string of objects:

  * •

$A_{F}$ is the set of equivalence classes on $\mathcal{C}\cup\mathcal{D}$ for
the equivalence relation $c\cong_{F}d$ if and only if $(c=d)\in\Sigma_{F}$,

  * •

for $R\in\mathcal{D}$ $n$-ary relation symbol and
$c_{1},\ldots,c_{n}\in\mathcal{C}\cup\mathcal{D}$,
$R_{F}([c_{1}]_{F},\dots,[c_{n}]_{F})$ holds if and only if
$R(c_{1},\dots,c_{n})\in\Sigma_{F}$,

  * •

$d_{F}=[d]_{F}$ for any $d\in\mathcal{D}\cup\mathcal{C}$.

Consistency properties are so designed that $\mathcal{A}_{F}$ is a Tarski
structure for $\mathrm{L}(\mathcal{C})$:

######  Fact 5.5.

Let $\mathrm{L}=\mathcal{R}\cup\mathcal{D}$ be a relational
$\omega$-signature, $\mathcal{C}$ a fresh set of constants, $S$ a consistency
property for $\mathrm{L}(\mathcal{C})_{\infty\omega}$, $F$ a maximal filter
for $\mathbb{P}_{S}$. Then $\mathcal{A}_{F}$ is a Tarski structure for
$\mathrm{L}(\mathcal{C})$.

###### Proof.

We need to check that the definition of $A_{F}$ and of $R_{F}$ does not depend
on the chosen representatives $c_{1},\ldots,c_{n}$. Suppose
$c_{1}=d_{1},\ldots,c_{n}=d_{n},R(c_{1}\ldots c_{n})\in\Sigma_{F}$. By the
previous Fact $\\{c_{1}=d_{1},\ldots,c_{n}=d_{n},R(c_{1}\ldots c_{n})\\}\in
F$. Hence by Clause 4.2(Ind2) for any
$p\supseteq\\{c_{1}=d_{1},\ldots,c_{n}=d_{n},R(c_{1}\ldots c_{n})\\}$ in
$\mathbb{P}_{S}$,
$p\cup\left\\{R(d_{1},\dots,d_{n})\right\\}\in\mathbb{P}_{S}$. This combined
with Clause 4.2(Con) gives that no $p\in\mathbb{P}_{S}$ can contain
$\\{c_{1}=d_{1},\ldots,c_{n}=d_{n},R(c_{1}\ldots c_{n}),\neg
R(d_{1},\dots,d_{n})\\}$. Hence by maximality of $F$,

$\\{c_{1}=d_{1},\ldots,c_{n}=d_{n},R(c_{1}\ldots
c_{n}),R(d_{1},\dots,d_{n})\\}\in F$

must be the case. ∎

######  Lemma 5.6.

Let $\mathrm{L}$ be a relational $\omega$-signature and $\mathcal{C}$ an
infinite set of constants disjoint from $\mathrm{L}$. Assume
$S\subset[\mathrm{L}(\mathcal{C})_{\infty\infty}]^{<\omega}$ is a consistency
property. Let $F\subseteq\mathbb{P}_{S}$ be a maximal filter on
$\mathbb{P}_{S}$. Consider $\Sigma^{\prime}_{F}\subset\Sigma_{F}$ the set of
(quantifier free) formulae $\psi\in\Sigma_{F}$ which are either atomic,
negated atomic, or such that any subformula of $\psi$ which is neither atomic
nor negated atomic contains just the logical constant $\bigwedge$. Then
$\mathcal{A}_{F}\vDash\Sigma^{\prime}_{F}$.

###### Proof.

We do it by induction on the complexity of $\psi\in\Sigma^{\prime}_{F}$. First
note that $\mathbb{P}_{S}$ is a consistency property of which $S$ is a dense
subset. The atomic case follows by Def. 5.4. For the remaining inductive
clauses we proceed as follows:

$\neg$:

Suppose $\psi=\neg\phi\in\Sigma^{\prime}_{F}$ with $\phi$ an atomic formula.
Let’s see that

$\mathcal{A}_{F}\nvDash\phi.$

Since $\phi$ is atomic it is enough to check $\phi\notin\Sigma^{\prime}_{F}$.
Suppose otherwise. Then there exists $p\in F$ with $\phi\in p$. Also $\psi\in
q$ for some $q\in F$. By compatibility of filters there exists $r\leq p,q$.
But $\phi,\neg\phi\in r$ contradicts clause 4.2(Con) for $\mathbb{P}_{S}$.
Therefore

$\mathcal{A}_{F}\vDash\psi.$

$\bigwedge$:

Suppose $\psi=\bigwedge\Phi$ is in $\Sigma_{F}^{\prime}$. One needs to check

$\mathcal{A}_{F}\vDash\phi$

for any $\phi\in\Phi$. Fix such a $\phi\in\Phi$. We start by showing that if
$\bigwedge\Phi\in\Sigma^{\prime}_{F}$, $\phi$ is also in
$\Sigma^{\prime}_{F}$. It is enough to check $\phi\in\Sigma_{F}$, and then
apply the inductive assumptions on $\phi\in\Sigma^{\prime}_{F}$, to get that
$\mathcal{A}_{F}\models\phi$. Towards this aim we note the following:

> For any $q\in\mathbb{P}_{S}$ with $\bigwedge\Phi\in q$,
> $q\cup\left\\{\phi\right\\}\in\mathbb{P}_{S}$, while
> $q\cup\mathbb{\neg\phi}\not\in\mathbb{P}_{S}$.

###### Proof.

Take $q$ in $\mathbb{P}_{S}$ with $\bigwedge\Phi\in q$. By Clause 4.2(Ind.2),
$q\cup\\{\phi\\}\in\mathbb{P}_{S}$. Assume now that $\neg\phi\in q$. Then
$q\cup\\{\phi\\}\in\mathbb{P}_{S}$ would contradict Clause 4.2(Ind.2) for
$\mathbb{P}_{S}$. The thesis follows. ∎

By maximality of $F$ if some $q\in F$ is such that $\bigwedge\Phi\in q$, then
$q\cup\left\\{\phi\right\\}\in F$ as well, yielding that $\phi\in\Sigma_{F}$
as was to be shown.

∎

######  Theorem 5.7.

Let $\mathrm{L}$ be a relational $\omega$-signature, $\mathcal{C}$ an infinite
set of constants disjoint from $\mathrm{L}$, and $S$ be a consistency property
consisting of $\mathrm{L}(\mathcal{C})_{\infty\omega}$-sentences.

Assume that $F$ is a $V$-generic filter for $\mathbb{P}_{S}$. Then in $V[F]$
it holds that:

  1. (1)

The domain of $\mathcal{A}_{F}$ is exactly given by
$\left\\{[c]_{F}:c\in\mathcal{C}\right\\}$.

  2. (2)

For any $\mathrm{L}(\mathcal{C})_{\infty\omega}$-sentence $\psi$

$\mathcal{A}_{F}\vDash\psi\text{ if }\psi\in\Sigma_{F}.$

Note the following apparently trivial corollary of the above Theorem:

######  Corollary 5.8.

Assume $S$ is a consistency property on
$\mathrm{L}(\mathcal{C})_{\infty\omega}$ satisfying the assumptions of Thm.
5.7. Then for any $s\in S$ $s\not\vdash\emptyset$.

######  Remark 5.9.

We note that essentially the same Theorem and Corollary have been proved
independently by Ben De Bondt and Boban Velickovic (using the language of
forcing via partial orders to formulate them).

###### Proof.

Assume $s\vdash\emptyset$ for some $s\in S$. Note that if $F$ is $V$-generic
for $\mathbb{P}_{S}$ with $s\in F$, the same proof existing in $V$ of
$s\vdash\emptyset$ is a proof of the same sequent in $V[F]$. By Thm. 5.7
$\mathcal{A}_{F}\models\bigwedge s$ holds in $V[F]$. Hence by the soundness of
Tarski semantics for $\vdash$ in $V[F]$, we would get that
$\mathcal{A}_{F}\models\psi\wedge\neg\psi$ for some $\psi$ holds in $V[F]$.
This is clearly a contradictory statement for $V[F]$. ∎

We now prove Thm. 5.7:

###### Proof.

Let (in $V[F]$) $\mathcal{A}_{F}$ be the structure obtained from $F$ as in
Def. 5.4. Since $S$ is a dense subset of $\mathbb{P}_{S}$, $F\cap S$ is a
generic filter for $(S,\supseteq)$ as well. By Clause 4.2(Str.3)

$D_{d}=\\{p\in S:\exists c\in\mathcal{C},c=d\in p\\}$

is dense in $\mathbb{P}_{S}$ for any $d\in\mathcal{D}$. Let $p\in F\cap
D_{d}$. Then for some $c\in C$, $d=c\in p\subset\Sigma_{F}$ and
$[d]_{F}=[c]_{F}$. This proves part 1 of the Theorem.

We now establish part 2. We have to handle only the cases for $\neg$,
$\bigvee$, $\exists$, $\forall$ formulae, since the atomic case and the case
$\bigwedge$ can be treated exactly as we did in Fact 5.5 and Lemma 5.6. We
continue the induction as follows:

$\bigvee$:

Suppose $\bigvee\Phi\in\Sigma_{F}$. Let $p_{0}\in F$ be such that
$\bigvee\Phi\in p_{0}$. By Clause 4.2(Ind.4)

$D_{\bigvee\Phi}=\\{p\in S:\exists\phi\in\Phi,\phi\in p\\}$

is dense below $p_{0}$. Since $F$ is $V$-generic over $\mathbb{P}_{S}$ and
$p_{0}\in F$, there exists $p\in F\cap D_{\bigvee\Phi}$. Then for some
$\phi\in\Phi$, $\phi\in p\subset\Sigma_{F}$ and

$\mathcal{A}_{F}\vDash\phi,$

proving

$\mathcal{A}_{F}\vDash\bigvee\Phi.$

$\exists$:

Suppose $\exists\vec{v}\,\phi(\vec{v})\in\Sigma_{F}$. Let $p_{0}\in F$ such
that $\exists\vec{v}\,\phi(\vec{v})\in p_{0}$. By Clause 4.2(Ind.5)

$D_{\exists v\phi(\vec{v})}=\\{p\in
S:\exists\vec{c}\in\mathcal{C}^{\vec{v}},\phi(\vec{c})\in p\\}$

is dense below $p_{0}$. Since $F$ is $V$-generic over $\mathbb{P}_{S}$ and
$p_{0}\in F$, there exists $p\in F\cap D_{\exists\vec{v}\phi(\vec{v})}$. Then
for some $\vec{c}\in\mathcal{C}^{\vec{v}}$, $\phi(\vec{c})\in
p\subset\Sigma_{F}$. Therefore

$\mathcal{A}_{F}\vDash\phi(\vec{c}),$

hence

$\mathcal{A}_{F}\vDash\exists\vec{v}\phi(\vec{v}).$

$\forall$:

Suppose $\psi=\forall\vec{x}\phi(\vec{x})$ is in $\Sigma_{F}^{\prime}$. One
needs to check

$\mathcal{A}_{F}\vDash\phi(\vec{x})[\vec{x}/\vec{e}]$

for
$\vec{e}=\langle[e_{1}]_{F},\dots,[e_{n}]_{F}\rangle\in\mathcal{A}_{F}^{n}$.

Let $\mathcal{E}=\mathcal{C}\cup\mathcal{D}$. Then we have that

$\mathcal{A}_{F}=\left\\{[e]_{F}:\,e\in\mathcal{E}\right\\};$

hence

$\mathcal{A}_{F}^{<\omega}=\left\\{\langle[e_{1}]_{F},\dots,[e_{n}]_{F}\rangle:\,\langle
e_{1},\dots,e_{n}\rangle\in(\mathcal{E}^{<\omega})^{V[F]}\right\\}.$

A key observation is that

$(\mathcal{E}^{<\omega})^{V[F]}=(\mathcal{E}^{<\omega})^{V}.$

This gives that for any $\vec{e}\in\mathcal{A}_{F}^{<\omega}$

$\mathcal{A}_{F}\vDash\phi(\vec{x})[\vec{x}/\vec{e}]$

if and only if there are $e_{1}\dots e_{n}\in\mathcal{E}$ such that
$\vec{e}=\langle[e_{1}]_{F},\dots,[e_{n}]_{F}\rangle$ and

$\mathcal{A}_{F}\vDash\phi(e_{1},\dots,e_{n}).$

By Clause 4.2(Ind.3), assuming $\forall\vec{x}\phi(\vec{x})\in\Sigma_{F}$, we
get that $\phi(e_{1},\dots,e_{n})\in\Sigma_{F}$ for all
$e_{1},\dots,e_{n}\in\mathcal{E}$. Hence in $V[F]$ it holds that

$\mathcal{A}_{F}\vDash\phi(\vec{x})[\vec{x}/\vec{e}]$

for all $\vec{e}\in\mathcal{A}_{F}^{n}$, as was to be shown.

$\neg$:

Suppose $\neg\phi\in\Sigma_{F}$. Clause 4.2(Ind.1) ensures that
$F^{\prime}=[\Sigma_{F}\cup\\{\phi\neg\\}]$ is a prefilter on $\mathbb{P}_{S}$
containing $F$. By maximality of $F$, $\phi\neg\in F$. We know that $\phi\neg$
and $\neg\phi$ are equivalent (under any reasonable equivalence notion, for
example provability, or logical consequence for Boolean valued semantics).
Also the principal connective of $\phi\neg$ is of type
$\bigwedge,\forall,\bigvee$ or $\exists$, for which cases the proof has
already been given.

The above shows that for all
$\mathrm{L}(\mathcal{C})_{\infty\omega}$-sentences $\psi$, if
$\psi\in\Sigma_{F}$ then $\mathcal{A}_{F}\models\psi$. ∎

######  Remark 5.10.

One may wonder why the Theorem is proved just for consistency properties for
$\mathrm{L}_{\infty\omega}$ and not for arbitrary consistency properties on
$\mathrm{L}_{\infty\infty}$. Inspecting the proof one realizes that in the
case of $\forall$ we crucially used that $\mathcal{E}^{<\omega}$ is computed
the same way in $V[F]$ and in $V$. If instead we are working with
$\mathrm{L}_{\infty\lambda}$ for $\lambda>\omega$, it could be the case that
$\mathcal{E}^{<\lambda}$ as computed in $V[F]$ is a strict superset of
$\mathcal{E}^{<\lambda}$ as computed in $V$. In this case there is no reason
to expect that

$\mathcal{A}_{F}\models\phi(x_{i}:\,i<\alpha)[x_{i}/[e_{i}]_{F}:i<\alpha]$

when $\forall\vec{x}\phi(\vec{x})\in\Sigma_{F}$ but $\langle
e_{i}:i<\alpha\rangle\in\mathcal{E}^{<\lambda}\setminus V$.

Note that it may occur that for some
$\mathrm{L}(\mathcal{C})_{\infty\omega}$-sentence $\psi$, neither $\psi$ nor
$\neg\psi$ belongs to any $r\in S$, hence for some $V$-generic filter $F$ for
$\mathbb{P}_{s}$ it can be the case that $s\not\in F$ while
$\mathcal{A}_{F}\models\psi$. For example this occurs because $S$ is a set and
there are class many $\mathrm{L}(\mathcal{C})_{\infty\omega}$-sentence $\psi$.

We can prove a partial converse of the second conclusion of Thm. 5.7 which
requires a slight strengthening of the notion of consistency property:

######  Definition 5.11.

Let $\mathrm{L}=\mathcal{R}\cup\mathcal{D}$, $\mathcal{D}$, $S$ be as in Def.
4.2 and $\kappa$ be a cardinal greater than or equal to $|\mathcal{C}|$.

A consistency property $S$ is _$(\kappa,\lambda)$ -maximal_ if all its
elements consist of $\mathrm{L}(\mathcal{C})_{\kappa\lambda}$-sentences and
$S$ satisfies the following clause:

  1. (S-Max)

For any $p\in S$ and $\mathrm{L}(\mathcal{C})_{\kappa\lambda}$-sentence
$\phi$, either $p\cup\left\\{\phi\right\\}\in S$ or
$p\cup\left\\{\neg\phi\right\\}\in S$.

Example 2 (given by the finite sets of
$\mathrm{L}(M)_{\kappa\lambda}$-sentences which have positive value in some
fixed Boolean valued model with domain $M$) gives the standard case of a
$(\kappa,\lambda)$-maximal consistency property.

######  Proposition 5.12.

With the notation of Thm. 5.7 Assume $S$ is $(\kappa,\omega)$-maximal for some
$\kappa\geq|\mathcal{C}|$. Then for any
$\mathrm{L}(\mathcal{C})_{\kappa\omega}$-sentence $\psi$

$\mathcal{A}_{F}\vDash\psi\text{ if and only if }\psi\in\Sigma_{F}.$

###### Proof.

We need to prove the “only if” part of the implication assuming $S$ is
$(\kappa,\omega)$-maximal. Suppose $\psi$ is an
$\mathrm{L}(\mathcal{C})_{\kappa\omega}$-sentence not in $\Sigma_{F}$. By
$(\kappa,\omega)$-maximality of $S$ we get that

$D_{\psi}=\left\\{r\in S:\psi\in r\text{ or }\neg\psi\in r\right\\}$

is dense in $\mathbb{P}_{S}$. Since $F$ is $V$-generic for $\mathbb{P}_{S}$,
we get that $F\cap D_{\psi}$ is non-empty. Hence either $\psi\in\Sigma_{F}$ or
$\neg\psi\in\Sigma_{F}$, but the first is not the case by hypothesis. Then
$\neg\psi\in\Sigma_{F}$ and by Theorem 5.7 $\mathcal{A}_{F}\models\neg\psi$,
e.g. $\mathcal{A}_{F}\not\models\psi$.

The desired thesis follows. ∎

Let us recall one result about $<\kappa$-cc forcing notions. Proposition 5.13
appears in [2].

######  Proposition 5.13.

Let $\kappa$ be a regular cardinal and $\mathbb{P}\subset H_{\kappa}$ a
forcing notion with the $<\kappa$-cc. Suppose $p\in\mathbb{P}$ and
$\dot{\tau}$ is a $\mathbb{P}$-name such that $p\Vdash\dot{\tau}\in
H_{\check{\kappa}}$, then there exists $\dot{\sigma}\in H_{\kappa}$ such that
$p\Vdash\dot{\sigma}=\dot{\tau}$.

######  Definition 5.14.

Given a relational $\omega$-signature $\mathrm{L}=\mathcal{R}\cup\mathcal{D}$,
an infinite set of constants $\mathcal{C}$ disjoint from $\mathrm{L}$, and a
consistency property
$S\subset[\mathrm{L}(\mathcal{C})_{\infty\omega}]^{<\omega}$, let

$\mathcal{A}_{S}=(A_{S},R_{S}:R\in\mathcal{R},d_{S}:d\in\mathcal{D}\cup\mathcal{C})$

be defined as follows:

  * •

$A_{S}=\left\\{\sigma\in V^{\operatorname{RO}(\mathbb{P}_{S})}\cap
H_{\mu}:\,\left\llbracket\sigma\in
A_{\dot{G}}\right\rrbracket^{V^{\operatorname{RO}(\mathbb{P}_{S})}}_{\operatorname{RO}(\mathbb{P}_{S})}=1_{\operatorname{RO}(\mathbb{P}_{S})}\right\\}$,
where $\mu$ is a regular cardinal big enough so that $\mathrm{L}\subseteq
H_{\mu}$ and for any $\sigma\in V^{\operatorname{RO}(\mathbb{P}_{S})}$ such
that

$\left\llbracket\sigma\in
A_{\dot{G}}\right\rrbracket^{V^{\operatorname{RO}(\mathbb{P}_{S})}}_{\operatorname{RO}(\mathbb{P}_{S})}=1_{\operatorname{RO}(\mathbb{P}_{S})},$

one can find $\tau\in V^{\operatorname{RO}(\mathbb{P}_{S})}\cap H_{\mu}$ with

$\left\llbracket\tau=\sigma\right\rrbracket^{V^{\operatorname{RO}(\mathbb{P}_{S})}}_{\operatorname{RO}(\mathbb{P}_{S})}=1_{\operatorname{RO}(\mathbb{P}_{S})};$

  * •

$\left\llbracket
R_{S}(\sigma_{1},\dots,\sigma_{n})\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{\mathcal{A}_{S}}=\left\llbracket\mathcal{A}_{\dot{G}}\models
R_{\dot{G}}(\sigma_{1},\dots,\sigma_{n})\right\rrbracket^{V^{\operatorname{RO}(\mathbb{P}_{S})}}_{\operatorname{RO}(\mathbb{P}_{S})}$
for $R\in\mathcal{R}$;

  * •

for $d\in\mathcal{D}\cup\mathcal{C}$, $d_{S}=\check{d}$.

######  Theorem 5.15.

Let $\mathrm{L}$ be a relational $\omega$-signature, $\mathcal{C}$ be a set of
constants disjoint from $\mathrm{L}$ of size at most $\kappa$ and
$S\subset[\mathrm{L}(\mathcal{C})_{\kappa\omega}]^{<\omega}$ be a consistency
property. Then $\mathcal{A}_{S}$ is a
$\operatorname{RO}(\mathbb{P}_{S})$-valued model with the mixing property, and
for every $s\in S$

$\left\llbracket\bigwedge
s\right\rrbracket^{\mathcal{A}_{S}}_{\operatorname{RO}(\mathbb{P}_{S})}=\left\llbracket\mathcal{A}_{\dot{G}}\models\bigwedge
s\right\rrbracket^{V^{\operatorname{RO}(\mathbb{P}_{S})}}_{\operatorname{RO}(\mathbb{P}_{S})}.$

######  Corollary 5.16.

Let $\mathrm{L}$ be a relational $\omega$-signature, $\mathcal{C}$ be a set of
constants disjoint from $\mathrm{L}$ of size at most $\kappa$ and
$S\subset[\mathrm{L}(\mathcal{C})_{\kappa\omega}]^{<\omega}$ be a consistency
property. Then for any $s\in S$ there is a $\mathsf{B}$-Boolean valued model
$\mathcal{M}$ with the mixing property in which

$\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}=1_{\mathsf{B}}.$

We first prove the Corollary assuming the Theorem.

###### Proof.

Given $s\in S$, we let
$\mathsf{B}=\operatorname{RO}(\mathbb{P}_{S})\restriction\text{Reg}\left(N_{s}\right)$.
Since

$s\Vdash_{\mathbb{P}_{S}}\mathcal{A}_{\dot{G}}\models\bigwedge s,$

we get that $\text{Reg}\left(N_{s}\right)\leq\left\llbracket\bigwedge
s\right\rrbracket^{\mathcal{A}_{S}}_{\operatorname{RO}(\mathbb{P}_{S})}$. In
particular if we consider $\mathcal{A}_{S}$ as a $\mathsf{B}$-valued model by
evaluating all atomic formulae $R(\vec{\sigma})$ by $\left\llbracket
R(\vec{\sigma}\right\rrbracket^{\mathcal{A}_{S}}_{\operatorname{RO}(\mathbb{P}_{S})}\wedge\text{Reg}\left(N_{s}\right)$,
we get that $\left\llbracket\bigwedge
s\right\rrbracket^{\mathcal{A}_{S}}_{\mathsf{B}}=1_{\mathsf{B}}$.

Note that $\mathsf{B}$ is not the one point Boolean algebra, since
$\text{Reg}\left(N_{s}\right)\neq\emptyset=0_{\operatorname{RO}(\mathbb{P}_{S})}$
for all $s\in S$.

It is also immediate to check that $\mathcal{A}_{S}$ retains the mixing
property also when seen as a $\mathsf{B}$-valued model.

∎

We now prove Thm. 5.15. We need beforehand to extend the forcing relation to
formulae of infinitary logic.

######  Remark 5.17.

Given a complete Boolean algebra $\mathsf{B}$, an $\in$-formula
$\phi(v_{1},\ldots,v_{n})$ for $\mathrm{L}_{\infty\omega}$ (for
$\mathrm{L}=\left\\{\in\right\\}$), and any family
$\tau_{1},\ldots,\tau_{n}\in V^{\mathsf{B}}$,
$\left\llbracket\phi(\tau_{1},\ldots,\tau_{n})\right\rrbracket^{V^{\mathsf{B}}}_{\mathsf{B}}$
denotes the $\mathsf{B}$-value of $\phi(\tau_{1},\ldots,\tau_{n})$ in the
Boolean valued model $V^{\mathsf{B}}$.

The definition of
$\left\llbracket\phi(\tau_{1},\ldots,\tau_{n})\right\rrbracket^{V^{\mathsf{B}}}_{\mathsf{B}}$
is by induction on the complexity of $\phi$. It is the standard one for the
atomic formulae
$\left\llbracket\tau\in\sigma\right\rrbracket^{V^{\mathsf{B}}}_{\mathsf{B}}$
and
$\left\llbracket\tau=\sigma\right\rrbracket^{V^{\mathsf{B}}}_{\mathsf{B}}$. We
extend it to all $\mathrm{L}_{\infty\omega}$ according to Def. 2.5.

###### Proof.

We first establish that $\mathcal{A}_{S}$ has the mixing property. Let
$\left\\{\sigma_{a}:a\in A\right\\}$ be a family of elements of $A_{S}$
indexed by an antichain $A$ of $\operatorname{RO}(\mathbb{P}_{S})$. Find (by
the mixing property of $V^{\operatorname{RO}(\mathbb{P}_{S})}$) $\sigma\in
V^{\operatorname{RO}(\mathbb{P}_{S})}$ such that
$\left\llbracket\sigma=\sigma_{a}\right\rrbracket^{V^{\operatorname{RO}(\mathbb{P}_{S})}}_{\operatorname{RO}(\mathbb{P}_{S})}\geq
a$ for all $a\in A$. By choice of $A_{s}$ we can suppose that $\sigma\in
A_{s}$. By definition of $\mathcal{A}_{S}$

$\left\llbracket\sigma=\sigma_{a}\right\rrbracket^{\mathcal{A}_{S}}_{\operatorname{RO}(\mathbb{P}_{S})}=\left\llbracket\sigma=\sigma_{a}\right\rrbracket^{V^{\operatorname{RO}(\mathbb{P}_{S})}}_{\operatorname{RO}(\mathbb{P}_{S})}\geq
a$

for all $a\in A$. Hence $\sigma$ is a mixing element for the family
$\left\\{\sigma_{a}:a\in A\right\\}$.

Now we prove the second part of the Theorem. One needs to check that for any
$\mathrm{L}_{\kappa\omega}$-formula $\phi(\vec{v})$ and
$\sigma_{1},\ldots,\sigma_{n}\in A_{S}$,

$\left\llbracket\phi(\vec{\sigma})\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{\mathcal{A}_{S}}=\left\llbracket\mathcal{A}_{\dot{G}}\vDash\phi(\vec{\sigma})\right\rrbracket^{V^{\operatorname{RO}(\mathbb{P}_{S})}}_{\operatorname{RO}(\mathbb{P}_{S})}.$

It is clear that this allows one to prove

$\left\llbracket\bigwedge
s\right\rrbracket^{\mathcal{A}_{S}}_{\operatorname{RO}(\mathbb{P}_{S})}=\left\llbracket\mathcal{A}_{\dot{G}}\vDash\bigwedge
s\right\rrbracket^{V^{\operatorname{RO}(\mathbb{P}_{S})}}_{\operatorname{RO}(\mathbb{P}_{S})},$

letting $\phi=\bigwedge s$.

We can prove the equality by induction on the complexity of formulae.

  * •

For atomic sentences this follows by definition.

  * •

For $\neg$,

$\left\llbracket\neg\phi\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{\mathcal{A}_{S}}=\neg\left\llbracket\phi\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{\mathcal{A}_{S}}=\neg\left\llbracket\mathcal{A}_{\dot{G}}\vDash\phi\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{V^{\operatorname{RO}(\mathbb{P}_{S})}}=\left\llbracket\mathcal{A}_{\dot{G}}\not\vDash\phi\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{V^{\operatorname{RO}(\mathbb{P}_{S})}}=\left\llbracket\mathcal{A}_{\dot{G}}\vDash\neg\phi\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{V^{\operatorname{RO}(\mathbb{P}_{S})}}.$

  * •

For $\bigwedge$,

$\left\llbracket\bigwedge\Phi\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{\mathcal{A}_{S}}=\bigwedge_{\phi\in\Phi}\left\llbracket\phi\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{\mathcal{A}_{S}}=\bigwedge_{\phi\in\Phi}\left\llbracket\mathcal{A}_{\dot{G}}\vDash\phi\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{V^{\operatorname{RO}(\mathbb{P}_{S})}}=\left\llbracket\mathcal{A}_{\dot{G}}\vDash\bigwedge\Phi\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{V^{\operatorname{RO}(\mathbb{P}_{S})}}.$

  * •

For $\exists$,

$\displaystyle\left\llbracket\exists
v\phi(v,\vec{\sigma})\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{\mathcal{A}_{S}}=\bigvee_{\tau\in
A_{S}}\left\llbracket\phi(\tau,\vec{\sigma})\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{\mathcal{A}_{S}}=\bigvee_{\tau\in
A_{S}}\left\llbracket\mathcal{A}_{\dot{G}}\vDash\phi(\tau,\vec{\sigma})\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{V^{\operatorname{RO}(\mathbb{P}_{S})}}\leq$
$\displaystyle\bigvee_{\tau\in
V^{\operatorname{RO}(\mathbb{P}_{S})}}\left\llbracket\mathcal{A}_{\dot{G}}\vDash\phi(\tau,\vec{\sigma})\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{V^{\operatorname{RO}(\mathbb{P}_{S})}}=\left\llbracket\mathcal{A}_{\dot{G}}\vDash\exists
v\phi(v,\vec{\sigma})\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{V^{\operatorname{RO}(\mathbb{P}_{S})}}=$
$\displaystyle\left\llbracket\mathcal{A}_{\dot{G}}\vDash\phi(\tau_{0},\vec{\sigma})\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{V^{\operatorname{RO}(\mathbb{P}_{S})}}=\left\llbracket\phi(\tau_{0},\vec{\sigma})\right\rrbracket^{\mathcal{A}_{S}}_{\operatorname{RO}(\mathbb{P}_{S})}\leq\left\llbracket\exists
v\phi(v,\vec{\sigma})\right\rrbracket_{\operatorname{RO}(\mathbb{P}_{S})}^{\mathcal{A}_{S}},$

where $\tau_{0}\in\mathcal{A}_{S}$ is obtained by fullness of
$V^{\operatorname{RO}(\mathbb{P}_{S})}$ and can be supposed in $H_{\mu}$ by
Proposition 5.13; while the equality in the last line holds by inductive
assumptions.

∎

Let us briefly remark why genericity is needed for dealing with formulae of
type $\neg$, $\bigvee$ and $\exists$ in proving the model existence Theorem.

The case of negated formulae is dealt with by taking advantage of Def. 4.1. If
the negated formula is atomic, its truth value follows by the definition of
$\mathcal{A}_{F}$. For negated formulae $\neg\phi$ with $\phi$ non-atomic, by
moving a negation inside repeatedly, we find a logically equivalent formula
$\psi$ where negations appear only at the atomic level of the structural tree
of $\psi$; at this level there is control. In particular the operation
$\phi\mapsto\phi\neg$ allows to prove Thm. 5.15 by an induction in which one
only deals with the logical symbols $\bigwedge,\forall,\bigvee$ and $\exists$.

Genericity comes to play when dealing with formulae whose principal connective
is $\bigvee$ or $\exists$. For both connectives the role of genericity in the
proof of the corresponding inductive step is similar, so we only analyze the
first one. The key point is that the structure $\mathcal{A}_{F}$ associated to
a maximal filter $F$ on $\mathbb{P}_{S}$ is decided by which atomic formulae
belong to $\Sigma_{F}$: any maximal consistent set of atomic formulae for
$\mathrm{L}$ defines an $\mathrm{L}$-structure $\mathcal{A}_{F}$ by Fact 5.5.
Now if $F$ is maximal but not $V$-generic, it may miss some $D_{\bigvee\Phi}$
for some $\Phi\in\Sigma_{F}$. In which case
$[\Sigma_{F}\cup\\{\phi\\}]^{<\omega}$ is not a prefilter on $\mathbb{P}_{S}$
for any $\phi\in\Phi$, by maximality of $F$. Supposing this occurs for some
$\Phi$ which is a disjunction of atomic or negated atomic formulae, we get
that $\left\\{\phi\neg:\phi\in\Phi\right\\}\subseteq F$, again by maximality
of $F$. Hence $\mathcal{A}_{F}\not\vDash\bigvee\Phi$ even if $\bigvee\Phi\in
F$.

######  Remark 5.18.

When working with a consistency property $S$ for
$\mathrm{L}(\mathcal{C})_{\kappa\omega}$, there is a canonical way of
extending it to a $(\kappa,\omega)$-maximal one. Consider the Boolean valued
model $\mathcal{A}_{S}$ of Def. 5.14, let also
$\mathsf{B}=\operatorname{RO}(\mathbb{P}_{S})$. Then

$S\subset
M_{S}=\\{t\in[\mathrm{L}(\mathcal{C}\cup\mathcal{A}_{S})_{\kappa\omega}]^{<\omega}:\left\llbracket
t\right\rrbracket^{\mathcal{A}_{S}}_{\mathsf{B}}>0_{\mathsf{B}}\\}$

and $M_{S}$ is a $(\kappa,\omega)$-maximal consistency property for
$\mathrm{L}(\mathcal{C}\cup\mathcal{A}_{S})_{\kappa\omega}$.

######  Remark 5.19.

Note that in Def. 4.2 the size of $\mathcal{C}$ can vary. While Thm. 5.7 holds
for any size of $\mathcal{C}$, some sizes automatically collapse cardinals.
Consider for example $\mathrm{L}=\\{d_{\alpha}:\alpha<\omega_{1}\\}$ and
$\mathcal{C}=\\{c_{n}:n<\omega\\}$ countable. Let $S$ denote the set whose
elements are the $s\in[\mathrm{L}(\mathcal{C})_{\omega_{2}\omega}]^{<\omega}$
such that for some injective interpretation

$c_{i_{1}}\mapsto\alpha_{i_{1}},\ldots,c_{i_{n}}\mapsto\alpha_{i_{n}},\
\alpha_{i_{j}}<\omega_{1},$

of the constants from $\mathcal{C}$ appearing in $s$,

$(\omega_{1},=,c_{i_{k}}\mapsto\alpha_{i_{k}},d_{\alpha}\mapsto\alpha)\vDash
s.$

$S$ is readily checked to be a consistency property. Consider
$\mathcal{A}_{G}\in V[G]$ for $G$ $V$-generic for $\mathbb{P}_{S}$. It is a
model of $\bigwedge_{\alpha\neq\beta\in\omega_{1}^{V}}d_{\alpha}\neq
d_{\beta}$, furthermore the interpretation maps

$\displaystyle f:\omega_{1}^{V}$
$\displaystyle\rightarrow\left\\{[d]_{G}:d\in\mathcal{D}\right\\}$
$\displaystyle\alpha$ $\displaystyle\mapsto d_{\alpha}^{\mathcal{A}_{G}}$
$\displaystyle g:\omega$
$\displaystyle\rightarrow\left\\{[c_{n}]_{G}:n<\omega\right\\}$ $\displaystyle
n$ $\displaystyle\mapsto[c_{n}]_{G}$

are both injective. This entails that the map $\alpha\mapsto n$ if
$\left\\{d_{\alpha}=c_{n}\right\\}\in G$ is also injective. Therefore
$\omega_{1}^{V}$ is collapsed.

##  6\. Mansfield’s Model Existence Theorem

We now prove Mansfield’s Model Existence Theorem.

######  Theorem 6.1.

Let $\mathrm{L}$ be an $\omega$-signature and $S\subset
P(\mathrm{L}(\mathcal{C})_{\kappa\lambda})$ a consistency property. Then for
any $s\in S$ there exists a Boolean valued model $\mathcal{M}$ in which all
sentences from $s$ are valid.

###### Proof.

Fix $s_{0}\in S$. Consider $\mathbb{P}_{S}$ the forcing notion associated to
$S$, $\mathsf{B}=\operatorname{RO}(\mathbb{P}_{S})$ the corresponding Boolean
completion, $\mathbb{P}_{S}\upharpoonleft s_{0}=\\{t\in S:t\leq s_{0}\\}$ the
restriction to conditions extending $s_{0}$ and $\mathsf{B}\upharpoonleft
s_{0}=\\{t\in\mathsf{B}:t\leq\text{Reg}\left(N_{s_{0}}\right)\\}=\operatorname{RO}(\mathbb{P}_{S}\upharpoonleft
s_{0})$. The Boolean valued model $\mathcal{M}$ is constructed with truth
values in $\mathsf{B}\upharpoonleft s_{0}$ and base set the set of constants
$\mathcal{C}\cup\mathcal{D}$. The interpretations of constants are given by
themselves. Mimicking Mansfield’s proof we keep his notation whenever
possible. For any sentence $\phi$ define

$L(\phi)=\bigvee\\{\text{Reg}\left(N_{t}\right):\phi\in t\\},$

and for $\phi$ atomic set

$\left\llbracket\phi\right\rrbracket=L(\phi).$

The main technical result in [10] is Lemma 3. Its equivalent (according to our
notion of consistency property) goes as follows:

######  Claim 1.

If for any $t\leq s$, $t\cup\\{\phi\\}\in S$, then $N_{s}\subseteq L(\phi)$.
In particular, $\text{Reg}\left(N_{s}\right)\leq L(\phi)$.

###### Proof.

Suppose $N_{s}\not\subseteq L(\phi)$. Since the family
$\\{N_{t}:t\in\mathbb{P}_{S}\upharpoonleft s_{0}\\}$ is a basis of
$\mathsf{B}$, basic topological facts bring that there exists some
$t\in\mathbb{P}_{S}\upharpoonleft s_{0}$ such that

$N_{t}\subseteq N_{s}\cap\neg
L(\phi)=\text{Reg}\left(\bigcup\left\\{N_{t}:N_{t}\cap
L(\phi)=\emptyset\right\\}\right).$

  * •

Since $N_{t}\subseteq N_{s}$ we have $t\leq s$ and the hypothesis ensures
$t\cup\\{\phi\\}\in S$. Then, by definition of $L$,
$N_{t\cup\\{\phi\\}}\subseteq L(\phi)$.

  * •

Since $N_{t}\subseteq\neg L(\phi)$ and $N_{t\cup\\{\phi\\}}\subseteq N_{t}$,
$N_{t\cup\\{\phi\\}}\subseteq\neg L(\phi)$.

The two statements are incompatible. Hence $N_{s}\leq L(\phi)$. ∎

######  Claim 2.

For any $\phi$, $L(\phi)\leq\left\llbracket\phi\right\rrbracket$.

###### Proof.

We proceed by induction on the complexity of formulae. The thesis holds for
atomic formulae by definition. The other cases are dealt with as follows:

$\neg$:

Since $S$ is closed under moving a negation inside, we only need to care about
negations acting on atomic formulae. Suppose $\phi$ is atomic. We have

$\left\llbracket\neg\phi\right\rrbracket_{\mathsf{B}_{S}\upharpoonleft
s_{0}}=\neg\left\llbracket\phi\right\rrbracket_{\mathsf{B}_{S}\upharpoonleft
s_{0}}=\neg L(\phi)=\bigwedge\\{\neg\text{Reg}\left(N_{t}\right):\phi\in
t\\}.$

We need to prove

$L(\neg\phi)=\bigvee\\{\text{Reg}\left(N_{p}\right):\neg\phi\in
p\\}\leq\bigwedge\\{\neg\text{Reg}\left(N_{t}\right):\phi\in t\\}.$

Fix $t$ containing $\phi$. For any $p\ni\neg\phi$, $p$ and $t$ are
incompatible by Clause (Con)(Con). Remark 2.3 ensures
$\text{Reg}\left(N_{p}\right)\leq\neg\text{Reg}\left(N_{t}\right)$. Then

$\bigvee_{p\ni\neg\phi}\text{Reg}\left(N_{p}\right)\leq\neg\text{Reg}\left(N_{t}\right).$

Since this is true for any $t\ni\phi$,

$\bigvee_{p\ni\neg\phi}\text{Reg}\left(N_{p}\right)\leq\bigwedge_{t\ni\phi}\neg\text{Reg}\left(N_{t}\right).$

$\bigwedge$:

Suppose by induction that the result holds for any $\phi\in\Phi$. Let
$\bigwedge\Phi\in s$. Then for any $t$ extending $s$ and any $\phi\in\Phi$,
$t\cup\\{\phi\\}\in S$. By Claim 1, $\text{Reg}\left(N_{s}\right)\leq L(\phi)$
for any $\phi\in\Phi$. By the induction hypothesis
$\text{Reg}\left(N_{s}\right)\leq
L(\phi)\leq\left\llbracket\phi\right\rrbracket$. As this holds for any
$\phi\in\Phi$,
$\text{Reg}\left(N_{s}\right)\leq\bigwedge_{\phi\in\Phi}\left\llbracket\phi\right\rrbracket=\left\llbracket\bigwedge\Phi\right\rrbracket$.
This holds for any $s$ such that $\bigwedge\Phi\in s$, hence

$L(\bigwedge\Phi)=\bigvee\\{\text{Reg}\left(N_{s}\right):\bigwedge\Phi\in
s\\}\leq\left\llbracket\bigwedge\Phi\right\rrbracket.$

$\bigvee$:

Suppose the result true for any $\phi\in\Phi$. If
$L(\bigvee\Phi)\nleq\left\llbracket\bigvee\Phi\right\rrbracket$,
$L(\bigvee\Phi)\wedge\neg\left\llbracket\bigvee\Phi\right\rrbracket\neq\emptyset$.
Therefore there exists some $t\leq s_{0}$ such that $N_{t}\subseteq
L(\bigvee\Phi)$ and
$N_{t}\subseteq\neg\left\llbracket\bigvee\Phi\right\rrbracket$.

  * •:

By the first inclusion, since $N_{t}$ is open and
$\bigcup\\{\text{Reg}\left(N_{p}\right):\bigvee\Phi\in p\\}$ is dense in
$\bigvee\\{\text{Reg}\left(N_{p}\right):\bigvee\Phi\in p\\}$, there exists
some $p^{\prime}$ containing $\bigvee\Phi$ such that
$N_{t}\cap\text{Reg}\left(N_{p^{\prime}}\right)$ is non-empty. Hence we can
find $p\leq t,p^{\prime}$. Since $\bigvee\Phi\in p^{\prime}$, $\bigvee\Phi\in
p$.

  * •:

By the second inclusion (and $p\leq t$),

$N_{p}\subseteq
N_{t}\subseteq\neg\bigvee_{\phi\in\Phi}\left\llbracket\phi\right\rrbracket=\bigwedge_{\phi\in\Phi}\neg\left\llbracket\phi\right\rrbracket.$

(Ind.4) ensures that for some $\phi_{0}\in\Phi$, $q=p\cup\\{\phi_{0}\\}\in S$.
By induction hypothesis,
$L(\phi_{0})\leq\left\llbracket\phi_{0}\right\rrbracket$, hence
$\neg\left\llbracket\phi_{0}\right\rrbracket\leq\neg L(\phi_{0})$. Therefore

$\hskip 36.98866ptN_{q}\subseteq
N_{p}\subseteq\bigwedge_{\phi\in\Phi}\neg\left\llbracket\phi\right\rrbracket\leq\neg\left\llbracket\phi_{0}\right\rrbracket\leq\neg
L(\phi_{0})=\bigwedge\\{\neg\text{Reg}\left(N_{t}\right):\phi_{0}\in
t\\}\subseteq\neg\text{Reg}\left(N_{q}\right),$

a contradiction.

$\forall$:

Let $s\in S$ contain $\forall\vec{v}\phi(\vec{v})$. Then for any $t$ extending
$s$ and any $\vec{c}\in(C\cup D)^{\vec{v}}$, $t\cup\\{\phi(\vec{c})\\}\in S$.
By Claim 1 and the induction hypothesis,

$\text{Reg}\left(N_{s}\right)\leq
L(\phi(\vec{c}))\leq\left\llbracket(\phi(\vec{c}))\right\rrbracket$

for any $\vec{c}\in(C\cup D)^{\vec{v}}$. Then
$\text{Reg}\left(N_{s}\right)\leq\bigwedge_{\vec{c}\in(C\cup
D)^{\vec{v}}}\left\llbracket\phi(\vec{c})\right\rrbracket=\left\llbracket\forall\vec{v}\phi(\vec{v})\right\rrbracket$.
All this was done for any $s$ such that $\forall\vec{v}\phi(\vec{v})\in s$.
Then we may take the sup over such sets to obtain

$L(\forall\vec{v}\phi(\vec{v}))=\bigvee\\{\text{Reg}\left(N_{s}\right):\forall\vec{v}\phi(\vec{v})\in
s\\}\leq\left\llbracket\forall\vec{v}\phi(\vec{v})\right\rrbracket.$

$\exists$:

Suppose the result true for any $\phi(\vec{c})$. If
$L(\exists\vec{v}\phi(\vec{v}))\nleq\left\llbracket\exists\vec{v}\phi(\vec{v})\right\rrbracket$,
$L(\exists\vec{v}\phi(\vec{v}))\wedge\neg\left\llbracket\exists\vec{v}\phi(\vec{v})\right\rrbracket\neq\emptyset$.
Then there exists some $t\leq s_{0}$ such that $N_{t}\subseteq
L(\exists\vec{v}\phi(\vec{v}))$ and
$N_{t}\subseteq\neg\left\llbracket\exists\vec{v}\phi(\vec{v})\right\rrbracket$.

  * •:

By the first inclusion, since $N_{t}$ is open and
$\bigcup\\{\text{Reg}\left(N_{p}\right):\exists\vec{v}\phi(\overline{v})\in
p\\}$ is dense in
$\bigvee\\{\text{Reg}\left(N_{p}\right):\exists\vec{v}\phi(\vec{v})\in p\\}$,
there exists some $p^{\prime}$ containing $\exists\vec{v}\phi(\vec{v})$ such
that $N_{t}\cap\text{Reg}\left(N_{p^{\prime}}\right)$ is non-empty. Hence we
can find $p\leq t,p^{\prime}$ with $\exists\vec{v}\phi(\vec{v})\in p$.

  * •:

By the second inclusion,

$N_{p}\subseteq
N_{t}\subseteq\neg\bigvee_{\vec{c}\in(\mathcal{C}\cup\mathcal{D})^{\vec{v}}}\left\llbracket\phi(\vec{c})\right\rrbracket=\bigwedge_{\vec{c}\in(\mathcal{C}\cup\mathcal{D})^{\vec{v}}}\neg\left\llbracket\phi(\vec{c})\right\rrbracket.$

(Ind.5) ensures that for some
$\vec{c}_{0}\in(\mathcal{C}\cup\mathcal{D})^{\vec{v}}$,
$q=p\cup\\{\phi(\overline{c}_{0})\\}\in S$. By the induction hypothesis,
$L(\phi(\vec{c}_{0}))\leq\left\llbracket\phi(\vec{c}_{0})\right\rrbracket_{\mathsf{B}}$,
hence $\neg\left\llbracket\phi(\vec{c}_{0})\right\rrbracket\leq\neg
L(\phi(\vec{c}_{0}))$. Therefore

$N_{q}\subseteq
N_{p}\subseteq\bigwedge_{\overline{c}\subseteq\mathcal{C}}\neg\left\llbracket\phi(\overline{c})\right\rrbracket_{\mathsf{B}}\subseteq\neg\left\llbracket\phi(\overline{c}_{0})\right\rrbracket_{\mathsf{B}}\subseteq\neg
L(\phi(\overline{c}_{0}))=\bigwedge\\{\neg\text{Reg}\left(N_{t}\right):\phi(\overline{c}_{0})\in
t\\}\subseteq\neg\text{Reg}\left(N_{q}\right),$

a contradiction.

∎

Now we can check that $\mathcal{M}$ is a Boolean valued model.

  * •

Since for any $t$ in $S$ and any $c\in\mathcal{C}\cup\mathcal{D}$,
$t\cup\\{c=c\\}\in S$, $L(c=c)=1_{\mathsf{B}_{S}\upharpoonleft s_{0}}$.

  * •

Let $c=d\in s$. Then for any $t$ extending $s$, $t\cup\\{d=c\\}\in S$, hence
$\text{Reg}\left(N_{s}\right)\leq L(d=c)$. Since the previous holds for any
$s$ containing $c=d$,

$\bigvee\\{\text{Reg}\left(N_{s}\right):c=d\in s\\}\leq L(d=c).$

Since $c=d$ and $d=c$ are atomic,

$\left\llbracket
c=d\right\rrbracket=L(c=d)=\bigvee\\{\text{Reg}\left(N_{s}\right):c=d\in
s\\}\leq L(d=c)=\left\llbracket d=c\right\rrbracket.$

  * •

Let $c_{1}=d_{1},\ldots,c_{n}=d_{n},\phi(c_{1},\ldots,c_{n})\in s$ with $\phi$
atomic. Then for any $t$ extending $s$,
$t\cup\\{\phi(d_{1},\ldots,d_{n})\\}\in S$, hence
$\text{Reg}\left(N_{s}\right)\leq L(\phi(d_{1},\ldots,d_{n}))$. Since the
previous holds for any $s$ containing
$c_{1}=d_{1},\ldots,c_{n}=d_{n},\phi(c_{1},\ldots,c_{n})$,

$\bigvee\\{\text{Reg}\left(N_{s}\right):c_{1}=d_{1},\ldots,c_{n}=d_{n},\phi(c_{1},\ldots,c_{n})\in
s\\}\leq L(\phi(d_{1},\ldots,d_{n})).$

Since $\phi$ and $c_{i}=d_{i}$ are atomic,

$\displaystyle\left\llbracket c_{1}=d_{1}\right\rrbracket\wedge\ldots\wedge$
$\displaystyle\left\llbracket
c_{n}=d_{n}\right\rrbracket\wedge\left\llbracket\phi(c_{1},\ldots,c_{n})\right\rrbracket=$
$\displaystyle L(c_{1}=d_{1})\wedge\ldots\wedge$ $\displaystyle\
L(c_{n}=d_{n})\wedge L(\phi(c_{1},\ldots,c_{n}))=$
$\displaystyle\bigvee\\{\text{Reg}\left(N_{s}\right):c_{1}=d_{1},\ldots,c_{n}=d_{n},\
$ $\displaystyle\phi(c_{1},\ldots,c_{n})\in s\\}\leq
L(\phi(d_{1},\ldots,d_{n}))=$
$\displaystyle\left\llbracket\phi(d_{1},\ldots,d_{n})\right\rrbracket.$

To prove the equality between lines two and three it is enough to check
$L(\phi)\wedge L(\psi)=\bigvee\\{\text{Reg}\left(N_{s}\right):\phi,\psi\in
s\\}$. By definition of $L$,

$\displaystyle L(\phi)\wedge L(\psi)$ $\displaystyle=$
$\displaystyle\bigvee\\{\text{Reg}\left(N_{s}\right):\phi\in
s\\}\wedge\bigvee\\{\text{Reg}\left(N_{t}\right):\psi\in t\\}$
$\displaystyle=\bigvee\\{\bigvee\\{\text{Reg}\left(N_{s}\right):\phi\in
s\\}\wedge\text{Reg}\left(N_{t}\right):\psi\in t\\}=$
$\displaystyle\bigvee\\{\bigvee\\{\text{Reg}\left(N_{s}\right)\wedge\text{Reg}\left(N_{t}\right):\phi\in
s\\}:\psi\in t\\}$
$\displaystyle=\bigvee\\{\text{Reg}\left(N_{s}\right)\wedge\text{Reg}\left(N_{t}\right):\phi\in
s\wedge\psi\in t\\}=$ $\displaystyle\bigvee\\{\text{Reg}\left(N_{q}\right):\ $
$\displaystyle\phi,\psi\in q\\}.$

It remains to conclude that $s_{0}$ is valid in $\mathcal{M}$. Now for any
$\phi\in s_{0}$ and $t\leq s_{0}$, $\phi\in t$, hence Claim 2 ensures
$1_{\mathsf{B}_{S}\upharpoonleft s_{0}}=\text{Reg}\left(N_{s_{0}}\right)\leq
L(\phi)\leq\left\llbracket\phi\right\rrbracket_{\mathsf{B}}$ for any such
$\phi$. ∎

######  Remark 6.2.

We note that a key assumption for Mansfield’s result is that $\mathrm{L}$ is a
first order signature. This is crucially used in the proof that B holds for
the relation symbols of $\mathrm{L}$ in the structure $\mathcal{M}$: since
these relation symbols are finitary, in the proof above we just had to
distribute finitely many infinite disjunctions; this distributive law holds
for any complete Boolean algebra. If we dealt with a relational
$\omega_{1}$-signature we might have had to distribute countably many
infinitary disjunctions to establish B; this is possible only under very
special circumstances on $\mathbb{P}_{S}$.

We do not know whether this result can be established for arbitrary
$\lambda$-signatures. We conjecture this is not the case.

######  Remark 6.3.

The model produced by Mansfield’s theorem does not verify the mixing property
in general. Consider again the setting in remark 5.19.

Fix $\alpha<\omega_{1}$. For each $n<\omega$ define

$\phi_{n}:\hskip 8.5359ptc_{n}=d_{\alpha}\ \wedge\bigwedge_{m<n}c_{m}\neq
d_{\alpha}.$

Then $\\{\text{Reg}\left(N_{\\{\phi_{n}\\}}\right):n<\omega\\}$ is an
antichain. Assign $c_{n-1}$ to each
$\text{Reg}\left(N_{\\{\phi_{n}\\}}\right)$. We show that for no element $m$
in the structure provided by Mansfield theorem with respect to the collapsing
consistency property we have

$\text{Reg}\left(N_{\\{\phi_{n}\\}}\right)\leq\left\llbracket
m=c_{n-1}\right\rrbracket=\bigvee\\{\text{Reg}\left(N_{s}\right):m=c_{n-1}\in
s\\}.$

There are three possibilities for $m$.

  * •

$m=c_{n_{0}}$: Note that because the interpretations generating the
consistency property are injective the set

$\\{t\in S:\bigwedge_{n\neq m}c_{n}\neq c_{m}\in t\\}$

is dense. Then $\left\llbracket c_{n_{0}}=c_{n_{0}-1}\right\rrbracket=0$ and
it cannot be that

$\text{Reg}\left(N_{\\{\phi_{n_{0}}\\}}\right)\leq\left\llbracket
c_{n_{0}}=c_{n_{0}-1}\right\rrbracket.$

  * •

$m=d_{\alpha}$: Take any $n<\omega$. The set $\\{t\in S:c_{n-1}\neq
d_{\alpha}\in t\\}$ is dense below $\\{\phi_{n}\\}$. Then we cannot have
$\text{Reg}\left(N_{\\{\phi_{n}\\}}\right)\leq\bigvee\\{\text{Reg}\left(N_{t}\right):d_{\alpha}=c_{n-1}\in
t\\}$.

  * •

$m=d_{\beta}$, $\beta\neq\alpha$: Take any $n<\omega$. Because the constant
$d_{\beta}$ does not appear in $\\{\phi_{n}\\}$ and the sentence $\phi_{n}$
only forces $c_{n-1}$ not be $d_{\alpha}$, we can suppose that the
interpretation that generates $\\{\phi_{n}\\}$ is such that $c_{n-1}$ is
interpreted differently from $d_{\beta}$, proving $\\{\phi_{n},c_{n-1}\neq
d_{\beta}\\}\in S$. Then we cannot have
$\text{Reg}\left(N_{\\{\phi_{n}\\}}\right)\leq\bigvee\\{\text{Reg}\left(N_{t}\right):d_{\beta}=c_{n-1}\\}$
since $\text{Reg}\left(N_{\\{\phi_{n},c_{n-1}\neq
d_{\beta}\\}}\right)\leq\text{Reg}\left(N_{\\{\phi_{n}\\}}\right)$.

##  7\. Proofs of model theoretic results

In this section we prove that $\mathrm{L}_{\infty\omega}$ with Boolean valued
semantics has a completeness theorem, the Craig interpolation property and
also an omitting types theorem. The last two results generalize to
$\mathrm{L}_{\infty\omega}$ results obtained in [6] for
$\mathrm{L}_{\omega_{1}\omega}$ by replacing Tarski semantics with Boolean
valued semantics. We also provide the missing details for the general
$\mathrm{L}_{\infty\infty}$ results.

###  7.1. Proof of Thm. 3.1

###### Proof.

3 implies 2 and 2 implies 1 are either standard or trivial.

Assume 3 fails, we show that 1 fails as well. Assume $T\not\vdash S$ with
$T,S$ sets of $\mathrm{L}_{\infty\omega}$-formulae. Let $\mathcal{C}$ be an
infinite set of fresh constants and let $R$ be the family of finite sets
$r\subseteq\mathrm{L}(\mathcal{C})$ such that

  * •

$r\cup T\not\vdash S$,

  * •

any $\phi\in r$ contains only finitely many constants from $\mathcal{C}$.

Provided $R$ is a consistency property, this gives that $\mathcal{A}_{R}$
witnesses that $T\not\models_{\mathrm{Sh}}S$ as:

  * •

$\left\llbracket\psi\right\rrbracket^{\mathcal{A}_{R}}=1_{\operatorname{RO}(\mathbb{P}_{R})}$
for all $\psi\in T$, since for any $\psi\in T$

$E_{\psi}=\left\\{r\in R:\,\psi\in r\right\\}$

is dense in $\mathbb{P}_{R}$;

  * •

$\left\llbracket\phi\right\rrbracket^{\mathcal{A}_{R}}=0_{\operatorname{RO}(\mathbb{P}_{R})}$
for all $\phi\in S$, since for any such $\phi$

$F_{\phi}=\left\\{r\in R:\,\neg\phi\in r\right\\}$

is dense in $\mathbb{P}_{R}$: note that $r\cup\left\\{\neg\phi\right\\}\cup
T\vdash S$ if and only if $r\cup T\vdash S\cup\left\\{\phi\right\\}$, which
-if $\phi\in S$\- amounts to say that $r\not\in R$.

Now we show that $R$ is a consistency property:

  * (Con)

Trivial by definition of $R$, since the calculus is sound.

  * (Ind.1)

Trivial since for any $\neg\phi$ in $r$, $\bigwedge
r\vdash\bigwedge(r\cup\left\\{\phi\neg\right\\})$ and conversely.

  * (Ind.2)

Trivial since $r\vdash\bigwedge(r\cup\left\\{\phi\right\\})$ and conversely if
$\bigwedge\Phi\in r$ and $\phi\in\Phi$.

  * (Ind.3)

Trivial since $r\vdash\bigwedge(r\cup\left\\{\phi(c)\right\\})$ and conversely
if $\forall v\,\phi(v)\in r$.

  * (Ind.4)

Let $\bigvee\Sigma\in r\in R$. Since $r\in R$, $r\cup T\not\vdash S$. By
contradiction suppose that for all $\sigma\in\Sigma$,
$r\cup\left\\{\sigma\right\\}\cup T\vdash S$. Then, by the left $\bigvee$-rule
of the calculus $r\cup\left\\{\bigvee\Sigma\right\\}\cup T\vdash S$. This
contradicts $r\in R$, since $r=r\cup\left\\{\bigvee\Sigma\right\\}$.

  * (Ind.5)

Suppose $\exists v\,\varphi(v)\in r$. Pick $c\in\mathcal{C}$ which does not
appear in any formula in $r$. It exists by definition of $R$. Suppose
$r\cup\left\\{\varphi(c)\right\\}\cup T\vdash S$. Since $c$ does not appear in
any formula of $r\cup S$, $r\cup\left\\{\exists x\,\varphi(x)\right\\}\vdash
S$ (applying the rules of the calculus). This contradicts $r\in R$, since
$r=r\cup\left\\{\exists x\,\varphi(x)\right\\}$.

  * (Str.1,2,3)

All three cases follow from standard applications of the rules of the calculus
for equality.

∎

###  7.2. Proof of Thm. 3.2

###### Proof.

Fix a set $\mathcal{C}$ of fresh constants for $\mathrm{L}$ of size $\kappa$.
Consider $X_{\phi}$ the set of all
$\mathrm{L}(\mathcal{C})_{\kappa\omega}$-sentences $\chi$ such that:

  * •

all non logical symbols from $\mathrm{L}$ appearing in $\chi$ also appear in
$\phi$,

  * •

only a finite number of constants from $\mathcal{C}$ are in $\chi$.

Define $X_{\psi}$ similarly. Consider $S$ the set of finite sets of
$\mathrm{L}(\mathcal{C})_{\kappa\omega}$-sentences $s$ such that:

  * •

$s=s_{1}\cup s_{2}$,

  * •

$s_{1}\subset X_{\phi}$,

  * •

$s_{2}\subset X_{\psi}$,

  * •

if $\theta,\sigma\in X_{\phi}\cap X_{\psi}$ are such that

    * –

no constant symbols of $\mathcal{C}$ appears in either $\theta$ or $\sigma$,

    * –

$\vDash_{\mathrm{BVM}}\bigwedge s_{1}\rightarrow\theta$ and
$\vDash_{\mathrm{BVM}}\bigwedge s_{2}\rightarrow\sigma$,

then $\theta\wedge\sigma$ is Boolean consistent.

We will later show that $S$ is a consistency property. Assuming this fact as
granted, we now show why this provides the interpolant. The Model Existence
Theorem 5.15 grants that any $s\in S$ has a Boolean valued model. By
hypothesis $\vDash_{\mathrm{BVM}}\phi\rightarrow\psi$, thus the set
$\\{\phi,\neg\psi\\}$ is not consistent and it cannot belong to $S$.

Now we search what property the set $\\{\phi,\neg\psi\\}$ misses. They have no
constant from $\mathcal{C}$ since they are $\mathrm{L}$-sentences. The sets
$s_{1}$ and $s_{2}$ are given by $\\{\phi\\}$ and $\\{\neg\psi\\}$. So, the
last property must fail. This means that there exist $\theta,\sigma\in
X_{\phi}\cap X_{\psi}$ with no constant symbols of $\mathcal{C}$ in either of
them and such that $\vDash_{\mathrm{B}}\phi\rightarrow\theta$,
$\vDash_{\mathrm{BVM}}\neg\psi\rightarrow\sigma$ and $\theta\wedge\sigma$ is
not consistent. The last assertion gives

$\vDash_{\mathrm{BVM}}\theta\rightarrow\neg\sigma.$

This together with

$\vDash_{\mathrm{BVM}}\neg\sigma\rightarrow\psi$

implies

$\vDash_{\mathrm{BVM}}\theta\rightarrow\psi.$

Recall that $\theta,\sigma$ have no constant symbol from $\mathcal{C}$, hence
the interpolant is given by the $\mathrm{L}_{\kappa\omega}$-sentence $\theta$.

It remains to check that $S$ is a consistency property.

  * (Con)

The very definition of $S$ then gives that if some $s\in S$ is such that
$\theta,\neg\theta\in S$, then $\theta,\neg\theta\in s_{1}\subseteq X_{\phi}$
or $\theta,\neg\theta\in s_{2}\subseteq X_{\psi}$. Towards a contradiction
w.l.o.g. we can suppose that for some $s=s_{1}\cup s_{2}\in S$ and $\theta\in
X_{\phi}$, $\theta,\neg\theta\in s_{1}$. Consider any sentence
$\chi^{\prime}\in X_{\phi}\cap X_{\psi}$ such that
$\vDash_{\mathrm{BVM}}\bigwedge s_{2}\rightarrow\chi^{\prime}$. Because
$s_{1}$ is contradictory we have $\vDash_{\mathrm{BVM}}\bigwedge
s_{1}\rightarrow\neg\chi^{\prime}$. But $\chi^{\prime}\wedge\neg\chi^{\prime}$
is not Boolean consistent, a contradiction.

  * (Ind.1)

Suppose $\neg\chi\in s_{1}\subseteq s$. Because $s_{1}\cup\\{\chi\neg\\}$ and
$s_{1}$ are equivalent, any sentence $\chi^{\prime}$ such that
$\vDash_{\mathrm{BVM}}\bigwedge
s_{1}\cup\\{\chi\neg\\}\rightarrow\chi^{\prime}$ also verifies
$\vDash_{\mathrm{BVM}}\bigwedge s_{1}\rightarrow\chi^{\prime}$. Then,
$s\cup\\{\chi\neg\\}\in S$.

  * (Ind.2)

Suppose $\chi\in\Phi$ and $\bigwedge\Phi\in s_{1}\subseteq s$. Because
$\bigwedge s_{1}$ and $\bigwedge s_{1}\cup\\{\chi\\}$ are equivalent,
$s\cup\\{\chi\\}\in S$.

  * (Ind.3)

Suppose $\forall v\chi(v)\in s_{1}\subseteq s$ and
$c\in\mathcal{C}\cup\mathcal{D}$. Because $\bigwedge s_{1}$ and $\bigwedge
s_{1}\cup\\{\chi(c)\\}$ are equivalent, $s\cup\\{\chi(c)\\}\in S$.

  * (Ind.4)

Let $\bigvee\Sigma\in s_{1}\subseteq s$. By contradiction we suppose that for
no $\sigma\in\Sigma$, $s\cup\\{\sigma\\}\in S$. This means that for each
$\sigma\in\Sigma$ there exist $\chi_{\sigma}^{1},\chi_{\sigma}^{2}\in
X_{\phi}\cap X_{\psi}$ such that

$\vDash_{\mathrm{BVM}}\bigwedge(s_{1}\cup\\{\sigma\\})\rightarrow\chi_{\sigma}^{1}\emph{
and }\vDash_{\mathrm{BVM}}\bigwedge s_{2}\rightarrow\chi_{\sigma}^{2},$

but $\chi_{\sigma}^{1}\wedge\chi_{\sigma}^{2}$ is inconsistent. Then

$\displaystyle\vDash_{\mathrm{BVM}}\bigwedge(s_{1}\cup\\{\bigvee\Sigma\\})\rightarrow\bigvee\\{\chi_{\sigma}^{1}:\sigma\in\Sigma\\}\emph{
and }$ $\displaystyle\vDash_{\mathrm{BVM}}\bigwedge
s_{2}\rightarrow\bigwedge\\{\chi_{\sigma}^{2}:\sigma\in\Sigma\\}.$

Note that $s_{1}\cup\left\\{\bigvee\Sigma\right\\}=s_{1}$. Because
$\chi_{\sigma}^{1}\wedge\chi_{\sigma}^{2}$ is Boolean inconsistent for each
$\sigma\in\Sigma$, so is

$\displaystyle\bigvee\\{\chi_{\sigma}^{1}:\sigma\in\Sigma\\}\wedge\bigwedge\\{\chi_{\sigma}^{2}:\sigma\in\Sigma\\}\equiv_{\mathrm{BVM}}$
$\displaystyle\bigvee\\{\chi_{\sigma^{\prime}}^{1}\wedge\bigwedge\\{\chi_{\sigma}^{2}:\sigma\in\Sigma\\}:\sigma^{\prime}\in\Sigma\\}\equiv_{\mathrm{BVM}}$
$\displaystyle\bigvee\bigwedge\\{\chi_{\sigma^{\prime}}^{1}\wedge\chi_{\sigma}^{2}:\sigma\in\Sigma\wedge\sigma^{\prime}\in\Sigma\\}\models_{\mathrm{BVM}}$
$\displaystyle\bigvee\bigwedge\\{\chi_{\sigma}^{1}\wedge\chi_{\sigma}^{2}:\sigma\in\Sigma\\},$

since the latter is Boolean inconsistent.

Then $\theta$ being $\bigvee\\{\chi_{\sigma}^{1}:\sigma\in\Sigma\\}$ and
$\sigma$ being $\bigwedge\\{\chi_{\sigma}^{2}:\sigma\in\Sigma\\}$ witness that
$s=s_{1}\cup s_{2}\not\in S$.

  * (Ind.5)

Suppose $\exists v\chi(v)\in s_{1}\subseteq s$ and consider $c\in\mathcal{C}$
a constant not appearing in $s$, which exists by the clause on the number of
constants from $\mathcal{C}$ in sentences in $X_{\phi}$. Let us check
$s\cup\\{\chi(c)\\}\in S$. For this take $\theta,\sigma\in X_{\phi}\cap
X_{\psi}$ such that $\vDash_{\mathrm{Sh}}\bigwedge
s_{1}\cup\\{\chi(c)\\}\rightarrow\theta$ and $\vDash_{\mathrm{Sh}}\bigwedge
s_{2}\rightarrow\sigma$ with no constants from $\mathcal{C}$ either in
$\theta$ or in $\sigma$. We must show that $\theta\wedge\sigma$ is Boolean
satisfiable. It is enough to prove
$\vDash_{\mathrm{Sh}}s_{1}\rightarrow\theta$. Consider $\mathcal{M}$ a Boolean
valued model for $\mathrm{L}\cup\left\\{c\right\\}$ with the mixing property
such that $\mathcal{M}\vDash s_{1}$. Since $\exists v\chi(v)\in s_{1}$,
$\left\llbracket\exists
v\chi(v)\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}=1_{\mathsf{B}}$; since
$\mathcal{M}$ is full, we can find $\tau\in M$ such that

$\left\llbracket\exists
v\chi(v)\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}=\left\llbracket\chi(\tau)\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}=1_{\mathsf{B}}.$

Consider $\mathcal{M}^{\prime}$ to be the model obtained from $\mathcal{M}$
reinterpreting all symbols of $\mathrm{L}$ the same way, but mapping now $c$
to $\tau$.

Then $\mathcal{M}^{\prime}\models\bigwedge s_{1}\cup\left\\{\phi(c)\right\\}$,
hence
$\left\llbracket\theta\right\rrbracket^{\mathcal{M}^{\prime}}_{\mathsf{B}}=1_{\mathsf{B}}$
as well. Since $c$ does not appear in $\theta$ we get that
$\left\llbracket\theta\right\rrbracket^{\mathcal{M}}=\left\llbracket\theta\right\rrbracket^{\mathcal{M}^{\prime}}=1_{\mathsf{B}}$.

  * (Str.1,2,3)

All three cases follow from $\bigwedge s_{1}$ and $\bigwedge
s_{1}\cup\\{\chi\\}$ being $\mathrm{BVM}$-equivalent when $\chi$ is the
relevant formula of each clause.

∎

###  7.3. Proof of Thm. 3.5

###### Proof.

Fix a set $\mathcal{C}=\left\\{c_{i}:i<\kappa\right\\}$ of constants. Consider
$\mathrm{L}(\mathcal{C})_{T,\mathcal{F}}$ the set of all sentences obtained by
replacing in the $\mathrm{L}_{T,\mathcal{F}}$-formulae with free variables in
$\left\\{v_{i}:i\in\omega\right\\}$ all occurrences of these finitely many
free variables by constants from $\mathcal{C}$. The consistency property $S$
has as elements the sets

$\displaystyle
s=s_{0}\cup\left\\{\bigvee\left\\{\phi[c_{\sigma_{\Phi}(0)},\ldots,c_{\sigma_{\Phi}(n_{\Phi}-1)}]:\phi\in\Phi\right\\}:\,\Phi\in\mathcal{F}_{0}\right\\},$

where:

  * •

$s_{0}$ is a finite set of $\mathrm{L}(\mathcal{C})_{T,\mathcal{F}}$
sentences,

  * •

only finitely many constants from $\mathcal{C}$ appear in $s_{0}$,

  * •

$\mathcal{F}_{0}$ is a finite subset of $\mathcal{F}$,

  * •

$\sigma_{\Phi}:\omega\to\mathcal{C}$ for all $\Phi\in\mathcal{F}_{0}$, and

  * •

$T\cup s_{0}$ has a Boolean valued model.

We first check that $S$ is a consistency property: consider $s\in S$ and
$\psi\in s$. First of all, by definition of $S$ and the Completeness Thm. 3.1
we can fix a mixing model $\mathcal{M}$ of $s_{0}\cup T$. We deal with two
cases. If $\psi\in s_{0}\cup T$, then $\mathcal{M}\vDash\psi$ allows to find
the correspondent formula (here one also uses that only finitely many
constants from $\mathcal{C}$ occur in $s_{0}$). Thus we only need to deal with
the case

$\psi=\bigvee\left\\{\phi[c_{\sigma_{\Phi}(0)},\ldots,c_{\sigma_{\Phi}(n_{\Phi}-1)}]:\phi\in\Phi\right\\}$

for some $\Phi\in\mathcal{F}$ and $\sigma_{\Phi}:\omega\to\mathcal{C}$. We
need to find some $\phi\in\Phi$ such that $s\cup\\{\phi\\}\in s$. Denote
$d_{0},\ldots,d_{m}\in\mathcal{C}$ the constants in $s_{0}$ from $\mathcal{C}$
that are not $c_{\sigma(0)},\ldots,c_{\sigma(n_{\Phi}-1)}$ and write $s_{0}$
as

$s_{0}[c_{\sigma_{\Phi}(0)},\ldots,c_{\sigma_{\Phi}(n_{\Phi}-1)},d_{0},\ldots,d_{m}]$

with its constant symbols displayed. Since

$\mathcal{M}\vDash T\cup s_{0},$

we have

$\mathcal{M}\vDash\exists v_{0}\ldots v_{n_{\Phi}-1}\exists w_{0}\ldots
w_{m}\bigwedge s_{0}[v_{0},\ldots,v_{n_{\Phi}-1},w_{0},\ldots,w_{m}].$

By the Theorem assumptions, since

$\exists v_{0}\ldots v_{n_{\Phi}-1}\exists w_{0}\ldots w_{m}\bigwedge
s_{0}[v_{0},\ldots,v_{n_{\Phi}-1},w_{0},\ldots,w_{m}]$

is an $\mathrm{L}_{T,\mathcal{F}}$-formula, we get that for some
$\phi\in\Phi$,

$T\cup\left\\{\exists v_{0}\ldots v_{n_{\Phi}-1}\exists w_{0}\ldots
w_{m}\bigwedge
s_{0}[v_{0},\ldots,v_{n_{\Phi}-1},w_{0},\ldots,w_{m}]\wedge\phi[v_{0},\ldots,v_{n_{\Phi}-1}]\right\\}$

has an $\mathrm{L}_{T,\mathcal{F}}$-model $\mathcal{N}$, which again by
completeness can be supposed to be mixing. Make $\mathcal{N}$ an
$\mathrm{L}(\mathcal{C})_{T,\mathcal{F}}$-structure by choosing an
interpretation of the constants from $\mathcal{C}$ such that
$c_{\sigma_{\Phi}(0)},\ldots,c_{\sigma_{\Phi}(n_{\Phi}-1)}$ are assigned to
$v_{0},\ldots,v_{n_{\Phi}-1}$ and $d_{0},\ldots,d_{m}$ are assigned to
$w_{0},\ldots,w_{m}$. Then

$s_{0}\cup\\{\phi\\}\cup\left\\{\bigvee\left\\{\phi[c_{\sigma_{\Phi}(0)},\ldots,c_{\sigma_{\Phi}(n_{\Phi}-1)}]:\phi\in\Phi\right\\}:\,\Phi\in\mathcal{F}_{0}\right\\}\in
S.$

This concludes the proof that $S$ is a consistency property.

It is now straightforward to check that for all $\Phi\in\mathcal{F}$ and
$\sigma:\omega\to\mathcal{C}$

$D_{\Phi,\sigma}=\left\\{s\in
S:\bigvee\left\\{\phi[c_{\sigma(0)},\ldots,c_{\sigma(n_{\Phi}-1)}]:\phi\in\Phi\right\\}\in
s\right\\}$

is dense in $\mathbb{P}_{S}$ and that for all $\phi\in T$ so is
$D_{\phi}=\left\\{s\in S:\phi\in S\right\\}$.

By the Model Existence Theorem there is a model $\mathcal{M}$ of

$T\cup\left\\{\bigvee\left\\{\phi[c_{\sigma(0)},\ldots,c_{\sigma(n_{\Phi}-1)}]:\phi\in\Phi\right\\}:\,\Phi\in\mathcal{F},\sigma:\omega\to\mathcal{C}\right\\}$

in which all the elements are the interpretation of some constant from
$\mathcal{C}$. Thus $\mathcal{M}$ models the theory

$T\cup\left\\{\bigwedge_{\Phi\in\mathcal{F}}\forall v_{0}\ldots
v_{n_{\Phi}-1}\bigvee\Phi(v_{0},\ldots,v_{n_{\Phi}-1}\right\\},$

as required.

∎

###  7.4. Proof of Thm. 3.7

###### Proof.

The proof is a small twist of the proof of Thm. 3.2 with two differences.
First, when obtaining a Boolean valued model for an element of the consistency
property one needs to apply Thm. 6.1 instead of 5.7. Secondly, when proving in
the proof of Thm. 3.2 that $S$ is a consistency property, the existential case
strongly uses the Boolean valued models being full, thus also this part of
that proof needs a revision.

  * (Ind.5)

Suppose $\exists\vec{v}\varphi(\vec{v})\in s_{1}\subseteq s$ and consider
$\vec{c}\in\mathcal{C}^{\vec{v}}$ a sequence of constants not appearing in
$s$, which exists by the clause on the number of constants from $\mathcal{C}$
in sentences444Note that $\vec{v}$ can be an infinite string of variables of
length less than $\lambda$, nonetheless in $\varphi(\vec{v})$ only finitely
many constants from $\mathcal{C}$ appears in it as well as in any other
formula of $s$. in $X_{\phi}$. Let us check $s\cup\\{\varphi(\vec{c})\\}\in
S$. For this take $\theta,\sigma\in X_{\phi}\cap X_{\psi}$ such that
$\vDash_{\mathrm{BVM}}\bigwedge
s_{1}\cup\\{\varphi(\vec{c})\\}\rightarrow\theta$ and
$\vDash_{\mathrm{BVM}}\bigwedge s_{2}\rightarrow\sigma$. We must show that
$\theta\wedge\sigma$ is Boolean consistent. It is enough to prove
$\vDash_{\mathrm{BVM}}s_{1}\rightarrow\theta$. Consider $\mathcal{M}$ a
Boolean valued model such that $\mathcal{M}\vDash s_{1}$. Since
$\exists\vec{v}\varphi(\vec{v})\in s_{1}$,
$\left\llbracket\exists\vec{v}\varphi(\vec{v})\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}=1_{\mathsf{B}}$;
therefore we can find a maximal antichain $A\subset\mathsf{B}$ and a family
$\\{\vec{\tau}_{a}:a\in A\\}\subset M$ such that

$\left\llbracket\varphi(\vec{\tau}_{a})\right\rrbracket_{\mathsf{B}}\geq a$

and

$\left\llbracket\exists\vec{v}\varphi(\vec{v})\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}=\bigvee_{a\in
A}\left\llbracket\varphi(\vec{\tau}_{a})\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}=1_{\mathsf{B}}.$

If we are able to check
$\left\llbracket\theta\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}\geq a$ for
any $a\in A$, we will conclude since

$\left\llbracket\theta\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}\geq\bigvee_{a\in
A}a=1_{\mathsf{B}}.$

Consider $a\in A$ and the structure $\mathcal{M}$ together with the assignment
$\vec{c}\mapsto\vec{\tau}_{a}$. Consider also the Boolean algebra
$\mathsf{B}\upharpoonleft a$. For any $m_{1},\ldots,m_{n}\in\mathcal{M}$ and
any $n$-ary relational symbol $R$ of the relational $\omega$-signature
$\mathrm{L}$ define

$\left\llbracket
R(m_{1},\ldots,m_{n})\right\rrbracket_{\mathsf{B}\upharpoonleft
a}^{(\mathcal{M},\vec{c}\ \mapsto\ \vec{\tau}_{a})}=a\wedge\left\llbracket
R(m_{1},\ldots,m_{n})\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}.$

Then one makes $\mathcal{M}$ a $\mathsf{B}\upharpoonleft a$-Boolean valued
model for $\mathrm{L}\cup\\{\vec{c}\\}$ letting

$\left\llbracket\vartheta[\vec{c}\ \mapsto\
\vec{\tau}_{a}]\right\rrbracket_{\mathsf{B}\upharpoonleft
a}^{\mathcal{M}}=a\wedge\left\llbracket\vartheta[\vec{c}\ \mapsto\
\vec{\tau}_{a}]\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}.$

In particular the $\mathsf{B}\upharpoonleft a$-value of $\bigwedge
s_{1}\cup\\{\varphi(\vec{c})\\}$ in
$(\mathcal{M},\vec{c}\mapsto\vec{\tau}_{a})$ is $1_{\mathsf{B}\upharpoonleft
a}=a$. Finally, the hypothesis $\vDash_{\mathrm{BVM}}\bigwedge
s_{1}\cup\\{\varphi(\vec{c})\\}\rightarrow\theta$ ensures
$\left\llbracket\theta\right\rrbracket_{\mathsf{B}\upharpoonleft
a}^{(\mathcal{M},\vec{c}\ \mapsto\ \vec{\tau}_{a})}=a$ and since

$a=\left\llbracket\theta\right\rrbracket_{\mathsf{B}\upharpoonleft
a}^{(\mathcal{M},\vec{c}\ \mapsto\
\vec{\tau}_{a})}=a\wedge\left\llbracket\theta\right\rrbracket_{\mathsf{B}}^{\mathcal{M}},$

we conclude
$a\leq\left\llbracket\theta\right\rrbracket_{\mathsf{B}}^{\mathcal{M}}$.

∎

##  8\. Forcing notions as consistency properties

By the results of Section 5 a consistency property $S$ for
$\mathrm{L}_{\kappa\omega}$ can be naturally seen as a forcing notion
$\mathbb{P}_{S}$; then, using the forcing machinery on $\mathbb{P}_{S}$, we
can produce a Boolean valued model with the mixing property of $\bigwedge p$
for any $p\in S$. In this section we show that it is possible to go the other
way round: we prove that any forcing notion $\mathbb{P}$ has a consistency
property $S_{\mathbb{P}}$ associated to it, so that it is equivalent to force
with $\mathbb{P}$ or with $\mathbb{P}_{S_{\mathbb{P}}}$.

From now on we deal with forcing notions given both by partial orders or by
complete Boolean algebras.

Given a complete Boolean algebra $\mathsf{B}$, we show that for some regular
$\kappa$ large enough in $V$, $\mathsf{B}$ is forcing equivalent to a
consistency property describing the $\in$-theory of $H_{\kappa}$ as computed
in a $V$-generic extension by $\mathsf{B}$.

######  Notation 8.1.

Let $\mathsf{B}$ be a complete Boolean algebra of cardinality $\kappa$. The
signature $\mathrm{L}$ is $\left\\{\in\right\\}$.
$\mathrm{L}(\mathcal{C})_{\infty\omega}$ is produced by the set of constants
$\mathcal{C}=V^{\mathsf{B}}\cap H_{\kappa^{+}}$. We use
$\phi^{H_{\kappa^{+}}}$ to denote that all quantifiers from $\phi$ are
restricted to $H_{\kappa^{+}}$. We write
$\left\llbracket\phi\right\rrbracket_{\mathsf{B}}$ rather than
$\left\llbracket\phi\right\rrbracket^{V^{\mathsf{B}}}_{\mathsf{B}}$.

######  Theorem 8.2.

For any complete Boolean algebra $\mathsf{B}$ of size less or equal than
$\kappa$ and any regular cardinal $\lambda$ the following holds:

  1. (i)

$S_{\mathsf{B}}=\\{s\in[\mathrm{L}(\mathcal{C})_{\lambda\omega}]^{<\omega}:\left\llbracket(\bigwedge
s)^{H_{\check{\kappa}^{+}}}\right\rrbracket_{\mathsf{B}}>0_{\mathsf{B}}\\}$ is
a consistency property,

  2. (ii)

the map

$\displaystyle\pi_{\mathsf{B}}:(S_{\mathsf{B}},\leq)$
$\displaystyle\rightarrow(\mathsf{B}^{+},\leq_{\mathsf{B}})$ $\displaystyle s$
$\displaystyle\mapsto\left\llbracket(\bigwedge
s)^{H_{\check{\kappa}^{+}}}\right\rrbracket_{\mathsf{B}}$

is a dense embedding. In particular $\mathsf{B}$ and $S_{\mathsf{B}}$ are
equivalent forcing notions.

###### Proof.

We first prove $(ii)$.

  * •

If $p\leq q$, then $q\subseteq p$ and $\pi(p)=\left\llbracket\bigwedge
p\right\rrbracket_{\mathsf{B}}\leq\left\llbracket\bigwedge
q\right\rrbracket_{\mathsf{B}}=\pi(q)$.

  * •

We have $p\perp q\Leftrightarrow p\cup q\notin
S_{\mathsf{B}}\Leftrightarrow\left\llbracket\bigwedge(p\cup
q)\right\rrbracket_{\mathsf{B}}=0_{\mathsf{B}}\Leftrightarrow\left\llbracket\bigwedge
p\right\rrbracket_{\mathsf{B}}\wedge\left\llbracket\bigwedge
q\right\rrbracket_{\mathsf{B}}=0_{\mathsf{B}}\Leftrightarrow\pi(p)\perp\pi(q)$.

  * •

Let $\dot{G}=\left\\{(\check{b},b):\,b\in\mathsf{B}\right\\}$ be the canonical
$\mathsf{B}$-name for a $V$-generic filter. Since for any $b\in\mathsf{B}^{+}$
the $\mathsf{B}$-value of $\check{b}\in\dot{G}$ is $b$, the map $\pi$ is
surjective and in particular $\pi[S_{\mathsf{B}}]$ is dense in
$\mathsf{B}^{+}$.

Now we prove $(i)$. We have to check that $S_{\mathsf{B}}$ satisfies the
clauses of Def. 4.2. Note that by choice of $\kappa$,

$\left\llbracket\forall v\in
H_{\check{\kappa}^{+}}\phi^{H_{\check{\kappa}^{+}}}(v)\right\rrbracket^{V^{\mathsf{B}}}_{\mathsf{B}}=\bigwedge_{\tau\in\mathcal{C}}\left\llbracket\phi^{H_{\check{\kappa}^{+}}}(v)\right\rrbracket^{V^{\mathsf{B}}}_{\mathsf{B}}.$

In view of the above observation, for notational simplicity we use
$\left\llbracket\phi\right\rrbracket_{\mathsf{B}}$ instead of
$\left\llbracket\phi^{H_{\check{\kappa}^{+}}}\right\rrbracket^{V^{\mathsf{B}}}_{\mathsf{B}}$.
Note also that in the proof below we will only be interested in formulae where
quantifiers range over (and constants belong to) $H_{\kappa^{+}}\cap
V^{\mathsf{B}}$.

(Con):

Consider $s\in S_{\mathsf{B}}$ and
$\phi\in\mathrm{L}(\mathcal{C})_{\infty\omega}$. If $\phi$ and $\neg\phi$ are
both in $s$, $\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}\leq\left\llbracket\phi\wedge\neg\phi\right\rrbracket_{\mathsf{B}}=0_{\mathsf{B}}$,
a contradiction since $\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}>0_{\mathsf{B}}$. Then for any $\phi$, either
$\phi\notin s$ or $\neg\phi\notin s$.

(Ind.1):

Consider $s\in S_{\mathsf{B}}$ and $\neg\phi\in s$. Since
$\left\llbracket\neg\phi\right\rrbracket_{\mathsf{B}}=\left\llbracket\phi\neg\right\rrbracket_{\mathsf{B}}$,
$\left\llbracket\bigwedge(s\cup\\{\phi\neg\\})\right\rrbracket_{\mathsf{B}}=\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}>0_{\mathsf{B}}$ and $s\cup\\{\phi\neg\\}\in
S_{\mathsf{B}}$.

(Ind.2):

Consider $s\in S_{\mathsf{B}}$ and $\bigwedge\Phi\in s$. For any
$\phi\in\Phi$,
$\left\llbracket\bigwedge(s\cup\\{\phi\\})\right\rrbracket_{\mathsf{B}}=\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}>0_{\mathsf{B}}$ and $s\cup\\{\phi\\}\in
S_{\mathsf{B}}$.

(Ind.3):

Consider $s\in S_{\mathsf{B}}$, $\forall v\phi(v)\in s$ and
$\tau\in\mathcal{C}$. We have

$\left\llbracket\forall
v\phi(v)\right\rrbracket_{\mathsf{B}}=\bigwedge_{\sigma\in V^{\mathsf{B}}\cap
H_{\kappa^{+}}}\left\llbracket\phi(\sigma)\right\rrbracket_{\mathsf{B}}\leq\left\llbracket\phi(\tau)\right\rrbracket_{\mathsf{B}}.$

Therefore

$\left\llbracket\bigwedge(s\cup\\{\phi(\tau)\\})\right\rrbracket_{\mathsf{B}}=\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}>0_{\mathsf{B}},$

and $s\cup\\{\phi(\tau)\\}\in S_{\mathsf{B}}$.

(Ind.4):

Consider $s\in S_{\mathsf{B}}$ and $\bigvee\Phi\in s$. Suppose that for no
$\phi\in\Phi$, $s\cup\\{\phi\\}\in S_{\mathsf{B}}$. Then for any
$\phi\in\Phi$,
$\left\llbracket\bigwedge(s\cup\\{\phi\\})\right\rrbracket_{\mathsf{B}}=\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}\wedge\left\llbracket\phi\right\rrbracket_{\mathsf{B}}=0_{\mathsf{B}}$.
Therefore $\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}\leq\left\llbracket\neg\phi\right\rrbracket_{\mathsf{B}}$
for any $\phi\in\Phi$. Since
$\bigwedge_{\phi\in\Phi}\left\llbracket\neg\phi\right\rrbracket_{\mathsf{B}}$
is the greatest lower bound of
$\\{\left\llbracket\neg\phi\right\rrbracket_{\mathsf{B}}:\phi\in\Phi\\}$, we
have $\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}\leq\bigwedge_{\phi\in\Phi}\left\llbracket\neg\phi\right\rrbracket_{\mathsf{B}}=\left\llbracket\neg\bigvee\Phi\right\rrbracket_{\mathsf{B}}$.
Then
$\left\llbracket\bigwedge(s\cup\\{\neg\bigvee\Phi\\})\right\rrbracket_{\mathsf{B}}=\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}>0_{\mathsf{B}}$, but since $\bigvee\Phi$ and
$\neg\bigvee\Phi$ are both in $s\cup\\{\neg\bigvee\Phi\\}$,
$\left\llbracket\bigwedge(s\cup\\{\neg\bigvee\Phi\\})\right\rrbracket_{\mathsf{B}}=0_{\mathsf{B}}$,
a contradiction.

(Ind.5):

Consider $s\in S_{\mathsf{B}}$ and $\exists v\phi(v)\in s$. Suppose that for
no $\tau\in\mathcal{C}$, $s\cup\\{\phi(\tau)\\}\in S_{\mathsf{B}}$. Then for
any $\tau\in\mathcal{C}$,

$\left\llbracket\bigwedge(s\cup\\{\phi(\tau)\\})\right\rrbracket_{\mathsf{B}}=\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}\wedge\left\llbracket\phi(\tau)\right\rrbracket_{\mathsf{B}}=0_{\mathsf{B}}.$

This gives that

$\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}\leq\left\llbracket\neg\phi(\tau)\right\rrbracket_{\mathsf{B}}$

for any $\tau\in V^{\mathsf{B}}\cap H_{\kappa^{+}}$. Therefore

$\left\llbracket\bigwedge s\right\rrbracket_{\mathsf{B}}\leq\bigwedge_{\tau\in
V^{\mathsf{B}}\cap
H_{\kappa^{+}}}\left\llbracket\neg\phi(\tau)\right\rrbracket_{\mathsf{B}}=\left\llbracket\forall
v\neg\phi(v)\right\rrbracket_{\mathsf{B}}=\left\llbracket\neg\exists
v\phi(v)\right\rrbracket_{\mathsf{B}}.$

Note that the equality

$\bigwedge_{\tau\in V^{\mathsf{B}}\cap
H_{\kappa^{+}}}\left\llbracket\neg\phi(\tau)\right\rrbracket_{\mathsf{B}}=\left\llbracket\forall
v\neg\phi(v)\right\rrbracket_{\mathsf{B}}$

only holds because the quantifiers from $\phi$ are restricted to
$H_{\check{\kappa}^{+}}$. Therefore

$\left\llbracket\bigwedge(s\cup\\{\neg\exists
v\phi(v)\\})\right\rrbracket_{\mathsf{B}}=\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}>0_{\mathsf{B}}.$

But now $\exists v\phi(v)$ and $\neg\exists v\phi(v)$ are both in
$s\cup\\{\neg\exists v\phi(v)\\}$, hence

$\left\llbracket\bigwedge(s\cup\\{\neg\exists
v\phi(v)\\})\right\rrbracket_{\mathsf{B}}=0_{\mathsf{B}}.$

We reached a contradiction.

(Str.1):

Suppose $s\in S_{\mathsf{B}}$ and $\tau=\sigma\in s$; since
$\mathsf{B}$-valued models for set theory verify
$\left\llbracket\tau=\sigma\right\rrbracket_{\mathsf{B}}=\left\llbracket\sigma=\tau\right\rrbracket_{\mathsf{B}}$,
$\left\llbracket\bigwedge(s\cup\\{\sigma=\tau\\})\right\rrbracket_{\mathsf{B}}=\left\llbracket\bigwedge
s\right\rrbracket_{\mathsf{B}}>0_{\mathsf{B}}$ and $s\cup\\{\sigma=\tau\\}\in
S_{\mathsf{B}}$.

(Str.2):

Suppose $s\in S_{\mathsf{B}}$ and $\\{\sigma=\tau,\phi(\tau)\\}\subset s$. We
have
$\left\llbracket\sigma=\tau\right\rrbracket_{\mathsf{B}}\wedge\left\llbracket\phi(\tau)\right\rrbracket_{\mathsf{B}}\leq\left\llbracket\phi(\sigma)\right\rrbracket_{\mathsf{B}}$;
therefore
$\left\llbracket\bigwedge(s\cup\\{\phi(\sigma)\\})\right\rrbracket_{\mathsf{B}}>0_{\mathsf{B}}$
and $s\cup\\{\phi(\sigma)\\}\in S_{\mathsf{B}}$.

(Str.3):

Trivial since $\mathrm{L}=\left\\{\in\right\\}$ has no constant symbol.

∎

Note that the only formulae one needs to keep in $S_{\mathsf{B}}$ in order to
ensure that there is a dense embedding between both forcing notions are
$\check{b}\in\dot{G}$. This is because in order to prove that the embedding
has dense image, one only uses that the $\mathsf{B}$-value of
$\check{b}\in\dot{G}$ is $b$. In particular one can consider various choices
of constants $\mathcal{C}$ to produce the desired consistency property
$S_{\mathsf{B}}$, other than the one we made.

##  9\. Appendix

We collect here some results which are useful to clarify several concepts but
not central.

###  9.1. Separating Tarski satisfiability from boolean satisfiability

In this section we show that being satisfiable in the ordinary sense (e.g.
with respect to Tarski semantics) is strictly stronger than being boolean
satisfiable, which is also strictly stronger than being weakly boolean
satisfiable.

We first show that there is a boolean satisfiable theory which has no Tarski
model.

######  Fact 9.1.

Let
$\mathrm{L}=\left\\{F\right\\}\cup\left\\{d_{n}:n\in\omega\right\\}\cup\left\\{e_{\alpha}:\alpha<\omega_{1}^{V}\right\\}$
with $F$ a binary predicate. Consider the
$\mathrm{L}_{\omega_{2}^{V}\omega}$-theory $S$ given by

$\forall x\,\exists!y\,F(x,y)$ $\exists
y\,F(x,y)\leftrightarrow\bigvee_{n\in\omega}x=d_{n}$
$\bigvee_{n\in\omega}F(d_{n},e_{\alpha})$

for all $\alpha<\omega_{1}^{V}$.

Then every countable fragment of $S$ has a Tarski model, while $S$ has no
Tarski model.

Furthermore $S$ has a boolean valued model.

Note that $S$ witnesses the failure of the compactness and completeness
theorems for Tarski semantics for $\mathrm{L}_{\omega_{2}^{V}\omega}$; it is
clearly a counterexample to compactness for this semantics; it is also a
counterexample to completeness (using the axiom system we present in Section
2.4) since $S\not\vdash\emptyset$ in view of Thm. 3.1.

###### Proof.

Given a countable fragment $R$ of $S$, find $\beta$ countable and such that
any $e_{\alpha}$ occurring in some formula in $R$ has $\alpha<\beta$. Then
$(\beta,f)$ where $f$ is a surjection of $\omega$ onto $\beta$ can be extended
to a model of $R$ by mapping $d_{n}$ to $n$ and $e_{\alpha}$ to $\alpha$ for
any $n\in\omega,\alpha<\beta$.

Note that the interpretation of $F$ in any Tarski model of $S$ in $V$ is a map
with domain a countable set and range an uncountable set in $V$. Hence no such
model can exist in $V$.

Now if $G$ is $\operatorname{Coll}(\omega,\omega_{1})$-generic in $V[G]$ the
generic function $\omega\to\omega_{1}^{V}$ given by $\cup G$ gives in $V[G]$ a
Tarski model of $S$. Taking this into account, in $V$ consider the
$\operatorname{RO}(\operatorname{Coll}(\omega,\omega_{1}^{V}))$-valued model
$\mathcal{M}=(\omega_{1}^{V},F^{\mathcal{M}},d_{n}^{\mathcal{M}}:n\in\omega,e_{\alpha}^{\mathcal{M}}:\alpha<\omega_{1}^{V})$
given by

  * •

$R^{\mathcal{M}}(n,\alpha)=\text{Reg}\left(\left\\{q\in\operatorname{Coll}(\omega,\omega_{1}^{V}):\,\langle
n,\alpha\rangle\in q\right\\}\right)$ for $n\in\omega$ and
$\alpha<\omega_{1}^{V}$;
$R^{\mathcal{M}}(\beta,\alpha)=0_{\operatorname{Coll}(\omega,\omega_{1}^{V})}$
for $\beta\not\in\omega$ and $\alpha<\omega_{1}^{V}$;

  * •

$d_{n}^{\mathcal{M}}=n$ for all $n\in\omega$,

  * •

$e_{\alpha}^{\mathcal{M}}=\alpha$ for all $\alpha\in\omega_{1}^{V}$.

It can be checked that in $V$ it holds that $\mathcal{M}$ assigns value
$1_{\operatorname{Coll}(\omega,\omega_{1}^{V})}$ to all axioms of $S$. ∎

Now we exhibit a theory $T$ which is a counterexample to the compactness
theorem with respect to boolean satisfiability: all finite fragments of $T$
are boolean satisfiable while $T$ is not.

######  Fact 9.2.

Consider the first order $\mathrm{L}_{\omega_{1}\omega}$-theory $T$ for
$\mathrm{L}=\left\\{d_{n}:n\in\omega,c_{m}:\,m\in\omega\right\\}$ with axioms:

  * •

$\bigwedge_{n\in\omega}\bigvee_{m\in\omega}d_{n}=c_{m}$,

  * •

$\bigwedge_{n\neq m\in\omega}c_{n}\neq c_{m}$,

  * •

$d_{n}\neq c_{m}$ for $n,m\in\omega$.

The following holds:

  * •

Every finite fragment of $T$ is Tarski satisfiable.

  * •

$T$ is not boolean satisfiable.

  * •

$T$ is weakly boolean satisfiable.

###### Proof.

Let:

  * •

$\mathsf{B}$ be the boolean completion of the Cohen forcing
$\omega^{<\omega}$,

  * •

$M=\left\\{\sigma\in
V^{\mathsf{B}}:\left\llbracket\sigma\in\check{\omega}\right\rrbracket=1_{\mathsf{B}}\right\\}$,

  * •

$\dot{r}$ be the canonical $\mathsf{B}$-name for the Cohen generic real.

  * •

$\mathcal{M}$ be the $\mathsf{B}$-model for $\mathrm{L}$ with domain $M$,
$\left\llbracket\cdot=\cdot\right\rrbracket^{\mathcal{M}}=\left\llbracket\cdot=\cdot\right\rrbracket^{V^{\mathsf{B}}}$,
and interpretation of $d_{n}$ by $\dot{r}(\check{n})$ and $c_{m}$ by
$\check{m}$.

Then $\mathcal{M}$ witnesses that $T$ is weakly boolean satisfiable.

$T$ cannot be boolean satisfiable because in any boolean valued model it
cannot be that $d_{n}\neq c_{m}$ gets boolean value $1_{\mathsf{B}}$ for all
$m$ while also $\bigvee_{m\in\omega}d_{n}=c_{m}$ gets the same value.

$(\omega,c_{n}\mapsto n:n\in\omega)$ can be extended to a Tarski model of any
finite fragment of $T$. ∎

Our last example is a _finite_ weakly boolean satisfiable theory which is not
boolean satisfiable.

######  Fact 9.3.

Consider the finite $\mathrm{L}_{\omega\omega}$-theory $T$ for
$\mathrm{L}=\left\\{d,c_{0},c_{1}\right\\}$ with axioms:

  * •

$\bigvee_{m\in 2}d=c_{m}$,

  * •

$c_{0}\neq c_{1}$,

  * •

$d\neq c_{m}$ for $m\in 2$.

The following holds:

  * •

$T$ is not boolean satisfiable.

  * •

$T$ is weakly boolean satisfiable.

###### Proof.

Let $\mathsf{B}=\left\\{0,a,\neg a,1\right\\}$ be the four elements boolean
algebra, let $\mathcal{M}$ consists of the four possible functions
$f:\left\\{a,\neg a\right\\}\to 2$. Let $c_{i}$ be interpreted by the constant
functions with value $i$ and $d$ by one of the other two. Set $\left\llbracket
f=g\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}=\bigvee\\{b\in\left\\{a,\neg
a\right\\}:f(b)=g(b)\\}$. Then

$\left\llbracket\bigvee_{m\in
2}d=c_{m}\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}=\left\llbracket c_{0}\neq
c_{1}\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}=1_{\mathsf{B}}$

and $\left\llbracket d\neq
c_{i}\right\rrbracket^{\mathcal{M}}_{\mathsf{B}}>0_{\mathsf{B}}$ for both
$i=0,1$. Hence $T$ is weakly boolean satisfiable.

$T$ cannot be boolean satisfiable since

$\left\llbracket d\neq
c_{0}\right\rrbracket^{\mathcal{N}}_{\mathsf{C}}=\neg\left\llbracket
d=c_{1}\right\rrbracket^{\mathcal{N}}_{\mathsf{C}}$

in all $\mathsf{C}$-valued models $\mathcal{N}$ of $c_{0}\neq
c_{1}\wedge\bigvee_{m\in 2}d=c_{m}$. Hence we cannot have that

$\left\llbracket d\neq
c_{0}\right\rrbracket^{\mathcal{N}}_{\mathsf{C}}=\left\llbracket d_{n}\neq
c_{1}\right\rrbracket^{\mathcal{N}}_{\mathsf{C}}=1_{\mathsf{C}}$

in any boolean valued model model of the other axioms of $T$. ∎