# Outward compactness

Peter Holy  Peter Holy. Institut für Diskrete Mathematik und Geometrie, TU
Wien, Wiedner Hauptstraße 8–10/104, 1040 Wien, Austria
<EMAIL_ADDRESS>,  Philipp Lücke  Philipp Lücke. Fachbereich
Mathematik, Universität Hamburg, Bundesstraße 55, Hamburg, 20146, Germany
<EMAIL_ADDRESS>and  Sandra Müller  Sandra Müller. Institut für
Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstraße 8–10/104, 1040
Wien, Austria<EMAIL_ADDRESS>
###### Abstract.

We introduce and study a new type of compactness principle for strong logics
that, roughly speaking, infers the consistency of a theory from the
consistency of its small fragments in certain outer models of the set-
theoretic universe. We refer to this type of compactness property as _outward
compactness_ , and we show that instances of this type of principle for
second-order logic can be used to characterize various large cardinal notions
between measurability and extendibility, directly generalizing a classical
result of Magidor that characterizes extendible cardinals as the strong
compactness cardinals of second-order logic. In addition, we generalize a
result of Makowsky that shows that Vopěnka’s Principle is equivalent to the
existence of compactness cardinals for all abstract logics by characterizing
the principle “ $\operatorname{Ord}$ is Woodin” through outward compactness
properties of abstract logics.

###### Key words and phrases:

Large cardinals, compactness, second-order logic, abstract logics, elementary
embeddings

###### 2020 Mathematics Subject Classification:

(Primary) 03E55; (Secondary) 03B16, 03C55, 03C85

The authors would like to thank Menachem Magidor for very helpful discussions
on some of the topics of this paper. The first and third authors gratefully
acknowledge funding from Austrian Science Fund (FWF) Elise Richter grant
number V844. The second author gratefully acknowledges supported by the
Deutsche Forschungsgemeinschaft (Project number 522490605) and the Spanish
Government under grant EUR2022-134032. Furthermore, the third author
gratefully acknowledges support by her FWF START Prize Y1498 and her FWF
International Project I6087. This research was funded in whole or in part by
the Austrian Science Fund (FWF) [10.55776/Y1498]. For open access purposes,
the authors have applied a CC BY public copyright license to any author
accepted manuscript version arising from this submission.

##  1\. Introduction

The work presented in this paper contributes to the study of the deep
connections between large cardinals and strong logics. Its starting point is a
classical result of Magidor [13] that relates the existence of extendible
cardinals to the compactness properties of second-order logic
$\mathcal{L}^{2}$. Remember that an infinite cardinal $\kappa$ is _extendible_
if for every ordinal $\eta>\kappa$, there is an ordinal $\zeta$ and a non-
trivial elementary embedding ${j}:{V_{\eta}}\longrightarrow{V_{\zeta}}$
satisfying $\operatorname{crit}(j)=\kappa$ and $j(\kappa)>\eta$. Moreover,
given a cardinal $\kappa$, an $\mathcal{L}^{2}$-theory $T$ is _${
<}\kappa$-consistent_ if every subtheory of $T$ of cardinality less than
$\kappa$ is consistent. Finally, an infinite cardinal $\kappa$ is a _strong
compactness cardinal for $\mathcal{L}^{2}$_ if every ${<}\kappa$-consistent
$\mathcal{L}^{2}$-theory is consistent. The next theorem summarizes Magidor’s
characterization of extendible cardinals through compactness properties of
$\mathcal{L}^{2}$:

######  Theorem 1.1 (Magidor [13]).

A cardinal $\kappa$ is a strong compactness cardinal for $\mathcal{L}^{2}$ if
and only if there is an extendible cardinal less than or equal to $\kappa$.

The results of this paper are motivated by the aim to obtain analogous
characterizations for various well-studied large cardinal properties below
extendibility, e.g., measurability, strongness and supercompactness. More
precisely, for a given large cardinal property $\varphi$, we want to assign,
uniformly in a parameter $\kappa$, natural and rich classes
$\mathcal{T}_{\kappa}$ of ${<}\kappa$-consistent $\mathcal{L}^{2}$-theories to
infinite cardinals $\kappa$ in a way that ensures that for all such $\kappa$,
all theories in $\mathcal{T}_{\kappa}$ are consistent if and only some
$\lambda\leq\kappa$ satisfies $\varphi(\lambda)$. We start by presenting a
characterization of measurability through compactness properties of
$\mathcal{L}^{2}$ that illustrates the main concepts on which our more general
results are based.

More precisely, we show that for every uncountable cardinal $\kappa$, there is
a canonical ${<}\kappa$-consistent $\mathcal{L}^{2}$-theory $T_{\kappa}$ whose
consistency is equivalent to the existence of a measurable cardinal less than
or equal to $\kappa$. To obtain this theory, first recall that there is an
$\mathcal{L}^{2}$-sentence in the language of set theory that holds in a model
using this language if and only if its $\in$-relation is well-founded. Next,
work in the language that extends the language of set theory by constant
symbols $c_{x}$ for every $x\in V_{\kappa+1}$ and constant symbols
$d_{\gamma}$ for all $\gamma\leq\kappa$, and let $T_{\kappa}$ be the
$\mathcal{L}^{2}$-theory consisting of the union of the following theories:

  * •

The elementary first-order diagram of $V_{\kappa+1}$, making use of the
constant symbols $c_{x}$.

  * •

All (first-order) sentences of the form $d_{\beta}\in d_{\gamma}\in
c_{\kappa}$ for $\beta<\gamma\leq\kappa$.

  * •

The (second-order) sentence stating that the $\in$-relation is well-founded.

The theory $T_{\kappa}$ is obviously ${<}\kappa$-consistent because for every
fragment $F$ of $T_{\kappa}$ of cardinality less than $\kappa$, we can easily
find a model of $F$ with underlying set $V_{\kappa+1}$. Moreover, if there is
a measurable cardinal less than or equal to $\kappa$, then $T_{\kappa}$ is
consistent because this assumption allows us to use iterated ultrapowers (see
[12, Section 19]) to find a transitive class $M$ and a non-trivial elementary
embedding ${j}:{V}\longrightarrow{M}$ with $j(\operatorname{crit}(j))>\kappa$,
and this allows us to construct a model $N$ of $T_{\kappa}$ with underlying
set $j(V_{\kappa+1})$, $c_{x}^{N}=j(x)$ for all $x\in V_{\kappa+1}$ and
$d_{\gamma}^{N}=\gamma$ for all $\gamma\leq\kappa$. Finally, if $T_{\kappa}$
is consistent, then there exists a transitive set $N$ and an elementary
embedding ${j}:{V_{\kappa+1}}\longrightarrow{N}$ with $j(\kappa)>\kappa$, and
the existence of such an embedding directly implies that some cardinal less
than or equal to $\kappa$ is measurable.

Following the approach outlined above, we now want to isolate a natural
consistency property of a theory $T$ with respect to infinite cardinals
$\kappa$, that strengthens ${<}\kappa$-consistency, is possessed by
$T_{\kappa}$ at every infinite cardinal $\kappa$, and implies the consistency
of $T$ at cardinals $\kappa$ that are greater than or equal to a measurable
cardinal. Our definition of this property is motivated by the observation that
for every cardinal $\lambda<\kappa$, the theory $T_{\kappa}$ is not only
${<}\lambda$-consistent in our ground model $V$, but remains
${<}\lambda$-consistent when we pass to outer models111I.e., models of
$\operatorname{ZFC}$ in which $V$ is a transitive class containing all
ordinals. in which $\lambda$ is still a cardinal, as witnessed by the
structure $V_{\kappa+1}^{V}$,222Whenever $M$ is an inner or outer model of our
set-theoretic universe $V$ and $\alpha$ is an ordinal, we write
$V_{\alpha}^{M}=\\{x\in M~{}|~{}\operatorname{rank}(x)<\alpha\\}$. In
particular, when working in some outer model of $V$, we use the notation
$V_{\alpha}^{V}$ for initial segments of (the ground model) $V$. by the
absoluteness of well-foundedness. In addition, if we naïvely assumed that the
property that an $\mathcal{L}^{2}$-theory $T$ is ${<}\lambda$-consistent in
every outer model in which $\lambda$ is a cardinal could be uniformly
expressed by a first-order formula with parameters $T$ and $\lambda$, it would
follow that for every cardinal $\kappa$ greater than or equal to a measurable
cardinal, every $\mathcal{L}^{2}$-theory $T$ with this property is in fact
consistent in $V$. We would argue as follows: By our assumptions, there is a
cardinal $\lambda$ with $T\in H_{\lambda}$ and an elementary embedding
${j}:{V}\longrightarrow{M}$ with $\operatorname{crit}(j)\leq\kappa$ and
$j(\kappa)>\lambda$. Using our naïve assumption, we apply the elementarity of
$j$ to conclude that the $\mathcal{L}^{2}$-theory $j(T)$ is
${<}\lambda$-consistent in every outer model of $M$ in which $\lambda$ is
still a cardinal. In particular, since $V$ is such an outer model of $M$, it
follows that $j(T)$ is a ${<}\lambda$-consistent $\mathcal{L}^{2}$-theory in
$V$, and this implies that the pointwise image $j[T]\subseteq j(T)$ of $T$
under $j$, which is of size less than $\lambda$, is a consistent
$\mathcal{L}^{2}$-theory in $V$. But this conclusion also shows that $T$ is
consistent in $V$, because the finitary character of
$\mathcal{L}^{2}$-formulae allows us to identify $j[T]$ with $T$ via the
renaming of the symbols of the given language induced by $j$.

Since the above assumption on the uniform first-order definability of
statements about truth in outer models is easily seen to be too naïve, we will
introduce several concepts that allow us to turn the above approach into a
mathematically sound argument.

######  Definition 1.2.

Given a fragment $F$ of $\operatorname{ZFC}$ and a transitive set $M$, a
transitive set $N\supseteq M$ is an _outer $F$-model of $M$_ if $F$ holds in
$N$ and the sets $M$ and $N$ have the same ordinals.

In the following, we let $\operatorname{ZFC}^{*}$ denote the fragment of
$\operatorname{ZFC}$ that contains all the single axioms of
$\operatorname{ZFC}$, together with the replacement and separation schemes
restricted to $\Sigma_{2}$-formulae. Note that this theory proves the
_$\Sigma_{2}$ -Recursion Theorem_ and therefore proves that all levels of the
_von Neumann hierarchy_
$\langle{V_{\alpha}}~{}|~{}{\alpha\in\operatorname{Ord}}\rangle$ are sets.
Moreover, it is strong enough to yield the existence of the second-order
satisfaction relation $\models_{\mathcal{L}^{2}}$ for set-sized models,333This
is because the second-order satisfaction relation can be defined by a
recursion over subformulae that is based on a $\Sigma_{2}$-function: we
proceed as for the first-order satisfaction relation, however we have to
essentially assert that $\exists W~{}W=\mathcal{P}(M)$ holds for each set-
sized model with underlying set $M$, which is already a
$\Sigma_{2}$-statement, and then we can bound all other relevant quantifiers
by $W$. To prove that such a form of recursion can successfully be performed
only requires replacement for $\Sigma_{2}$-formulae (a version of this
argument for $\Sigma_{1}$-recursion can be found in [6, Theorem 6.4]). in a
way that for every second-order formula
$\varphi(v_{0},\ldots,v_{m-1},W_{0},\ldots,W_{n-1})$ with first-order
variables $v_{0},\ldots,v_{m-1}$ and second-order variables
$W_{0},\ldots,W_{n-1}$,444In the following, we will usually use uppercase
letters for second-order variables and lowercase letters for first-order
variables in $\mathcal{L}^{2}$-formulas. $\operatorname{ZFC}^{*}$ proves that
for every non-empty set $M$, all $x_{0},\ldots,x_{m-1}\in M$ and all
$Y_{0},\ldots,Y_{n-1}\in\mathcal{P}(M)$, the statement

$\langle
M,\in\rangle\models_{\mathcal{L}^{2}}\varphi(x_{0}\ldots,x_{m-1},Y_{0},\ldots,Y_{n-1})$

holds if and only if the first order formula

$\varphi^{M}(x_{0}\ldots,x_{m-1},Y_{0},\ldots,Y_{n-1})$

holds, where $\varphi^{M}$ denotes the _first-order relativization_ of
$\varphi$ to $M$.555Inductively defined by $(v_{0}\in v_{1})^{M}\equiv
v_{0}\in v_{1}$, $(v\in W)^{M}\equiv v\in W$,
$(\neg\varphi)^{M}\equiv\neg\varphi^{M}$,
$(\varphi\wedge\psi)^{M}\equiv\varphi^{M}\wedge\psi^{M}$, $(\forall
x~{}\varphi(x))^{M}\equiv\forall x\in M~{}\varphi^{M}(x)$ and $(\forall
X~{}\varphi(X))^{M}\equiv\forall x\subseteq M~{}\varphi^{M}(x)$. Note that,
violating our above convention, we consider $W$ to be a first order variable
after relativizing $v\in W$ to $M$ (alternatively, we could replace $W$ by a
new and otherwise unused variable symbol $w$). In particular, the theory
$\operatorname{ZFC}^{*}$ allows us to uniformly speak about the consistency of
$\mathcal{L}^{2}$-theories.

######  Definition 1.3.

  1. (1)

Given an infinite cardinal $\kappa$, an $\mathcal{L}^{2}$-theory $T$ is _${
<}\kappa$-outward consistent_ if for all cardinals $\lambda<\kappa$ and all
sufficiently large cardinals $\vartheta>\kappa$ with $T\in V_{\vartheta}$, the
partial order $\mathrm{Col}(\omega,\vartheta)$ forces that $T$ is
${<}\lambda$-consistent in every outer $\operatorname{ZFC}^{*}$-model of
$V_{\vartheta}^{V}$ in which $\lambda$ is a cardinal.

  2. (2)

A cardinal $\kappa$ is an _outward compactness cardinal for $\mathcal{L}^{2}$_
if all ${<}\kappa$-outward consistent $\mathcal{L}^{2}$-theories are
consistent.

In this definition, note that in any $\mathrm{Col}(\omega,\vartheta)$-generic
extension $V[G]$, any outer $\operatorname{ZFC}^{*}$-model $N$ of
$V_{\vartheta}^{V}$ is countable in $V[G]$: $N$ provides any of its levels
$V_{\alpha}^{N}$ with some cardinality less than $\vartheta$, and since
$\vartheta$ is countable in $V[G]$, it follows that $N$ is a countable union
of countable sets, and thus itself countable. Before we state our
characterization of measurable cardinals, we make some easy observations that
relate outward compactness cardinals to compactness cardinals.

######  Proposition 1.4.

  1. (1)

If $\kappa$ is an outward compactness cardinal for $\mathcal{L}^{2}$, then
every cardinal greater than $\kappa$ is an outward compactness cardinal for
$\mathcal{L}^{2}$.

  2. (2)

Given a limit cardinal $\kappa$, every ${<}\kappa$-outward consistent
$\mathcal{L}^{2}$-theory is ${<}\kappa$-consistent.

  3. (3)

Every strong compactness cardinal for $\mathcal{L}^{2}$ is an outward
compactness cardinal for $\mathcal{L}^{2}$.

###### Proof.

(1) Let $\rho>\kappa$ be a cardinal and let $T$ be a ${<}\rho$-outward
consistent $\mathcal{L}^{2}$-theory. Let $\lambda<\kappa$ be a cardinal, let
$\vartheta>\rho$ be a sufficiently large cardinal with $T\in V_{\vartheta}$,
let $G$ be $\mathrm{Col}(\omega,\vartheta)$-generic over $V$, and let $N$ be
an outer $\operatorname{ZFC}^{*}$-model of $V_{\vartheta}^{V}$ in $V[G]$ in
which $\lambda$ is a cardinal. Then, $T$ is ${<}\lambda$-consistent in $N$.
This shows that $T$ is ${<}\kappa$-outward consistent, and hence $T$ is
consistent.

(2) Let $T$ be an $\mathcal{L}^{2}$-theory that is ${<}\kappa$-outward
consistent, and assume, for the sake of a contradiction, that it is not
${<}\kappa$-consistent. Fix an inconsistent subtheory $T_{0}$ of $T$ of
cardinality less than $\kappa$ and a cardinal $\lambda<\kappa$ with
$|T_{0}|<\lambda$. Pick a cardinal $\vartheta>\kappa$ such that
$V_{\vartheta}$ is a model of $\operatorname{ZFC}^{*}$ and $T$ is an element
of $V_{\vartheta}$. Let $G$ be $\mathrm{Col}(\omega,\vartheta)$-generic over
$V$. Then, in $V[G]$, the set $V_{\vartheta}^{V}$ is an outer
$\operatorname{ZFC}^{*}$-model of itself in which $\lambda$ is a cardinal and
hence $T$ is ${<}\lambda$-consistent in $V_{\vartheta}^{V}$. This shows that
$T_{0}$ is consistent in $V_{\vartheta}^{V}$ and, by the nature of the
satisfaction relation of $\mathcal{L}^{2}$, we can conclude that $T_{0}$ is
consistent in $V$, contradicting our assumption.

(3) Let $\rho$ be a strong compactness cardinal for $\mathcal{L}^{2}$. By
Theorem 1.1, there exists an extendible cardinal $\kappa\leq\rho$. In this
situation, Theorem 1.1 together with (2) implies that $\kappa$ is an outward
compactness cardinal for $\mathcal{L}^{2}$, and we can apply (1) to conclude
that $\rho$ also has this property. ∎

We now present the characterization of measurable cardinals that realizes the
approach outlined above. The arguments appearing in its proof already contain
many of the key ideas utilized in the later sections of this paper. Some parts
of these arguments already appeared in a slightly different form in the naïve
approach outlined above, but will be repeated within the proof below for the
sake of clarity of presentation.

######  Theorem 1.5.

A cardinal $\kappa$ is an outward compactness cardinal for $\mathcal{L}^{2}$
if and only if there exists a measurable cardinal less than or equal to
$\kappa$.

###### Proof.

First, assume that $\kappa$ is a measurable cardinal and $T$ is a
${<}\kappa$-outward consistent $\mathcal{L}^{2}$-theory. Pick a cardinal
$\lambda>\kappa$ with $T\in H_{\lambda}$. Using standard iteration arguments
(see [12, Corollary 19.7(b)]), we find an inner model $M$ and an elementary
embedding ${j}:{V}\longrightarrow{M}$ such that
$\operatorname{crit}(j)=\kappa$ and $j(\kappa)>\lambda$. Pick a sufficiently
large cardinal $\vartheta>j(\kappa)$ such that $j(\vartheta)=\vartheta$ and
$V_{\vartheta}$ is a model of $\operatorname{ZFC}^{*}$. Elementarity ensures
that in $M$, the $\mathcal{L}^{2}$-theory $j(T)$ is ${<}j(\kappa)$-outward
consistent. Let $G$ be $\mathrm{Col}(\omega,\vartheta)$-generic over $V$.
Since $V_{\vartheta}^{M}$ is a model of $\operatorname{ZFC}^{*}$, the
observation after Definition 1.3 shows that $V_{\vartheta}^{M}$ is countable
in $M[G]$. In this situation, our setup ensures that, in $M[G]$, the
$\mathcal{L}^{2}$-theory $j(T)$ is ${<}\lambda$-consistent in every countable
outer $\operatorname{ZFC}^{*}$-model of $V_{\vartheta}^{M}$ in which $\lambda$
is a cardinal. This statement can be formulated by a $\Pi_{1}$-formula using
parameters contained in $H_{\aleph_{1}}^{M[G]}$ and is therefore provably
equivalent to a $\mathbf{\Pi}^{1}_{2}$-statement whose parameters are real
numbers coding the original parameters (see [11, Lemma 25.25]). Therefore,
_Shoenfield absoluteness_ implies that the given statement also holds in
$V[G]$. But, in $V[G]$, the set $V_{\vartheta}^{V}$ is a countable outer
$\operatorname{ZFC}^{*}$-model of $V_{\vartheta}^{M}$ in which $\lambda$ is a
cardinal, and hence $j(T)$ is ${<}\lambda$-consistent in $V_{\vartheta}^{V}$.
Now, note that $j[T]\subseteq j(T)$ is an element of $V_{\vartheta}^{V}$ and
has cardinality less than $\lambda$ in $V_{\vartheta}^{V}$. Therefore, we can
conclude that $j[T]$ is consistent in $V_{\vartheta}^{V}$, and this implies
that $j[T]$ is a consistent $\mathcal{L}^{2}$-theory in $V$. But this also
shows that $T$ is consistent in $V$, because the finitary character of
$\mathcal{L}^{2}$-formulae ensures that we can identify $j[T]$ and $T$ via a
renaming of symbols. These computations show that $\kappa$ is an outward
compactness cardinal for $\mathcal{L}^{2}$ and we can now apply Proposition
1.4.(1) to see that every cardinal greater than $\kappa$ is also an outward
compactness cardinal for $\mathcal{L}^{2}$.

Next, assume that $\kappa$ is an outward compactness cardinal for
$\mathcal{L}^{2}$. Consider the $\mathcal{L}^{2}$-theory $T_{\kappa}$ defined
earlier, i.e.,, we extend the language of set theory by constant symbols
$c_{x}$ for every $x\in V_{\kappa+1}$ and constant symbols $d_{\gamma}$ for
all $\gamma\leq\kappa$, and define $T_{\kappa}$ to consist of:

  * •

The elementary first-order diagram of $V_{\kappa+1}$, making use of the
constant symbols $c_{x}$.

  * •

All (first-order) sentences of the form $d_{\beta}\in d_{\gamma}\in
c_{\kappa}$ for $\beta<\gamma\leq\kappa$.

  * •

The (second-order) sentence stating that the $\in$-relation is well-founded.

######  Claim.

The $\mathcal{L}^{2}$-theory $T_{\kappa}$ is ${<}\kappa$-outward consistent.

###### Proof of the Claim.

Let $\lambda<\kappa$ be a cardinal, let $\vartheta>\kappa$ be a cardinal with
$T_{\kappa}\in V_{\vartheta}$, let $G$ be
$\mathrm{Col}(\omega,\vartheta)$-generic over $V$ and let $N\in V[G]$ be an
outer $\operatorname{ZFC}^{*}$-model of $V_{\vartheta}^{V}$ in which $\lambda$
is a cardinal. Then $V_{\kappa+1}^{V}$ is an element of $N$. Now, work in $N$
and fix a subtheory $T_{0}$ of $T_{\kappa}$ of cardinality less than
$\lambda$. Since $\lambda$ is a cardinal, the axioms of
$\operatorname{ZFC}^{*}$ allow us to construct a model of $T_{0}$ with domain
$V_{\kappa+1}^{V}$ that interprets all constant symbols $c_{x}$ appearing in
$T_{0}$ by the corresponding element $x$, and interprets the (less than
$\lambda$-many) constant symbols $d_{\beta}$ appearing in $T_{0}$ as suitable
elements of $\lambda$. ∎

Using the fact that $\kappa$ is an outward compactness cardinal for
$\mathcal{L}^{2}$, we thus obtain that $T_{\kappa}$ is consistent. The
definition of $T_{\kappa}$ now ensures that $T_{\kappa}$ has a transitive
model $M$ and, by sending $x\in V_{\kappa+1}$ to $c_{x}^{M}$, we obtain a non-
trivial elementary embedding ${j}:{V_{\kappa+1}}\longrightarrow{M}$ with
$j(\kappa)>d_{\kappa}^{M}\geq\kappa$. As above, the existence of such an
embedding implies that there is a measurable cardinal less than or equal to
$\kappa$. ∎

In Sections 2 and 3, we will refine the notion of outward compactness to
isolate analogous characterizations for other types of large cardinal notions
below extendibility. The main result of these sections provides a general
correspondence between objects in this region of the large cardinal hierarchy
and outward compactness principles for second-order logic (see Theorem 3.5)
that we will afterwards apply to several well-studied notions (see Corollary
3.6). This correspondence generalizes both Magidor’s characterization of
extendibility in Theorem 1.1 and the characterization of measurability in
Theorem 1.5. More specifically, using the more general notion of _$\Psi$
-outward compactness_ that will be introduced in Definition 3.3 below, we will
obtain the following characterizations:

######  Theorem 1.6.

  1. (1)

A cardinal $\kappa$ is a $\Psi_{str}$-outward compactness cardinal for
$\mathcal{L}^{2}$ if and only if there is a strong cardinal less than or equal
to $\kappa$.

  2. (2)

A cardinal $\kappa$ is a $\Psi_{sc}$-outward compactness cardinal for
$\mathcal{L}^{2}$ if and only if there is a supercompact cardinal less than or
equal to $\kappa$.

  3. (3)

A cardinal $\kappa$ is a $\Psi_{stc}$-outward compactness cardinal for
$\mathcal{L}^{2}$ if and only if $\kappa$ is $\omega_{1}$-strongly compact.

  4. (4)

A cardinal $\kappa$ is a $\Psi_{ext}$-outward compactness cardinal for
$\mathcal{L}^{2}$ if and only if there is an extendible cardinal less than or
equal to $\kappa$.

We then discuss the naturalness of the obtained large cardinal
characterizations in Section 4, and consider the question which other types of
large cardinal properties below extendibility can be characterized through
outward compactness principles. Finally, in Sections 5 and 6, we extend our
concepts to arbitrary abstract logics and large cardinal properties beyond
extendibility. The main results of these sections will provide a general
correspondence between outward compactness for abstract logics and fragments
of _Vopěnka’s Principle_ (see Theorems 6.9 and 6.10) that directly leads to
outward compactness characterizations for Vopěnka’s Principle and the
principle “  _$\operatorname{Ord}$ is Woodin_” (see Corollaries 6.11 and
6.12). The given correspondence directly generalizes a classical result of
Makowsky in [15] that characterizes the validity of Vopěnka’s Principle
through the existence of strong compactness cardinals for all abstract logics
(see Theorem 6.3). These results will be based on the (even more general)
notion of _$F$ -$\Psi$-outward compactness_ introduced in Definition 6.4
below.

######  Theorem 1.7.

  1. (1)

The following schemes are equivalent over $\operatorname{ZFC}$:

    1. (a)

Vopěnka’s Principle.

    2. (b)

For every natural number $n$ and every abstract logic $\mathcal{L}$, there
exists a $\operatorname{ZFC}_{n}$-$\Psi_{ext}$-outward compactness cardinal
for $\mathcal{L}$.

    3. (c)

For every natural number $n$ and every abstract logic $\mathcal{L}$ with
simple formulas, there exists a $\operatorname{ZFC}_{n}$-$\Psi_{sc}$-outward
compactness cardinal for $\mathcal{L}$.

  2. (2)

The following schemes are equivalent over $\operatorname{ZFC}$:

    1. (a)

$\operatorname{Ord}$ is Woodin.

    2. (b)

For every natural number $n$ and every abstract logic $\mathcal{L}$ with
simple formulas, there exists a $\operatorname{ZFC}_{n}$-$\Psi_{str}$-outward
compactness cardinal for $\mathcal{L}$.

##  2\. $\Psi$-large cardinals

We now introduce a broad framework for large cardinal properties between
measurability and extendibility. Our results will later show that all notions
that fall into this framework can be characterized through compactness
properties of $\mathcal{L}^{2}$. In order to achieve this goal, we first
define a property to express a certain amount of closeness between two
transitive models $M\subseteq N$ of set theory. In the following, we say that
a class is _closed under basic set operations_ if it is closed under taking
pairs, products and intersections. Note that for every class $M$ closed under
basic set operations and every limit ordinal $\lambda$, the set $M\cap
V_{\lambda}$ is closed under basic set operations.

######  Definition 2.1.

  1. (1)

A tuple $\langle N,M,\mu,\nu,\rho\rangle$ of sets is _suitable_ if $M\subseteq
N$ are transitive sets closed under basic set operations with
$M\cap\operatorname{Ord}=N\cap\operatorname{Ord}\in\operatorname{Lim}$ and
$\mu<\nu<\rho\in M$ are limit ordinals with the property that $\mu$ is a
cardinal in $N$.

  2. (2)

A formula $\Psi(v_{0},\ldots,v_{4})$ in the language of set theory is a
_measure of closeness_ if $\operatorname{ZFC}$ proves the following
statements:

    1. (A)

If $\Psi(N,M,\mu,\nu,\rho)$ holds, then the tuple $\langle
N,M,\mu,\nu,\rho\rangle$ is suitable.

    2. (B)

If $\mu$ is a cardinal and $\nu<\rho<\theta$ are limit ordinals strictly
greater than $\mu$, then $\Psi(V_{\theta},V_{\theta},\mu,\nu,\rho)$ holds.

    3. (C)

Given transitive sets $M\subseteq N$ closed under basic set operations with
the property that
$M\cap\operatorname{Ord}=N\cap\operatorname{Ord}\in\operatorname{Lim}$ and
limit ordinals $\mu<\nu<\rho<\theta\in M$, we have

$\Psi(N,M,\mu,\nu,\rho)~{}\longleftrightarrow~{}\Psi(N\cap V_{\theta},M\cap
V_{\theta},\mu,\nu,\rho).$

As a first example of the concept introduced above, let
$\Psi_{ms}(v_{0},\ldots,v_{4})$ denote the canonical formula such that
$\Psi_{ms}(N,M,\mu,\nu,\rho)$ holds if and only if the tuple $\langle
N,M,\mu,\nu,\rho\rangle$ is suitable. It is easy to check that the formula
$\Psi_{ms}$ also satisfies the above properties 2(B) and 2(C):

######  Proposition 2.2.

The formula $\Psi_{ms}$ is a measure of closeness. ∎

We now continue by associating large cardinal properties to measures of
closeness.

######  Definition 2.3.

Given a formula $\Psi(v_{0},\ldots,v_{4})$ in the language of set theory, a
cardinal $\kappa$ is _$\Psi$ -large_ if for all limit ordinals
$\kappa<\eta<\theta$, there are unboundedly many cardinals $\lambda\geq\eta$
with the property that there is a transitive set $M$ and an elementary
embedding ${j}:{V_{\theta+1}}\longrightarrow{M}$ such that $j(\kappa)>\lambda$
and $\Psi(V_{j(\theta)},M\cap V_{j(\theta)},\lambda,j(\kappa),j(\eta))$ holds.

In the following, we want to show that several well-known large cardinal
properties can be characterized through the notion of $\Psi$-largeness, in the
sense that we can associate a measure of closeness $\Psi$ to the given large
cardinal property so that we can prove that some cardinal $\kappa$ is
$\Psi$-large if and only if there is a cardinal with the given property that
is less than or equal to $\kappa$. Before we start to derive these
characterizations, we observe that no large cardinal property that is
implication-wise stronger than extendibility can be characterized in this way.

######  Proposition 2.4.

If $\Psi(v_{0},\ldots,v_{4})$ is a measure of closeness, then every cardinal
greater than or equal to an extendible cardinal is $\Psi$-large.

###### Proof.

Assume that $\mu$ is an extendible cardinal, $\kappa\geq\mu$ is a cardinal,
$\kappa<\eta<\theta$ are limit ordinals and $\lambda>\theta$ is a cardinal.
Then there exists an ordinal $\rho$ and an elementary embedding
${i}:{V_{\lambda}}\longrightarrow{V_{\rho}}$ with $\operatorname{crit}(i)=\mu$
and $i(\mu)>\lambda$. If we now define

${j=i\restriction
V_{\theta+1}}:{V_{\theta+1}}\longrightarrow{V_{i(\theta)+1}},$

then $j(\kappa)>\lambda$ and 2(B) in Definition 2.1.2 ensures that
$\Psi(V_{j(\theta)},V_{j(\theta)},\lambda,j(\kappa),j(\eta))$ holds. These
computations allow us to conclude that $\kappa$ is $\Psi$-large. ∎

Next, we use standard arguments to show that $\Psi$-largeness can also be
characterized through the existence of elementary embeddings from $V$ into
inner models.

######  Lemma 2.5.

Given a measure of closeness $\Psi(v_{0},\ldots,v_{4})$, a cardinal $\kappa$
is $\Psi$-large if and only if for every limit ordinal $\eta>\kappa$, there
are unboundedly many cardinals $\lambda\geq\eta$ with the property that there
is an inner model $M$ and an elementary embedding ${j}:{V}\longrightarrow{M}$
such that $j(\kappa)>\lambda$ and
$\Psi(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds for
all limit ordinals $\vartheta>j(\eta)$.

###### Proof.

First, assume that $\kappa$ is $\Psi$-large and fix limit ordinals
$\zeta>\eta>\kappa$. Pick a sufficiently large fixed point $\nu>\zeta$ of the
$\beth$-function. By our assumption, we find a cardinal $\lambda\geq\zeta$, a
transitive set $M_{0}$, and an elementary embedding
${i}:{V_{\nu+1}}\longrightarrow{M_{0}}$, such that $i(\kappa)>\lambda$ and
$\Psi(V_{i(\nu)},M_{0}\cap V_{i(\nu)},\lambda,i(\kappa),i(\eta))$ holds. Since
$\nu$ is a strong limit cardinal, we can use standard arguments (see [12,
Lemma 26.1]) to find an inner model $M$ with $V_{i(\nu)}^{M}=M_{0}\cap
V_{i(\nu)}$, and an elementary embedding ${j}:{V}\longrightarrow{M}$
satisfying

$j\restriction(V_{\nu}\cup\\{\nu\\})~{}=~{}i\restriction(V_{\nu}\cup\\{\nu\\}).$

That is, we define $E$ to be the $(\kappa,i(\nu))$-extender derived from $i$,
defining $E_{a}$ for every $a\in[i(\nu)]^{<\omega}$ by

$X\in E_{a}\ \iff\ X\in\mathcal{P}([\nu]^{|a|})\,\land\,a\in i(X),$

and let ${j}:{V}\longrightarrow{M}$ be the ultrapower embedding of $V$ induced
by $E$. Let ${j^{\prime}}:{V_{\nu+1}}\longrightarrow{M}^{\prime}$ be the
ultrapower embedding of $V_{\nu+1}$ induced by $E$. Since $V$ and $V_{\nu+1}$
have the same subsets of $\nu$, it follows that $M$ and $M^{\prime}$ agree up
to $j(\nu)=j^{\prime}(\nu)$, and that $j$ and $j^{\prime}$ agree up to $\nu$.
Let ${k}:{M^{\prime}}\longrightarrow{M_{0}}$ be the factor embedding
satisfying that $i=k\circ j^{\prime}$. Then, by [12, Lemma 26.1], we first
obtain that $\operatorname{crit}(k)>j(\nu)$, yielding that
$i(\nu)=j^{\prime}(\nu)=j(\nu)$, and then consequently that
$V_{j(\nu)}^{M}=V_{j(\nu)}\cap M_{0}$. If $\vartheta>j(\eta)$ is a limit
ordinal, we apply 2(C) in Definition 2.1.2 to conclude that
$\Psi(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds.
Summing up, this shows that we have found our desired embedding
${j}:{V}\longrightarrow{M}$.

In the other direction, assume that for some limit ordinal $\eta>\kappa$,
there are unboundedly many cardinals $\lambda\geq\eta$ such that there exists
an inner model $M$ and an elementary embedding ${j}:{V}\longrightarrow{M}$ for
which $j(\kappa)>\lambda$ and
$\Psi(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds for
all limit ordinals $\vartheta>j(\eta)$. Fix a limit ordinal $\theta>\eta$ and
an ordinal $\zeta>\eta$. Pick a cardinal $\lambda\geq\zeta$, an inner model
$M$ and an elementary embedding ${j}:{V}\longrightarrow{M}$ for which
$j(\kappa)>\lambda$ and
$\Psi(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds for
all limit ordinals $\vartheta>j(\eta)$. Then, $M_{0}=V^{M}_{j(\theta)+1}$ is a
transitive set, and ${i=j\restriction
V_{\theta+1}}:{V_{\theta+1}}\longrightarrow{M_{0}}$ is an elementary embedding
for which $i(\kappa)>\lambda$ and $\Psi(V_{i(\theta)},M_{0}\cap
V_{i(\theta)},\lambda,i(\kappa),i(\eta))$ holds. These computations yield the
converse implication. ∎

Using the above equivalence, we can now easily show that the formula
$\Psi_{ms}$ canonically corresponds to the large cardinal notion of
measurability.

######  Corollary 2.6.

A cardinal $\kappa$ is $\Psi_{ms}$-large if and only if there is a measurable
cardinal less than or equal to $\kappa$.

###### Proof.

First, assume that $\kappa$ is $\Psi_{ms}$-large. By Lemma 2.5, there is a
transitive class $M$ and an elementary embedding ${j}:{V}\longrightarrow{M}$
with $j(\kappa)>\kappa$. In particular, we know that the critical point
$\operatorname{crit}(j)$ of $j$ is a measurable cardinal less than or equal to
$\kappa$. In the other direction, assume that there is a measurable cardinal
less than or equal to $\kappa$ and $\lambda>\kappa$ is a cardinal. Standard
arguments about iterated ultrapowers (see [12, Corollary 19.7(b)]) now allow
us to find an inner model $M$ and an elementary embedding
${j}:{V}\longrightarrow{M}$ with $j(\kappa)>\lambda$. Then it is easy to see
that $\Psi_{ms}(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$
holds for every limit ordinal $\kappa<\eta\leq\lambda$ and every limit ordinal
$\vartheta>j(\eta)$. By Lemma 2.5, this shows that the cardinal $\kappa$ is
$\Psi_{ms}$-large. ∎

In the remainder of this section, we will show that several other important
large cardinal notions in the region between measurability and extendibility
canonically correspond to measures of closeness.

###  2.1. Strong cardinals

Let $\Psi_{str}(v_{0},\ldots,v_{4})$ denote the canonical suitable formula in
the language of set theory with the property that
$\Psi_{str}(N,M,\mu,\nu,\rho)$ holds if and only if the tuple $\langle
N,M,\mu,\nu,\rho\rangle$ is suitable and $N\cap V_{\mu}\subseteq M$ holds. The
following statement is then immediate:

######  Proposition 2.7.

The formula $\Psi_{str}$ is a measure of closeness. ∎

In the following, we will show that the formula $\Psi_{str}$ corresponds to
the large cardinal concept of strongness. These and later arguments will make
crucial use of the following well-known consequence of the _Kunen
Inconsistency_ (see, for example, [12, Corollary 23.14]):

######  Lemma 2.8.

Let $M$ be a transitive class and let ${j}:{V}\longrightarrow{M}$ be a non-
trivial elementary embedding. If $\lambda$ is a limit ordinal of uncountable
cofinality with $V_{\lambda}\subseteq M$, then there exists a natural number
$n$ with $j^{n}(\operatorname{crit}(j))\geq\lambda$.

###### Proof.

Assume, towards a contradiction, that the above conclusion fails and define

$\zeta~{}=~{}\sup_{n<\omega}j^{n}(\operatorname{crit}(j)).$

Then, $\zeta$ is a limit ordinal of countable cofinality with
$j(\zeta)=\zeta$, and our assumptions ensure that $\zeta+2<\lambda$. But this
implies that the map ${j\restriction
V_{\zeta+2}}:{V_{\zeta+2}}\longrightarrow{V_{\zeta+2}}$ is a non-trivial
elementary embedding, contradicting the _Kunen Inconsistency_. ∎

######  Lemma 2.9.

A cardinal $\kappa$ is $\Psi_{str}$-large if and only if there is a strong
cardinal less than or equal to $\kappa$.

###### Proof.

First, assume that some cardinal $\mu\leq\kappa$ is strong, and
$\lambda>\kappa$ is a cardinal. In this situation, there exists an inner model
$M$ and an elementary embedding ${j}:{V}\longrightarrow{M}$ with
$\operatorname{crit}(j)=\mu$, $j(\mu)>\lambda$ and $V_{\lambda}\subseteq M$.
Then, $j(\kappa)>\lambda$, and it is easy to see that
$\Psi_{str}(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds
for all limit ordinals $\kappa<\eta\leq\lambda$ and all limit ordinals
$\vartheta>j(\eta)$. By Lemma 2.5, this shows that $\kappa$ is
$\Psi_{str}$-large.

In the other direction, assume that $\kappa$ is $\Psi_{str}$-large, and no
cardinal less than or equal to $\kappa$ is strong. Then, there exists a
regular cardinal $\eta>\kappa$ with the property that no cardinal less than or
equal to $\kappa$ is $\eta$-strong. Using Lemma 2.5, we find a cardinal
$\lambda>\eta$, an inner model $M$ with $V_{\lambda}\subseteq M$ and an
elementary embedding ${j}:{V}\longrightarrow{M}$ with $j(\kappa)>\lambda$. Set
$\mu=\operatorname{crit}(j)<\kappa$ and $\mu_{i}=j^{i}(\mu)$ for all
$i<\omega$. Lemma 2.8 yields a least natural number $n$ with $\mu_{n}>\eta$.
Since $\mu<\eta$, we know that $n>0$, and the fact that $\mu$ is not
$\eta$-strong then implies that $n>1$.

######  Claim.

In $M$, the cardinal $\mu_{1}$ is superstrong with target
$\mu_{n}$.666Remember that a cardinal $\kappa$ is _superstrong with target
$\lambda$_ if there is a transitive class $M$ with $V_{\lambda}\subseteq M$
and an elementary embedding ${j}:{V}\longrightarrow{M}$ with
$\operatorname{crit}(j)=\kappa$ and $j(\kappa)=\lambda$.

###### Proof of the Claim.

By induction, we show that, in $M$, for all $0<i<n$, the cardinal $\mu_{1}$ is
superstrong with target $\mu_{i+1}$. Since $n>1$, the embedding $j$ witnesses
that $\mu$ is superstrong with target $\mu_{1}$ in $V$ and hence elementarity
of $j$ ensures that, in $M$, the cardinal $\mu_{1}$ is superstrong with target
$\mu_{2}$. Next, assume that $i<n-1$ has the property that the cardinal
$\mu_{1}$ is superstrong with target $\mu_{i+1}$ in $M$. Then there exists a
transitive class $N\subseteq M$ with $V^{M}_{\mu_{i+1}}\subseteq N$ and an
elementary embedding ${i}:{M}\longrightarrow{N}$ with
$\operatorname{crit}(i)=\mu_{1}$ and $i(\mu_{1})=\mu_{i+1}$. Since $i+1<n$, we
have $\mu_{i+1}\leq\eta$ and hence $V_{\mu_{i+1}}\subseteq N$. Moreover, the
map ${i\circ j}:{V}\longrightarrow{N}$ is an elementary embedding with
$\operatorname{crit}(i\circ j)=\mu$ and $(i\circ j)(\mu)=\mu_{i+1}$. In
particular, this map witnesses that $\mu$ is superstrong with target
$\mu_{i+1}$ in $V$. This allows us to conclude that, in $M$, the cardinal
$\mu_{1}$ is superstrong with target $\mu_{i+2}$. This argument completes the
proof of the claim. ∎

By the above claim, there exists a transitive class $N\subseteq M$ with
$V^{M}_{\mu_{n}}\subseteq N$ and an elementary embedding
${i}:{M}\longrightarrow{N}$ with $\operatorname{crit}(i)=\mu_{1}$ and
$i(\mu_{1})=\mu_{n}$. But then, $V_{\lambda}\subseteq N$ and ${i\circ
j}:{V}\longrightarrow{N}$ is an elementary embedding with
$\operatorname{crit}(i\circ j)=\mu$ and $(i\circ j)(\mu)>\eta$, yielding that
$\mu$ is $\eta$-strong, a contradiction. ∎

###  2.2. Supercompact cardinals

Let $\Psi_{sc}(v_{0},\ldots,v_{4})$ denote the canonical formula in the
language of set theory with the property that $\Psi_{sc}(N,M,\mu,\nu,\rho)$
holds if and only if the tuple $\langle N,M,\mu,\nu,\rho\rangle$ is suitable
and ${}^{\mu}\rho\cap N\subseteq M$ holds.

######  Proposition 2.10.

The formula $\Psi_{sc}$ is a measure of closeness. ∎

We now show that this formula corresponds to the large cardinal concept of
supercompactness.

######  Lemma 2.11.

A cardinal $\kappa$ is $\Psi_{sc}$-large if and only if there is a
supercompact cardinal less than or equal to $\kappa$.

###### Proof.

First, assume that $\mu\leq\kappa$ is a supercompact cardinal, and
$\lambda>\kappa$ is a cardinal. Then, there exists a transitive class $M$ with
${}^{\lambda}M\subseteq M$ and an elementary embedding
${j}:{V}\longrightarrow{M}$ with $\operatorname{crit}(j)=\mu$ and
$j(\mu)>\lambda$. If $\kappa<\eta\leq\lambda$ is a limit ordinal, and
$\vartheta>j(\eta)$ is a limit ordinal, then ${}^{{<}\lambda}j(\eta)\subseteq
V_{\vartheta}^{M}$ and this shows that
$\Psi_{sc}(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds.
By Lemma 2.5, this shows that $\kappa$ is $\Psi_{sc}$-large.

Now, assume that $\kappa$ is $\Psi_{sc}$-large, and no cardinal less than or
equal to $\kappa$ is supercompact. Pick a fixed point $\eta>\kappa$ of the
$\beth$-function of uncountable cofinality with the property that no cardinal
less than or equal to $\kappa$ is $\eta$-supercompact, and apply Lemma 2.5 to
find a cardinal $\lambda>\eta$, a transitive class $M$ with
${}^{\lambda}j(\eta)\subseteq M$ and an elementary embedding
${j}:{V}\longrightarrow{M}$ with $j(\kappa)>\lambda$. Define
$\mu=\operatorname{crit}(j)<\kappa$, and $\mu_{i}=j^{i}(\mu)$ for all
$i<\omega$. Then, our choice of $\eta$ ensures that $V_{\eta}\subseteq M$, and
we can use Lemma 2.8 to find a minimal natural number $n>1$ with
$\mu_{n}>\eta$.

######  Claim.

The cardinal $\mu$ is (n-1)-huge with targets
$\mu_{1},\ldots,\mu_{n-1}$.777Remember that, given a natural number $n>0$, a
cardinal $\kappa$ is _$n$ -huge with targets $\lambda_{1},\ldots,\lambda_{n}$_
if there is a transitive class $M$ with ${}^{\lambda_{n}}M\subseteq M$ and an
elementary embedding ${j}:{V}\longrightarrow{M}$ with
$\operatorname{crit}(j)=\kappa$ and $j(\lambda_{m})=\lambda_{m+1}$ for all
$m<n$.

###### Proof.

Since we have $\mu_{n-1}\leq\eta<\lambda$, the fact that
${}^{\lambda}j(\eta)\subseteq M$ implies that $j[\mu_{n-1}]\in M$, and hence,
we know that

$\\{A\subseteq\mu_{n-1}~{}|~{}j[\mu_{n-1}]\in j(A)\\}$

is a ${<}\mu$-complete normal ultrafilter over $P(\mu_{n-1})$ that contains
the set

$\\{A\subseteq\mu_{n-1}~{}|~{}\text{$\operatorname{ot}(A\cap\mu_{i})=\mu_{i-1}$
for all $0<i<n$}\\}.$

By [12, Theorem 24.8], this implies the statement of the claim. ∎

By the above claim, we know that $\mu$ is huge with target $\mu_{n-1}$, and
hence, elementarity implies that, in $M$, the cardinal $\mu_{1}$ is huge with
target $\mu_{n}$. Therefore, we find a transitive class $N\subseteq M$
definable in $M$ with $M\cap{}^{\mu_{n}}N\subseteq N$, and an elementary
embedding ${i}:{M}\longrightarrow{N}$ definable in $M$ with
$\operatorname{crit}(i)=\mu_{1}$ and $i(\mu_{1})=\mu_{n}$. Since
$j[\eta]\in{}^{\eta}j(\eta)\subseteq M$ and $\eta<\mu_{n}$, we then know that

$(i\circ j)[\eta]~{}=~{}i[j[\eta]]~{}\in~{}M\cap{}^{\eta}N~{}\subseteq~{}N.$

This shows that ${i\circ j}:{V}\longrightarrow{N}$ is an elementary embedding
with $\operatorname{crit}(i\circ j)=\mu$, $(i\circ j)(\mu)>\eta$ and $(i\circ
j)[\eta]\in N$. But this contradicts the fact that $\mu$ is not
$\eta$-supercompact. ∎

###  2.3. Extendible cardinals

Let $\Psi_{ext}(v_{0},\ldots,v_{4})$ denote the canonical formula in the
language of set theory with the property that $\Psi_{ext}(N,M,\mu,\nu,\rho)$
holds if and only if the tuple $\langle N,M,\mu,\nu,\rho\rangle$ is suitable
and $N\cap V_{\rho}\subseteq M$ holds.

######  Proposition 2.12.

The formula $\Psi_{ext}$ is a measure of closeness. ∎

The following result shows that the formula $\Psi_{ext}$ corresponds to
extendibility. In order to motivate the concepts that will be introduced in
the next chapter, we use Theorem 1.1 to prove the backward implication of the
stated equivalence.888It would also be possible to establish this equivalence
with the help of [16, Theorem 2.28], which shows that a cardinal $\kappa$ is
extendible if and only if it is _jointly $\lambda$-supercompact and
$\lambda$-superstrong_ for all $\lambda>\kappa$ (see [16, Definition 2.24]).
However, the proof of the equivalence that we present, using the connections
between second-order logic and extendibility, provides an elegant alternative
to an adaptation of the proof of Lemma 2.11 to these large cardinal
properties.

######  Lemma 2.13.

A cardinal $\kappa$ is $\Psi_{ext}$-large if and only if there is an
extendible cardinal less than or equal to $\kappa$.

###### Proof.

First, note that by Proposition 2.4, every cardinal greater than or equal to
an extendible cardinal is $\Psi_{ext}$-large. In the other direction, assume
that $\kappa$ is $\Psi_{ext}$-large and let $T$ be a ${<}\kappa$-consistent
$\mathcal{L}^{2}$-theory. Pick a cardinal $\eta>\kappa$ with $T\in
V_{\eta}\prec_{\Sigma_{2}}V$. By our assumption, we can apply Lemma 2.5 to
find a cardinal $\lambda>\eta$, a transitive class $M$ and an elementary
embedding ${j}:{V}\longrightarrow{M}$ such that $j(\kappa)>\lambda$ and
$V_{j(\eta)}\subseteq M$. We then know that
$V_{j(\eta)}=V^{M}_{j(\eta)}\prec_{\Sigma_{2}}M$, and $j(T)\in V_{j(\eta)}$ is
a ${<}j(\kappa)$-consistent $\mathcal{L}^{2}$-theory in $M$. Moreover, our
assumptions on $M$ ensure that $j[T]$ is an element of $M$ and has cardinality
less than $j(\kappa)$ in $M$. But this implies that $j[T]$ is consistent in
$M$, and the above observations show that $V_{j(\eta)}$ contains a structure
that is a model of $j[T]$ in $M$. We conclude that $j[T]$ is also consistent
in $V$ and, since we can identify the theories $T$ and $j[T]$ through a
renaming of symbols, this shows that $T$ is consistent in $V$. These
computations show that $\kappa$ is a strong compactness cardinal for
$\mathcal{L}^{2}$, and hence, Theorem 1.1 ensures that there exists an
extendible cardinal less than or equal to $\kappa$. ∎

###  2.4. $\omega_{1}$-strongly compact cardinals

In [3] and [4], Bagaria and Magidor defined a cardinal $\kappa>\omega_{1}$ to
be _$\omega_{1}$ -strongly compact_ if for every set $I$, every
${<}\kappa$-complete filter on $I$ can be extended to a countably complete
ultrafilter on $I$. Note that every cardinal above an $\omega_{1}$-strongly
compact cardinal is also $\omega_{1}$-strongly compact. In the following, we
show that the naïve attempt to characterize strongly compact cardinals through
the concept of $\Psi$ -largeness (motivated by [12, Theorem 22.17]) actually
leads to a characterization of $\omega_{1}$-strongly compact cardinals. This
characterization differs strongly from the ones presented above, because e.g.,
the first $\omega_{1}$-strongly compact cardinal can be singular (see [3,
Theorem 6.1]). Let $\Psi_{stc}(v_{0},\ldots,v_{4})$ denote the canonical
formula in the language of set theory with the property that
$\Psi_{stc}(N,M,\mu,\nu,\rho)$ holds if and only if the tuple $\langle
N,M,\mu,\nu,\rho\rangle$ is suitable and for every $d\in N\cap P(\rho)$ with
the property that $N$ contains no injection of $\mu$ into $d$, there exists
$c\in M\cap P(\rho)$ such that $d\subseteq c$ and $M$ does not contain an
injection of $\nu$ into $c$.999Note that these assumptions are phrased in this
way because Definition 2.1.2 only makes minimal assumptions on the closure
properties of the sets $M$ and $N$.

######  Proposition 2.14.

The formula $\Psi_{stc}$ is a measure of closeness.

###### Proof.

The formula $\Psi_{stc}$ trivially satisfies 2(A) and 2(B) in Definition
2.1.2. Assume that $M\subseteq N$ are transitive sets closed under basic set
operations with
$M\cap\operatorname{Ord}=N\cap\operatorname{Ord}\in\operatorname{Lim}$ and
$\mu<\nu<\rho<\theta\in M$ are limit ordinals. Since $M$ and $M\cap
V_{\theta}$ (respectively, $N$ and $N\cap V_{\theta}$) contain the same
subsets of $\rho$, the same injections from $\mu$ into $\rho$ and the same
injections from $\nu$ into $\rho$, it follows directly that
$\Psi_{stc}(N,M,\mu,\nu,\rho)$ holds if and only if $\Psi_{stc}(N\cap
V_{\theta},M\cap V_{\theta},\mu,\nu,\rho)$ holds. This shows that $\Psi_{stc}$
also satisfies 2(C) in Definition 2.1.2. ∎

######  Lemma 2.15.

A cardinal is $\omega_{1}$-strongly compact if and only if it is
$\Psi_{stc}$-large.

###### Proof.

First, assume that $\kappa$ is an $\omega_{1}$-strongly compact cardinal, and
let $\lambda>\kappa$ be a cardinal. The proof of [3, Theorem 4.7] then shows
that there is a $\sigma$-complete fine ultrafilter $\mathcal{U}$ on
$P_{\kappa}(\lambda)$. Let ${j}:{V}\longrightarrow{M}$ denote the ultrapower
embedding induced by $\mathcal{U}$, and fix $d\subseteq j(\lambda)$ of
cardinality at most $\lambda$. Then there exists a sequence

$\langle{{f_{\gamma}}:{P_{\kappa}(\lambda)}\longrightarrow{\lambda}}~{}|~{}{\gamma<\lambda}\rangle$

of functions with $d=\\{[f_{\gamma}]_{\mathcal{U}}~{}|~{}\gamma<\lambda\\}$.
Define

${F}:{P_{\kappa}(\lambda)}\longrightarrow{P_{\kappa}(\lambda)};~{}{a}\longmapsto{\\{f_{\gamma}(a)~{}|~{}\gamma\in
a\\}}.$

Given $\gamma<\lambda$, we then have

$\\{a\in P_{\kappa}(\lambda)~{}|~{}f_{\gamma}(a)\in
F(a)\\}~{}\supseteq~{}\\{a\in P_{\kappa}(\lambda)~{}|~{}\gamma\in
a\\}~{}\in~{}\mathcal{U},$

and this implies that $[f_{\gamma}]_{\mathcal{U}}\in[F]_{\mathcal{U}}$.
Moreover, since $F(a)$ has cardinality less than $\kappa$ for each $a\in
P_{\kappa}(\lambda)$, we know that $[F]_{\mathcal{U}}$ has cardinality less
than $j(\kappa)$ in $M$. In particular, if $d$ has cardinality $\lambda$, then
$\lambda=|d|\leq|d|^{M}\leq|[F]_{\mathcal{U}}|^{M}<j(\kappa)$. These
computations show that
$\Psi_{stc}(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds
for every limit ordinal $\kappa<\eta\leq\lambda$ and every limit ordinal
$\vartheta>j(\eta)$. Using Lemma 2.5, this allows us to conclude that $\kappa$
is $\Psi_{stc}$-large.

Now, assume that $\kappa$ is $\Psi_{stc}$-large, and let $\eta>\kappa$ be a
limit ordinal. Lemma 2.5 allows us to find a cardinal $\lambda>\eta$, a
transitive class $M$ and an elementary embedding ${j}:{V}\longrightarrow{M}$
such that $j(\kappa)>\lambda$ and
$\Psi_{stc}(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds
for all limit ordinals $\vartheta>j(\eta)$. Since $j[\eta]$ is a subset of
$j(\eta)$ of cardinality less than $\lambda$, we can now find $b\in
P(j(\eta))^{M}$ such that $j[\eta]\subseteq b$ and $b$ has cardinality less
than $j(\kappa)$ in $M$. By [3, Theorem 4.7], these computations show that
$\kappa$ is $\omega_{1}$-strongly compact. ∎

##  3\. $\mathcal{L}^{2}$-characterizations

We now connect the concepts introduced in the previous section to compactness
properties of second-order logic. The next definition provides the first step
to establish these connections. In this definition, we consider simply
definable measures of closeness with the additional property that closeness of
a transitive model $M$ to $V_{M\cap\operatorname{Ord}}$ is expressible over
$M$ by an $\mathcal{L}^{2}$-statement. Remember that, given an $\in$-theory
$F$ and a natural number $n>0$, a set-theoretic formula
$\varphi(v_{0},\ldots,v_{m-1})$ is a _$\Delta_{n}^{F}$ -formula_ if there is a
$\Sigma_{n}$-formula $\psi_{0}(v_{0},\ldots,v_{m-1})$ and a $\Pi_{n}$-formula
$\psi_{1}(v_{0},\ldots,v_{m-1})$ with the property that

$F~{}\vdash~{}\forall
x_{0},\ldots,x_{m-1}~{}[\varphi(x_{0},\ldots,x_{m-1})\longleftrightarrow\psi_{i}(x_{0},\ldots,x_{m-1})]$

holds for all $i<2$.

######  Definition 3.1.

A formula $\Psi(v_{0},\ldots,v_{4})$ in the language of set theory is an
_$\mathcal{L}^{2}$ -measure of closeness_ if the following statements hold:

  1. (1)

$\Psi$ is a measure of closeness.

  2. (2)

$\Psi$ is both a $\Delta_{1}^{\operatorname{ZFC}^{*}}$\- and a
$\Delta_{1}^{\operatorname{ZFC}^{-}}$-formula.101010As usual, we let
$\operatorname{ZFC}^{-}$ denote the axioms of $\operatorname{ZFC}$ without the
Powerset axiom and with the Collection scheme instead of the Replacement
scheme.

  3. (3)

There exists an $\mathcal{L}^{2}$-formula $\Psi^{*}(v_{0},v_{1},v_{2})$ in the
language of set theory such that $\operatorname{ZFC}^{*}$ proves that for all
limit ordinals $\mu<\nu<\rho<\theta$ and every transitive set $M$ with
$\theta\subseteq M\subseteq V_{\theta}$, the statement
$\Psi(V_{\theta},M,\mu,\nu,\rho)$ holds if and only if $\langle
M,\in\rangle\models_{\mathcal{L}^{2}}\Psi^{*}(\mu,\nu,\rho)$ holds.

######  Proposition 3.2.

The formulas $\Psi_{ms}$, $\Psi_{str}$, $\Psi_{sc}$, $\Psi_{stc}$ and
$\Psi_{ext}$ are all $\mathcal{L}^{2}$-measures of closeness.

###### Proof.

First, note that there is a first-order sentence in the language of set theory
that holds in a transitive set if and only if the given set is closed under
basic set operations. Moreover, it is easy to find an
$\mathcal{L}^{2}$-formula $\varphi(v)$ with the property that
$\operatorname{ZFC}^{*}$ proves that for every transitive set $M$ closed under
basic set operations and every $a\in M$, we have $\langle
M,\in\rangle\models_{\mathcal{L}^{2}}\varphi(a)$ if and only if $a$ is an
element of $M\cap\operatorname{Ord}$ that is a infinite cardinal in $V$. In
combination with the fact that the formalized satisfaction relation can be
defined by both $\Delta_{1}^{\operatorname{ZFC}^{*}}$\- and
$\Delta_{1}^{\operatorname{ZFC}^{-}}$-formulas, this shows that $\Psi_{ms}$ is
an $\mathcal{L}^{2}$-measure of closeness. There is a canonical
$\mathcal{L}^{2}$-formula $\varphi(v_{0},v_{1})$ such that
$\operatorname{ZFC}^{*}$ proves that for every transitive set $M$ closed under
basic set operations and all $\alpha,\beta\in M\cap\operatorname{Ord}$, we
have $\langle M,\in\rangle\models_{\mathcal{L}^{2}}\varphi(\alpha,\beta)$ if
and only if the set ${}^{\alpha}\beta$ is a subset of $M$, and this shows that
$\Psi_{sc}$ is an $\mathcal{L}^{2}$-measure of closeness. There is an
$\mathcal{L}^{2}$-formula $\varphi(v_{0},v_{1},v_{2})$ such that
$\operatorname{ZFC}^{*}$ proves that for every transitive set $M$ closed under
basic set operations and all $\alpha,\beta,\gamma\in M\cap\operatorname{Ord}$,
we have $\langle
M,\in\rangle\models_{\mathcal{L}^{2}}\varphi(\alpha,\beta,\gamma)$ if and only
if for every subset $d$ of $\gamma$ of cardinality less than $\alpha$, there
exists $c\in M$ such that $d\subseteq c\subseteq\gamma$ and $M$ contains no
injection from $\beta$ into $c$, and this directly shows that $\Psi_{stc}$ is
an $\mathcal{L}^{2}$-measure of closeness.

For the remaining formulas, recall that an argument of Magidor in [13, Proof
of Theorem 2] shows that there is an $\mathcal{L}^{2}$-sentence $\psi$ in the
language of set theory with the property that for every transitive set $M$, we
have $\langle M,\in\rangle\models_{\mathcal{L}^{2}}\psi$ if and only if
$M=V_{M\cap\operatorname{Ord}}$. In addition, this $\mathcal{L}^{2}$-sentence
$\psi$ can be chosen in a way that this equivalence is provable in
$\operatorname{ZFC}^{*}$ (see [7, Proof of Fact 2.1]). This directly allows us
to find an $\mathcal{L}^{2}$-formula $\varphi(v_{0},v_{1})$ in the language of
set theory with the property that $\operatorname{ZFC}^{*}$ proves that for
every transitive set $M$ and all $a,b\in M$, we have $\langle
M,\in\rangle\models_{\mathcal{L}^{2}}\varphi(a,b)$ if and only if $a\in
M\cap\operatorname{Ord}$ and $b=V_{a}$. In combination with the fact that the
rank function is definable by both $\Delta_{1}^{\operatorname{ZFC}^{*}}$\- and
$\Delta_{1}^{\operatorname{ZFC}^{-}}$-formulas, we can conclude that the
formulas $\Psi_{str}$ and $\Psi_{ext}$ are both $\mathcal{L}^{2}$-measures of
closeness. ∎

The second step to connect $\Psi$-large cardinals to second-order logic is
given by the family of compactness properties introduced in the next
definition.

######  Definition 3.3.

Let $\Psi(v_{0},\ldots,v_{4})$ be a formula in the language of set theory and
let $\kappa$ be an infinite cardinal.

  1. (1)

An $\mathcal{L}^{2}$-theory $T$ is _$\Psi$ -outward consistent at $\kappa$_ if
there is a limit ordinal $\eta>\kappa$ such that for all infinite cardinals
$\lambda<\kappa$ and all cardinals $\vartheta>\eta$ with $T\in H_{\vartheta}$,
the partial order $\mathrm{Col}(\omega,\vartheta)$ forces that $T$ is
${<}\lambda$-consistent in every outer $\operatorname{ZFC}^{*}$-model $N$ of
$V_{\vartheta}^{V}$ with the property that
$\Psi(N,V_{\vartheta}^{V},\lambda,\kappa,\eta)$ holds.

  2. (2)

A cardinal $\kappa$ is a _$\Psi$ -outward compactness cardinal for
$\mathcal{L}^{2}$_ if all $\mathcal{L}^{2}$-theories that are $\Psi$-outward
consistent at $\kappa$ are consistent.

As noted after Definition 1.3 above, if $\vartheta$ is a limit ordinal, $G$ is
$\mathrm{Col}(\omega,\vartheta)$-generic over $V$ and $N$ is an outer
$\operatorname{ZFC}^{*}$-model of $V_{\vartheta}^{V}$ in $V[G]$, then both $N$
and $V_{\vartheta}^{V}$ are countable in $V[G]$. Moreover, it is easy to see
that, given an $\mathcal{L}^{2}$-theory $T$ and an infinite cardinal $\kappa$,
the theory $T$ is ${<}\kappa$-outward consistent (as defined in Definition
1.3) if and only if it is $\Psi_{ms}$-outward consistent at $\kappa$. In the
following, we make a further basic observation about the above notions that
relate them to strong compactness cardinals for $\mathcal{L}^{2}$.

######  Proposition 3.4.

Let $\Psi(v_{0},\ldots,v_{4})$ be a $\Delta_{1}^{\operatorname{ZFC}}$-formula
that is a measure of closeness. If $\kappa$ is a limit cardinal, then every
$\mathcal{L}^{2}$-theory that is $\Psi$-outward consistent at $\kappa$ is
${<}\kappa$-consistent. In particular, every cardinal greater than or equal to
an extendible cardinal is a $\Psi$-outward compactness cardinal for
$\mathcal{L}^{2}$.

###### Proof.

Let $T$ be an $\mathcal{L}^{2}$-theory that contains an inconsistent subtheory
$T_{0}$ of cardinality less than $\kappa$, and let $\eta>\kappa$ be a limit
ordinal. Pick a cardinal $\vartheta>\eta$ such that $T\in
H_{\vartheta}=V_{\vartheta}$ and $V_{\vartheta}$ is sufficiently elementary in
$V$. In addition, pick a cardinal $\lambda<\kappa$ with $|T_{0}|<\lambda$.
Then 2(B) in Definition 2.1.2 ensures that
$\Psi(V_{\vartheta},V_{\vartheta},\lambda,\kappa,\eta)$ holds. Moreover, since
$T$ is an element of $V_{\vartheta}$ and $V_{\vartheta}$ was chosen to be
sufficiently elementary in $V$, we also know that $T$ is not
${<}\lambda$-consistent in $V_{\vartheta}$. Now, let $G$ be
$\mathrm{Col}(\omega,\vartheta)$-generic over $V$. Then $V_{\vartheta}^{V}$ is
an outer $\operatorname{ZFC}^{*}$-model of $V_{\vartheta}^{V}$ in $V[G]$.
Moreover, since $\Psi$ is a $\Delta_{1}^{\operatorname{ZFC}}$-formula, we can
use $\Sigma_{1}$-upwards absoluteness to infer that
$\Psi(V_{\vartheta}^{V},V_{\vartheta}^{V},\lambda,\kappa,\eta)$ holds in
$V[G]$. Therefore, $T$ is not $\Psi$-outward consistent at $\kappa$ in $V$. ∎

We now combine the above two concepts to prove a general duality theorem that
connects $\Psi$-largeness with $\Psi$-outwards compactness. Together with our
earlier results, this can be seen as a generalization of Theorem 1.5 (see also
Corollary 3.6 below). For the proof of this result, we inductively define _the
second-order relativisation $\Psi|v$_ of an $\mathcal{L}^{2}$-formula in the
language of set theory to some first-order variable $v$ by setting ${(v_{0}\in
v_{1})|v}\equiv v_{0}\in v_{1}\in v$, ${(u\in W)|v}\equiv u\in W\subseteq v$,
${(\neg\varphi)|v}\equiv{\neg(\varphi|v)}$,
${(\varphi\wedge\psi)|v}\equiv{(\varphi|v)\wedge(\psi|v)}$, ${(\forall
x~{}\varphi(x))|v}\equiv{\forall x\in v~{}(\varphi|v)(x)}$ and ${(\forall
X~{}\varphi(X))|v}\equiv{\forall X\subseteq v~{}(\varphi|v)(X)}$. Note that we
consider the resulting formulas in this inductive definition as
$\mathcal{L}^{2}$-formulas in each case. Moreover, if
$\varphi(v_{0},\ldots,v_{n-1})$ is an $\mathcal{L}^{2}$-formula, then
$\operatorname{ZFC}^{*}$ proves that $\langle
M,\in\rangle\models_{\mathcal{L}^{2}}(\varphi|v_{n})(c_{0},\ldots,c_{n-1},d)$
is equivalent to $\langle
d,\in\rangle\models_{\mathcal{L}^{2}}\varphi(c_{0},\ldots,c_{n-1})$ whenever
$M$ is a transitive set, $d\in M$ is transitive and $c_{0},\ldots,c_{n-1}\in
d$.

######  Theorem 3.5.

Given an $\mathcal{L}^{2}$-measure of closeness $\Psi$, an infinite cardinal
$\kappa$ is $\Psi$-large if and only if it is a $\Psi$-outward compactness
cardinal for $\mathcal{L}^{2}$.

###### Proof.

First, assume that $\kappa$ is $\Psi$-large. Let $T$ be an
$\mathcal{L}^{2}$-theory that is $\Psi$-outward consistent at $\kappa$, and
let $\eta>\kappa$ be a limit ordinal witnessing this. Using Lemma 2.5, we can
find a cardinal $\lambda>\eta$ with $T\in H_{\lambda}$, an inner model $M$ and
an elementary embedding ${j}:{V}\longrightarrow{M}$ such that
$j(\kappa)>\lambda$ and
$\Psi(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds for
all limit ordinals $\vartheta>j(\eta)$. Pick a sufficiently large cardinal
$\vartheta>j(\lambda)$ such that $j(\vartheta)=\vartheta$,
$H_{\vartheta}=V_{\vartheta}$ and the set $V_{\vartheta}$ is a model of
$\operatorname{ZFC}^{*}$. Elementarity now implies that, in $M$, the cardinal
$j(\eta)$ witnesses that the $\mathcal{L}^{2}$-theory $j(T)$ is $\Psi$-outward
consistent at $j(\kappa)$. Let $G$ be $\mathrm{Col}(\omega,\vartheta)$-generic
over $V$. Since $V_{\vartheta}^{M}$ is a model of $\operatorname{ZFC}^{*}$,
$V_{\vartheta}^{M}$ is countable in $M[G]$. Moreover, our setup ensures that,
in $M[G]$, the $\mathcal{L}^{2}$-theory $j(T)$ is ${<}\lambda$-consistent in
every countable outer $\operatorname{ZFC}^{*}$-model $N$ of
$V_{\vartheta}^{M}$ with the property that
$\Psi(N,V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds. Since our
assumptions on $\Psi$ imply that this statement can be formulated by a
$\Pi_{1}$-formula with parameters in $H_{\aleph_{1}}^{M[G]}$ and is therefore
provably equivalent to a $\mathbf{\Pi}^{1}_{2}$-statement whose parameters are
real numbers canonically coding the original parameters (see [11, Lemma
25.25]), _Shoenfield absoluteness_ implies that the given statement also holds
in $V[G]$. But $\Sigma_{1}$-upwards absoluteness implies that
$\Psi(V_{\vartheta}^{V},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ also
holds in $V[G]$ and, since $V_{\vartheta}^{V}$ is a countable outer
$\operatorname{ZFC}^{*}$-model of $V_{\vartheta}^{M}$ in $V[G]$, we conclude
that $j(T)$ is ${<}\lambda$-consistent in $V_{\vartheta}^{V}$. Now, note that
$j[T]\subseteq j(T)$ is an element of $V_{\vartheta}^{V}$, and this set has
cardinality less than $\lambda$ in $V_{\vartheta}^{V}$. Thus, the theory
$j[T]$ is consistent in $V_{\vartheta}^{V}$ and, by the nature of
$\mathcal{L}^{2}$, we now know that $j[T]$ is a consistent
$\mathcal{L}^{2}$-theory in $V$. But this also shows that $T$ is consistent in
$V$, because the finitary character of $\mathcal{L}^{2}$-formulae ensures that
we can identify $j[T]$ and $T$ via a renaming of symbols induced by $j$.

Now, assume that $\kappa$ is a $\Psi$-outward compactness cardinal for
$\mathcal{L}^{2}$. Let $\theta>\eta>\kappa$ be limit ordinals and let
$\zeta>\eta$ be a cardinal. In addition, let $\Psi^{*}(v_{0},v_{1},v_{2})$ be
the $\mathcal{L}^{2}$-formula corresponding to $\Psi$ as in Definition 3.1.3
and let $\Psi^{\prime}(v_{0},\ldots,v_{3})$ denote the second-order
relativisation $\Psi^{*}|v_{3}$ of $\Psi^{*}(v_{0},v_{1},v_{2})$ to $v_{3}$.
Consider the language that extends the language of set theory by a constant
symbol $b$, constant symbols $c_{x}$ for all elements $x$ of $V_{\theta+1}$
and constant symbols $d_{\gamma}$ for all $\gamma\leq\zeta$. Let $T$ denote
the second-order theory consisting of the following:

  1. (1)

The first-order elementary diagram of $V_{\theta+1}$, using the constant
symbols $c_{x}$ for $x$ in $V_{\theta+1}$.

  2. (2)

All first-order sentences of the form ${\text{``}\hskip
1.29167pt{d_{\beta}<d_{\gamma}<c_{\kappa}}\text{''}}$ for
$\beta<\gamma\leq\zeta$.

  3. (3)

The second-order sentence stating that the $\in$-relation is well-founded.

  4. (4)

The first-order sentence ${\text{``}\hskip
1.29167pt{d_{\zeta}<b<c_{\kappa}}\text{''}}$.

  5. (5)

The second-order sentence
$\Psi^{\prime}(b,c_{\kappa},c_{\eta},c_{V_{\theta}})$.

######  Claim.

The ordinal $\eta$ witnesses that $T$ is $\Psi$-outwards consistent at
$\kappa$.

###### Proof of the Claim.

Let $\lambda<\kappa$ be a cardinal, let $\vartheta>\eta$ be a cardinal with
$T\in H_{\vartheta}$, let $G$ be $\mathrm{Col}(\omega,\vartheta)$-generic over
$V$ and let $N\in V[G]$ be an outer $\operatorname{ZFC}^{*}$-model of
$V_{\vartheta}^{V}$ with the property that
$\Psi(N,V_{\vartheta}^{V},\lambda,\kappa,\eta)$ holds in $V[G]$. Since $N$ is
a model of $\operatorname{ZFC}^{*}$ and $T\in H_{\vartheta}^{V}$ implies that
$\theta<\vartheta$, we can apply 2(C) in Definition 2.1.2 to infer that
$\Psi(V_{\theta}^{N},V_{\theta}^{V},\lambda,\kappa,\eta)$ holds in $V[G]$.
Moreover, since $\Psi$ is a $\Delta_{1}^{\operatorname{ZFC}^{*}}$-formula and
all occurring parameters are elements of $N$, we know that
$\Psi(V_{\theta}^{N},V_{\theta}^{V},\lambda,\kappa,\eta)$ also holds in $N$.
In addition, the fact that $N$ is a model of $\operatorname{ZFC}^{*}$ ensures
that

$\langle
V_{\theta}^{V},\in\rangle\models_{\mathcal{L}^{2}}\Psi^{*}(\lambda,\kappa,\eta)$

holds in $N$ and hence we know that

$\langle
V_{\theta+1}^{V},\in\rangle\models_{\mathcal{L}^{2}}\Psi^{\prime}(\lambda,\kappa,\eta,V_{\theta}^{V})$

holds in $N$.

Now, let $T_{0}$ be a subtheory of $T$ in $N$ that has cardinality less than
$\lambda$ in $N$. Since $\lambda$ is a cardinal in $N$, we can construct a
model of $T_{0}$ with domain $V_{\theta+1}^{V}$ in $N$ that interprets $b$ as
$\lambda$ and all constant symbols of the form $d_{\gamma}$ that appear in
sentences in $T_{0}$ as ordinals less than $\lambda$. ∎

Our setup now ensures that $T$ is consistent. Hence, we find a transitive set
$M$ and an elementary embedding ${j}:{V_{\theta+1}}\longrightarrow{M}$ such
that $j(\kappa)>\zeta$ and

$\langle
M,\in\rangle\models_{\mathcal{L}^{2}}\Psi^{\prime}(\zeta,j(\kappa),j(\eta),j(V_{\theta})).$

Elementarity then implies that $M\cap\operatorname{Ord}=j(\theta)+1$ and
$j(V_{\theta})=M\cap V_{j(\theta)}$ is transitive. In particular, we know that

$\langle M\cap
V_{j(\theta)},\in\rangle\models_{\mathcal{L}^{2}}\Psi^{*}(\zeta,j(\kappa),j(\eta)),$

and therefore we can conclude that $\Psi(V_{j(\theta)},M\cap
V_{j(\theta)},\zeta,j(\kappa),j(\eta))$ holds. These computations allow us to
conclude that $\kappa$ is $\Psi$-large. ∎

In combination with the results of the previous section, the above theorem now
shows that all of the large cardinal properties that we considered so far can
be characterized through compactness properties of second-order logic.

######  Corollary 3.6.

  1. (1)

A cardinal $\kappa$ is a $\Psi_{ms}$-outward compactness cardinal for
$\mathcal{L}^{2}$ if and only if there is a measurable cardinal less than or
equal to $\kappa$.

  2. (2)

A cardinal $\kappa$ is a $\Psi_{str}$-outward compactness cardinal for
$\mathcal{L}^{2}$ if and only if there is a strong cardinal less than or equal
to $\kappa$.

  3. (3)

A cardinal $\kappa$ is a $\Psi_{sc}$-outward compactness cardinal for
$\mathcal{L}^{2}$ if and only if there is a supercompact cardinal less than or
equal to $\kappa$.

  4. (4)

A cardinal $\kappa$ is a $\Psi_{stc}$-outward compactness cardinal for
$\mathcal{L}^{2}$ if and only if $\kappa$ is $\omega_{1}$-strongly compact.

  5. (5)

A cardinal $\kappa$ is a $\Psi_{ext}$-outward compactness cardinal for
$\mathcal{L}^{2}$ if and only if there is an extendible cardinal less than or
equal to $\kappa$. ∎

By Proposition 3.4, extendible cardinals are $\Psi$-outward compactness
cardinals for $\mathcal{L}^{2}$ whenever $\Psi$ is an
$\mathcal{L}^{2}$-measure of closeness. In particular, it is not possible to
use Theorem 3.5 to characterize large cardinal notions stronger than
extendibility. Therefore, we may view $\Psi_{ext}$ as the strongest
$\mathcal{L}^{2}$-measure of closeness. This view is further supported by the
following observation:

######  Proposition 3.7.

If $\kappa$ is an infinite cardinal, then every ${<}\kappa$-consistent
$\mathcal{L}^{2}$-theory is $\Psi_{ext}$-outward consistent.

###### Proof.

Let $T$ be a ${<}\kappa$-consistent $\mathcal{L}^{2}$-theory. Pick a cardinal
$\eta>\kappa$ with the property that $T\in H_{\eta}=V_{\eta}$ and $V_{\eta}$
is sufficiently elementary in $V$. Fix a cardinal $\lambda<\kappa$ and a
cardinal $\vartheta>\eta$, and let $G$ be
$\mathrm{Col}(\omega,\vartheta)$-generic over $V$. Pick an outer
$\operatorname{ZFC}^{*}$-model $N\in V[G]$ of $V_{\vartheta}^{V}$ with the
property that $\Psi_{ext}(N,V_{\vartheta}^{V},\lambda,\kappa,\eta)$ holds in
$V[G]$. We then have $V_{\eta}^{N}=V_{\eta}^{V}$. Let $T_{0}$ be a subtheory
of $T$ of cardinality less than $\lambda$ in $N$. Then, $T_{0}$ is contained
in $V$ and has cardinality less than $\lambda$ in $V$. Hence, we know that
$T_{0}$ is consistent in $V$ and our setup ensures that $T_{0}$ is also
consistent in $V_{\eta}^{V}$. Since $V_{\eta}^{N}=V_{\eta}^{V}$, this shows
that $T_{0}$ is also consistent in $N$. These computations prove that $T$ is
${<}\lambda$-consistent in $N$. ∎

##  4\. On the naturalness of $\mathcal{L}^{2}$-characterizations

The results of the previous two sections naturally raise the question which
other large cardinal notions between measurability and extendibility can be
characterized through outward compactness properties of $\mathcal{L}^{2}$. In
particular, Corollary 3.6.4 directly motivates the question whether strong
compactness can be characterized in this way. Below, we will show that this is
indeed possible. But, we will also argue that the presented characterization
lacks several desirable features that the characterizations listed in
Corollary 3.6 possess.

Given a formula $\psi(v_{0},v_{1},v_{2})$ in the language of set theory, we
let $\psi^{{}_{\uparrow}}(v_{0},\ldots,v_{4})$ denote the canonical formula in
the language of set theory with the property that
$\psi^{{}_{\uparrow}}(N,M,\mu,\nu,\rho)$ holds if and only if the tuple
$\langle N,M,\mu,\nu,\rho\rangle$ is suitable and either $N\cap
V_{\rho}\subseteq M$ or $\psi(\mu,\nu,\rho)$ holds in $M\cap V_{\rho+\omega}$.

######  Proposition 4.1.

Given a formula $\psi(v_{0},v_{1},v_{2})$ in the language of set theory, the
formula $\psi^{{}_{\uparrow}}$ is an $\mathcal{L}^{2}$-measure of closeness.

###### Proof.

The above definition directly ensures that $\psi^{{}_{\uparrow}}$ satisfies
2(A) and 2(B) in Definition 2.1.2. Fix transitive sets $M\subseteq N$ closed
under basic set operations with
$M\cap\operatorname{Ord}=N\cap\operatorname{Ord}\in\operatorname{Lim}$ and
limit ordinals $\mu<\nu<\rho<\theta\in M$. First, assume that
$\psi^{{}_{\uparrow}}(N,M,\mu,\nu,\rho)$ holds. In the first case, if $N\cap
V_{\rho}\subseteq M$ holds, then $N\cap V_{\theta}\cap V_{\rho}\subseteq M\cap
V_{\theta}$ holds and it follows that $\psi^{{}_{\uparrow}}(N\cap
V_{\theta},M\cap V_{\theta},\mu,\nu,\rho)$ is also true. Next, assume that
$\psi(\mu,\nu,\rho)$ holds in $V_{\rho+\omega}\cap M$. Since
$\rho+\omega\leq\theta$ and $V_{\rho+\omega}\cap(M\cap
V_{\theta})=V_{\rho+\omega}\cap M$, we then know that
$\psi^{{}_{\uparrow}}(N\cap V_{\theta},M\cap V_{\theta},\mu,\nu,\rho)$ also
holds in this case. Since an analogous case distinction also shows that
$\psi^{{}_{\uparrow}}(N\cap V_{\theta},M\cap V_{\theta},\mu,\nu,\rho)$ implies
that $\psi^{{}_{\uparrow}}(N,M,\mu,\nu,\rho)$ holds, we can now conclude that
$\psi^{{}_{\uparrow}}$ satisfies 2(C) in Definition 2.1.2 and hence we know
that (1) of Definition 3.1 holds. By arguing as in the proof of Proposition
3.2, we see that (2) of Definition 3.1 holds. Finally, it is easy to find an
$\mathcal{L}^{2}$-formula $\varphi(v,W)$ in the language of set theory such
that $\operatorname{ZFC}^{*}$ proves that for every transitive set $D$ and
every ordinal $\alpha$ in $D$, the set $D\cap V_{\alpha}$ is the unique
element $X$ of $\mathcal{P}(D)$ with the property that $\langle
D,\in\rangle\models_{\mathcal{L}^{2}}\varphi(\alpha,X)$ holds. Therefore, it
follows that $\psi^{{}_{\uparrow}}$ also satisfies (3) of Definition 3.1. ∎

We now show how the above concept can be used to obtain characterizations of
strongly compact cardinals through compactness properties of
$\mathcal{L}^{2}$. Let $\psi_{stc}(v_{0},v_{1},v_{2})$ denote the canonical
formula in the language of set theory with the property that
$\psi_{stc}(\mu,\nu,\rho)$ holds if and only if for some cardinal
$\mu<\kappa\leq\nu$, there exists a fine ultrafilter on
$\mathcal{P}_{\kappa}(\rho)$.

######  Proposition 4.2.

A cardinal $\kappa$ is $\psi_{stc}^{{}_{\uparrow}}$-large if and only if there
is a strongly compact cardinal less than or equal to $\kappa$. In particular,
a cardinal $\kappa$ is a $\psi_{stc}^{{}_{\uparrow}}$-outward compactness
cardinal for $\mathcal{L}^{2}$ if and only if there is a strongly compact
cardinal less than or equal to $\kappa$.

###### Proof.

First, assume that $\mu$ is a strongly compact cardinal and $\kappa\geq\mu$ is
a cardinal. Fix a limit ordinal $\eta>\kappa$ and a cardinal
$\lambda\geq\eta$. By [12, Corollary 22.18], there exists a fine ultrafilter
on $\mathcal{P}_{\mu}(\eta)$ and therefore $\psi_{stc}(\delta,\kappa,\eta)$
holds in $V_{\eta+\omega}$ for every cardinal $\delta<\mu$. Since $\mu$ is
measurable, we can find a transitive class $M$ and an elementary embedding
${j}:{V}\longrightarrow{M}$ with $j(\mu)>\lambda$. By elementarity, we know
that $\psi_{stc}(\lambda,j(\kappa),j(\eta))$ holds in
$V_{j(\eta)+\omega}^{M}$. This directly implies that
$\psi_{str}^{{}_{\uparrow}}(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$
holds for all limit ordinals $\vartheta>j(\eta)$. Using Lemma 2.5, we can now
conclude that $\kappa$ is $\psi_{stc}^{{}_{\uparrow}}$-large.

Now, assume, towards a contradiction, that $\kappa$ is a
$\psi_{stc}^{{}_{\uparrow}}$-large cardinal and no cardinal less than or equal
to $\kappa$ is strongly compact. Then there exists an ordinal $\alpha>\kappa$
with the property that for every cardinal $\mu\leq\kappa$, there is no fine
ultrafilter on $\mathcal{P}_{\mu}(\alpha)$. Fix a limit ordinal $\eta>\alpha$.
By Lemma 2.5, we can now find a cardinal $\lambda>\eta$, an inner model $M$
and an elementary embedding ${j}:{V}\longrightarrow{M}$ with
$j(\kappa)>\lambda$ and the property that
$\psi_{stc}^{{}_{\uparrow}}(V_{j(\eta)+\omega},V_{j(\eta)+\omega}^{M},\lambda,j(\kappa),j(\eta))$
holds. Since $\eta>\alpha$, we know that for all cardinals $\mu\leq\kappa$,
there is no fine ultrafilter on $\mathcal{P}_{\mu}(\eta)$. This shows that for
all $\delta<\kappa$, the statement $\psi_{stc}(\delta,\kappa,\eta)$ does not
hold in $V_{\eta+\omega}$. By elementarity,
$\psi_{stc}(\lambda,j(\kappa),j(\eta))$ fails in $V_{j(\eta)+\omega}^{M}$ and
it follows that
$\Psi_{ext}(V_{j(\eta)+\omega},V_{j(\eta)+\omega}^{M},\lambda,j(\kappa),j(\eta))$
holds. In particular, we have $V_{j(\eta)}\subseteq M$. These computations
show that for every limit ordinal $\eta>\kappa$, there is an inner model $M$
and an elementary embedding ${j}:{V}\longrightarrow{M}$ with $j(\kappa)>\eta$
and $V_{j(\eta)}\subseteq M$. This directly implies that $\kappa$ is
$\Psi_{ext}$-large and hence Lemma 2.13 ensures that there is an extendible
cardinal less than or equal to $\kappa$, contradicting our assumption. ∎

Note that the above arguments can be modified to obtain analogous
characterizations for various types of large cardinal properties defined
through the existence of certain filters. Nevertheless, we consider the
outward compactness characterizations obtained in this way as unsatisfying,
because, except for ensuring the suitability of certain tuples, the measures
of closeness $\psi^{{}_{\uparrow}}$ utilized in these characterization only
rely on properties of the ground model $V$ and are therefore almost completely
unaffected by the relationship between $V$ and the given outer model. In the
remainder of this section, we aim to isolate criteria for the naturalness of
outward compactness characterizations that allow us to separate the
characterizations obtained in Section 3 from the characterization of strong
compactness derived above.

Our first criterion is motivated by Propositions 3.4 and 3.7 which, in
conjunction, show that ${<}\kappa$-consistency provably coincides with
$\Psi_{ext}$-outward consistency at all limit cardinals $\kappa$. Moreover, if
$\Psi$ is an $\mathcal{L}^{2}$-measure of closeness and $\kappa$ is a
$\Psi$-large cardinal that is not extendible, then Theorems 1.1 and 3.5 show
that there is a ${<}\kappa$-consistent $\mathcal{L}^{2}$-theory that is not
$\Psi$-outward consistent at $\kappa$. Therefore, it seems reasonable to
expect that $\mathcal{L}^{2}$-measures of closeness $\Psi$ that canonically
correspond to large cardinal notions strictly weaker than extendibility have
the property that $\Psi$-outwards compactness differs from
${<}\kappa$-consistency at some cardinal $\kappa$. This intuition is captured
in the following definition:

######  Definition 4.3.

We say that a formula $\Psi(v_{0},\ldots,v_{4})$ in the language of set theory
_naturally induces a large cardinal property below extendibility_ if
$\operatorname{ZFC}$ proves that for some infinite cardinal $\kappa$, there is
a ${<}\kappa$-consistent $\mathcal{L}^{2}$-theory that is not $\Psi$-outward
consistent at $\kappa$.

A short argument now shows that the formulae introduced in Section 2 that
correspond to large cardinal notions below extendibility fulfill the above
criterion:

######  Proposition 4.4.

If $\Psi$ is one of the formulae $\Psi_{ms}$, $\Psi_{str}$, $\Psi_{stc}$ or
$\Psi_{sc}$, then there is an $\mathcal{L}^{2}$-sentence $\varphi$ in the
language of set theory such that $\\{\varphi\\}$ is a consistent
$\mathcal{L}^{2}$-theory that is not $\Psi$-outward consistent at any
uncountable cardinal $\kappa$.

###### Proof.

There are $\mathcal{L}^{2}$-sentences $\varphi_{0}$ and $\varphi_{1}$ in the
language of set theory such that $\operatorname{ZFC}^{*}$ proves the following
statements:

  * •

If $i<2$ and $\langle M,E\rangle\models_{\mathcal{L}^{2}}\varphi_{i}$, then
$E$ is a well-founded and extensional relation on $E$ and the corresponding
transitive collapse maps $M$ to $V_{\omega_{2}+1}$.

  * •

We have

$2^{\aleph_{1}}=\aleph_{2}~{}\Longleftrightarrow~{}\langle
V_{\omega_{2}+1},\in\rangle\models_{\mathcal{L}^{2}}\varphi_{0}~{}\Longleftrightarrow~{}\langle
V_{\omega_{2}+1},\in\rangle\models_{\mathcal{L}^{2}}\neg\varphi_{1}.$

Now, fix $i<2$ with $\langle
V_{\omega_{2}+1},\in\rangle\models_{\mathcal{L}^{2}}\varphi_{i}$. Note that
there is a $\sigma$-closed partial order $\mathbb{P}$ with the property that
whenever $G$ is $\mathbb{P}$-generic over $V$, then $\langle
V_{\omega_{2}+1}^{V[G]},\in\rangle\models\neg\varphi_{i}$ holds in $V[G]$.

Assume, towards a contradiction, that $\eta$ is a limit ordinal that witnesses
that the theory $\\{\varphi_{i}\\}$ is $\Psi$-outward consistent at some
uncountable cardinal $\kappa$. Pick a cardinal $\vartheta>\eta$ such that
$\mathbb{P}$ is an element of $H_{\vartheta}$ and $\mathbb{P}$ forces that
$V_{\vartheta}$ is a model of $\operatorname{ZFC}^{*}$. Let $H$ be
$\mathrm{Col}(\omega,\vartheta)$-generic over $V$. Then we can find $G\in
V[H]$ that is $\mathbb{P}$-generic over $V$ and our setup ensures that
$V_{\vartheta}^{V[G]}$ is an outer $\operatorname{ZFC}^{*}$-model of
$V_{\vartheta}^{V}$ in $V[H]$. Moreover, since $\mathbb{P}$ is $\sigma$-closed
in $V$, it is easy to check that
$\Psi(V_{\vartheta}^{V[G]},V_{\vartheta}^{V},\omega,\kappa,\eta)$ holds in
$V[H]$. It now follows that $\\{\varphi_{i}\\}$ is consistent in
$V_{\vartheta}^{V[G]}$ and this implies that $\langle
V^{V[G]}_{\omega_{2}+1},\in\rangle\models_{\mathcal{L}^{2}}\varphi_{i}$ holds
in $V[G]$, a contradiction. ∎

######  Corollary 4.5.

The formulae $\Psi_{ms}$, $\Psi_{str}$, $\Psi_{stc}$ and $\Psi_{sc}$ all
naturally induce a large cardinal property below extendibility. ∎

As a second criterion for the naturalness of outward compactness
characterizations, we want to demand that the given characterization actually
depends on the consistency of theories in outer models and does not only check
the validity of statements in initial segments of $V$. This criterion will be
given by the negation of the property introduced in the next definition. It
will turn out to be a weakening of our first criterion.

######  Definition 4.6.

Given a formula $\Psi(v_{0},\ldots,v_{4})$ in the language of set theory, we
say that _$\Psi$ -outward compactness trivializes at a cardinal $\kappa$_ if
there is a proper class of limit ordinals $\eta>\kappa$ with the property that
for all cardinals $\lambda<\kappa$ and all limit ordinals $\vartheta>\eta$,
the partial order $\mathrm{Col}(\omega,\vartheta)$ forces that $N\cap
V_{\eta}^{V[G]}\subseteq V$ holds for every outer
$\operatorname{ZFC}^{*}$-model $N$ of $V_{\vartheta}^{V}$ with the property
that $\Psi(N,V_{\vartheta}^{V},\lambda,\kappa,\eta)$ holds.

As a first observation, we show that the above property causes
$\Psi$-largeness to be equivalent to $\Psi_{ext}$-largeness:

######  Proposition 4.7.

Let $\Psi(v_{0},\ldots,v_{4})$ be an $\mathcal{L}^{2}$-measure of closeness.
If $\Psi$-outward compactness trivializes at a $\Psi$-large cardinal $\kappa$,
then there exists an extendible cardinal less than or equal to $\kappa$.

###### Proof.

Assume, towards a contradiction, that no cardinal less than or equal to
$\kappa$ is extendible. By [12, Proposition 23.15], we can find an ordinal
$\alpha>\kappa$ with the property that for every ordinal $\beta$, there is no
non-trivial elementary embedding ${j}:{V_{\alpha}}\longrightarrow{V_{\beta}}$
with $\operatorname{crit}(j)\leq\kappa$. Then there exists a limit ordinal
$\eta>\alpha$ such that for all cardinals $\lambda<\kappa$ and all limit
ordinals $\vartheta>\eta$, the partial order $\mathrm{Col}(\omega,\vartheta)$
forces that $N\cap V_{\eta}^{V[G]}\subseteq V$ holds for every outer
$\operatorname{ZFC}^{*}$-model $N$ of $V_{\vartheta}^{V}$ with the property
that $\Psi(N,V_{\vartheta}^{V},\lambda,\kappa,\eta)$ holds. Using Lemma 2.5,
we now find a cardinal $\lambda>\eta$, an inner model $M$ and an elementary
embedding ${j}:{V}\longrightarrow{M}$ such that $j(\kappa)>\lambda$ and
$\Psi(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds for
every limit ordinal $\vartheta>j(\eta)$. Pick a limit ordinal
$\vartheta>j(\eta)$ with the property that $j(\vartheta)=\vartheta$ and
$V_{\vartheta}$ is a model of $\operatorname{ZFC}^{*}$. Let $G$ be
$\mathrm{Col}(\omega,\vartheta)$-generic over $V$. By using the elementarity
of $j$ and Shoenfield absoluteness as in the proof of Theorem 3.5, it now
follows that, in $V[G]$, we have $N\cap V_{j(\eta)}^{V[G]}\subseteq
V_{\vartheta}^{M}$ whenever $N$ is an outer $\operatorname{ZFC}^{*}$-model of
$V_{\vartheta}^{M}$ with the property that
$\Psi(N,V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds. Since
$\alpha<j(\eta)$ and our assumptions on $\Psi$ ensure that
$\Psi(V_{\vartheta}^{V},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds in
both $V$ and $V[G]$, we can now conclude that $V_{j(\eta)}^{V}\subseteq M$ and
${j\restriction V_{\alpha}}:{V_{\alpha}}\longrightarrow{V_{j(\alpha)}}$ is a
non-trivial elementary embedding with $\operatorname{crit}(j\restriction
V_{\alpha})\leq\kappa$, contradicting our initial assumption. ∎

We now show how our two criteria are related:

######  Lemma 4.8.

Let $\Psi(v_{0},\ldots,v_{4})$ be a formula in the language of set theory. If
$\Psi$-outward compactness trivializes at an infinite cardinal $\kappa$, then
every ${<}\kappa$-consistent $\mathcal{L}^{2}$-theory is $\Psi$-outward
consistent at $\kappa$.

###### Proof.

Let $T$ be a ${<}\kappa$-consistent $\mathcal{L}^{2}$-theory. Then there
exists a limit ordinal $\eta>\kappa$ such that the following statements hold:

  * •

$T\in H_{\rho}$ for some cardinal $\rho<\eta$.

  * •

Every subtheory of $T$ of cardinality less than $\kappa$ has a model in
$V_{\eta}$.

  * •

For all cardinals $\lambda<\kappa$ and all limit ordinals $\vartheta>\eta$,
the partial order $\mathrm{Col}(\omega,\vartheta)$ forces that $N\cap
V_{\eta+1}^{V[G]}\subseteq V$ holds for every outer
$\operatorname{ZFC}^{*}$-model $N$ of $V_{\vartheta}^{V}$ with the property
that $\Psi(N,V_{\vartheta}^{V},\lambda,\kappa,\eta)$ holds.

Now, let $\lambda<\kappa$ be a cardinal, let $\vartheta>\eta$ be a cardinal,
let $G$ be $\mathrm{Col}(\omega,\vartheta)$-generic over $V$ and let $N$ be an
outer $\operatorname{ZFC}^{*}$-model of $V_{\vartheta}^{V}$ in $V[G]$ with the
property that $\Psi(N,V_{\vartheta}^{V},\lambda,\kappa,\eta)$ holds in $V[G]$.
We then know that $N\cap V_{\eta}^{V[G]}\subseteq V$ and therefore every
subtheory of $T$ of cardinality less than $\kappa$ in $N$ is contained in $V$
and has cardinality less than $\kappa$ in $V$. In particular, the theory $T$
is ${<}\lambda$-consistent in $N$. ∎

######  Corollary 4.9.

If $\Psi(v_{0},\ldots,v_{4})$ is a formula in the language of set theory that
naturally induces a large cardinal property below extendibility, then
$\operatorname{ZFC}$ proves that $\Psi$-outward compactness does not
trivialize at all infinite cardinals. ∎

We are now ready to show that, in contrast to the characterizations derived in
Section 3, the characterization of strong compactness given by Proposition 4.2
does not meet the above criteria.

######  Proposition 4.10.

If there are no strongly compact cardinals, then
$\psi_{stc}^{{}_{\uparrow}}$-outward compactness trivializes at all infinite
cardinals.

###### Proof.

Fix an infinite cardinal $\kappa$. Then there is a limit ordinal $\eta>\kappa$
with the property that for every infinite cardinal $\mu\leq\kappa$, there is
no fine ultrafilter on $\mathcal{P}_{\mu}(\eta)$. Let $\lambda<\kappa$ be a
cardinal, let $\vartheta>\kappa$ be a cardinal, let $G$ be
$\mathrm{Col}(\omega,\vartheta)$-generic over $V$ and let $N$ be an outer
$\operatorname{ZFC}^{*}$-model of $V_{\vartheta}^{V}$ in $V[G]$ with the
property that
$\psi_{stc}^{{}_{\uparrow}}(N,V_{\vartheta}^{V},\lambda,\kappa,\eta)$ holds in
$V[G]$. Since $\psi_{stc}(\lambda,\kappa,\eta)$ does not hold in
$V_{\eta+\omega}^{V}$, we can now conclude that $N\cap
V_{\eta}^{V[G]}\subseteq V$ holds. ∎

######  Corollary 4.11.

If the formula $\psi_{str}^{{}_{\uparrow}}$ naturally induces a large cardinal
property below extendibility, then $\operatorname{ZFC}$ is inconsistent. ∎

##  5\. Fragments of Vopěnka’s Principle

We now aim to extend the equivalence established in Theorem 3.5 to obtain
characterizations of stronger large cardinal assumptions. For this purpose, we
start by introducing a natural strengthening of the the notion of
$\Psi$-largeness that is motivated by classical characterizations of Woodin
cardinals (see [12, Theorem 26.14]) and related variations of supercompactness
and extendibility (see [12, p. 339]).

######  Definition 5.1.

Given a formula $\Psi(v_{0},\ldots,v_{4})$ in the language of set theory and a
class $A$,111111Note that, since we are working in a $\operatorname{ZFC}$
context, all classes are definable. a cardinal $\kappa$ is _$\Psi$ -large for
$A$_ if for all limit ordinals $\kappa<\eta<\theta$, there are unboundedly
many cardinals $\lambda\geq\eta$ with the property that there is a transitive
set $M$ and an elementary embedding ${j}:{V_{\theta+1}}\longrightarrow{M}$
such that $j(\kappa)>\lambda$,

$j(A\cap V_{\kappa})\cap V_{\lambda}~{}=~{}A\cap M\cap V_{\lambda}$

and $\Psi(V_{j(\theta)},M\cap V_{j(\theta)},\lambda,j(\kappa),j(\eta))$ holds.

In the following, we present the first example of a large cardinal principle
that we aim to represent through the above concept. Remember that _Vopěnka’s
Principle_ is a scheme of axioms stating that for every proper class
$\mathcal{C}$ of structures of the same type, there exist $A,B\in\mathcal{C}$
with $A\neq B$ and an elementary embedding from $A$ to $B$. Using results from
[1] and [2], it is easy to show that Vopěnka’s Principle provides various
examples of cardinals with the given property.

######  Proposition 5.2.

If Vopěnka’s Principle holds, then for every measure of closeness
$\Psi(v_{0},\ldots,v_{4})$ and every class $A$, there is a proper class of
cardinals that are $\Psi$-large for $A$.

###### Proof.

Pick a natural number $n>0$, a $\Sigma_{n}$-formula $\varphi(v_{0},v_{1})$,
and a set $y$ with the property that $A=\\{x~{}|~{}\varphi(x,y)\\}$, and pick
an ordinal $\xi$. Using [2, Corollary 6.9], we find a cardinal $\kappa>\xi$,
with $y\in V_{\kappa}$, that is _$C^{(n)}$ -extendible_, i.e., the cardinal
$\kappa$ has the property that for every $\lambda>\kappa$ with
$V_{\lambda}\prec_{\Sigma_{n}}V$, there exists an ordinal $\zeta$ and an
elementary embedding ${j}:{V_{\lambda}}\longrightarrow{V_{\zeta}}$ with
$\operatorname{crit}(j)=\kappa$, $j(\kappa)>\lambda$ and
$V_{j(\kappa)}\prec_{\Sigma_{n}}V$ (see [1, Definition 3.2]). We can now use
[2, Proposition 3.4] to see that $V_{\kappa}\prec_{\Sigma_{n+2}}V$ and hence
$A\cap V_{\kappa}=\\{x\in V_{\kappa}~{}|~{}V_{\kappa}\models\varphi(x,y)\\}$.
In the following, fix limit ordinals $\theta>\eta>\kappa$ and a cardinal
$\lambda>\theta$ with $V_{\lambda}\prec_{\Sigma_{n}}V$. In this situation, we
can find an ordinal $\zeta$ and an elementary embedding
${j}:{V_{\lambda}}\longrightarrow{V_{\zeta}}$ with
$\operatorname{crit}(j)=\kappa$, $j(\kappa)>\lambda$ and
$V_{j(\kappa)}\prec_{\Sigma_{n}}V$. Elementarity then implies that

$j(A\cap V_{\kappa})\cap V_{\lambda}~{}=~{}\\{x\in
V_{\lambda}~{}|~{}V_{j(\kappa)}\models\varphi(x,y)\\}~{}=~{}A\cap
V_{\lambda}.$

Since 2(B) in Definition 2.1.2 ensures that
$\Psi(V_{j(\theta)},V_{j(\theta)},\lambda,j(\kappa),j(\eta))$ holds, we can
conclude that the embedding ${j\restriction
V_{\theta+1}}:{V_{\theta+1}}\longrightarrow{V_{j(\theta)+1}}$ possesses the
desired properties with respect to $\eta$ and $\lambda$. ∎

Next, we prove the direct analog of Lemma 2.5 for Definition 5.1.

######  Lemma 5.3.

Given a measure of closeness $\Psi(v_{0},\ldots,v_{4})$ and a class $A$, a
cardinal $\kappa$ is $\Psi$-large for $A$ if and only if for every limit
ordinal $\eta>\kappa$, there exist unboundedly many cardinals
$\lambda\geq\eta$ with the property that there is an inner model $M$ and an
elementary embedding ${j}:{V}\longrightarrow{M}$ such that
$j(\kappa)>\lambda$, $j(A\cap V_{\kappa})\cap V_{\lambda}=A\cap M\cap
V_{\lambda}$ and
$\Psi(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds for
all limit ordinals $\vartheta>j(\eta)$.

###### Proof.

First, assume that $\kappa$ is $\Psi$-large for $A$ and $\zeta>\eta>\kappa$
are limit ordinals. We find a sufficiently large fixed point $\nu>\zeta$ of
the $\beth$-function, a cardinal $\lambda\geq\zeta$, a transitive set $M_{0}$
and an elementary embedding ${i}:{V_{\nu+1}}\longrightarrow{M_{0}}$ such that
$i(\kappa)>\lambda$, $i(A\cap V_{\kappa})\cap V_{\lambda}=A\cap M\cap
V_{\kappa}$ and $\Psi(V_{i(\nu)},M_{0}\cap
V_{i(\nu)},\lambda,i(\kappa),i(\eta))$ holds. By repeating the proof of Lemma
2.5, we can construct an inner model $M$ with $V_{i(\nu)}^{M}=M_{0}\cap
V_{i(\nu)}$ and an elementary embedding ${j}:{V}\longrightarrow{M}$ with
$j\restriction(V_{\nu}\cup\\{\nu\\})=i\restriction(V_{\nu}\cup\\{\nu\\})$. In
particular, we know that $j(\kappa)>\lambda$ and $j(A\cap V_{\kappa})\cap
V_{\lambda}=A\cap M\cap V_{\lambda}$. Moreover, for every limit ordinal
$\vartheta\geq j(\eta)$, we can use 2(C) in Definition 2.1.2 to show that
$\Psi(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds.

Now, assume that for every limit ordinal $\eta>\kappa$, there exist
unboundedly many cardinal $\lambda\geq\eta$ with the property that there is a
transitive class $M$ and an elementary embedding ${j}:{V}\longrightarrow{M}$
with $j(\kappa)>\lambda$, $j(A\cap V_{\kappa})\cap V_{\lambda}=A\cap
V_{\lambda}$ and the property that
$\Psi(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds for
all limit ordinals $\vartheta>j(\eta)$. Pick limit ordinals
$\theta>\eta>\kappa$ and an ordinal $\zeta>\eta$. Then there is a cardinal
$\lambda>\zeta$, an inner model $M$ and an elementary embedding
${j}:{V}\longrightarrow{M}$ such that the above statements hold with respect
to $\kappa$, $\eta$ and $\lambda$. We then know that $M\cap V_{j(\theta)+1}$
is a transitive set and the map ${j\restriction
V_{\theta+1}}:{V_{\theta+1}}\longrightarrow{M\cap V_{j(\theta)+1}}$ possesses
all of the desired properties. ∎

###  5.1. Vopěnka’s Principle

We now show how the validity of Vopěnka’s Principle can be characterized
through the existence of cardinals that are $\Psi$-large for certain classes.
Note that the results of [1, Section 4] show that, over $\operatorname{ZFC}$,
Vopěnka’s Principle is equivalent to the scheme that only requires the
instances of Vopěnka’s Principle for classes of finite signatures that are
defined by formulas without parameters to hold. The following lemma is a
direct adaptation of the results discussed in [12, p. 339] to our setting.

######  Lemma 5.4.

The following schemes are equivalent over $\operatorname{ZFC}$:

  1. (1)

Vopěnka’s Principle.

  2. (2)

For every class $A$, there is a proper class of cardinals that are
$\Psi_{ext}$-large for $A$.

  3. (3)

For every parameter-free formula $\varphi(v)$ in the language of set theory,
there is a cardinal that is $\Psi_{sc}$-large for the class
$\\{x~{}|~{}\varphi(x)\\}$.

###### Proof.

First, note that the implication from (1) to (2) already follows from
Proposition 5.2, and the implication from (2) to (3) is trivial. Now, assume
that (3) holds. Fix a proper class $\mathcal{C}$ of structures of the same
finite signature that is definable by a formula $\varphi(v)$ without
parameters. Let $\kappa$ be a cardinal that is $\Psi_{sc}$-large for
$\mathcal{C}$. By our assumptions, there is a structure $B$ in $\mathcal{C}$
that is not an element of $H_{\kappa}$. Fix a cardinal $\eta>\kappa$ with
$B\in H_{\eta}=V_{\eta}$ and a cardinal $\theta>\eta$ such that the formula
$\varphi(v)$ is absolute between $V_{\theta}$ and $V$. In this situation, we
can find a cardinal $\lambda\geq\eta$, a transitive set $M$ and an elementary
embedding ${j}:{V_{\theta+1}}\longrightarrow{M}$ such that
$j(\kappa)>\lambda$, $j(\mathcal{C}\cap V_{\kappa})\cap
V_{\lambda}=\mathcal{C}\cap V_{\lambda}$, and $\Psi_{sc}(V_{j(\theta)},M\cap
V_{j(\theta)},\lambda,j(\kappa),j(\eta))$ holds. We then know that $B\in
H_{j(\kappa)}^{M}$ and hence elementarity implies that $j(B)\neq B$. This
shows that $j(B)$ is a structure of the given signature and the embedding $j$
induces an elementary embedding of $B$ into $j(B)$. Moreover, our setup
ensures that the set $M$ contains both $B$ and the given elementary embedding
of $B$ into $j(B)$. In addition, we know that

$B~{}\in~{}\mathcal{C}\cap V_{\lambda}~{}\subseteq~{}j(\mathcal{C}\cap
V_{\kappa})~{}=~{}j(\\{x\in V_{\kappa}~{}|~{}V_{\theta}\models\varphi(x)\\}).$

By elementarity, this implies that, in $V_{\theta+1}$, there is an elementary
embedding of a structure $A$ of the given signature into $B$ such that $A\neq
B$ and $V_{\theta}\models\varphi(A)$. This shows that, in $V$, there is a
structure $A\in\mathcal{C}$ with $A\neq B$ and an elementary embedding of $A$
into $B$. As discussed earlier, the results of [1, Section 4] show that these
computations ensure that Vopěnka’s Principle holds. ∎

###  5.2. Ord is Woodin

Remember that “  _$\operatorname{Ord}$ is Woodin_” is the scheme of axioms
stating that for every class $A$, there exists a cardinal $\kappa$ that is
_strong for $A$_, i.e., for every ordinal $\lambda$, $\kappa$ is _$\lambda$
-strong for $A$_: there is an inner model $M$ with $V_{\lambda}\subseteq M$
and an elementary embedding ${j}:{V}\longrightarrow{M}$ with
$\operatorname{crit}(j)=\kappa$, $j(\kappa)>\lambda$ and $A\cap
V_{\lambda}=j(A\cap V_{\kappa})\cap V_{\lambda}$ (see [5] and [17]). The
following lemma shows how this principle is related to the concepts introduced
above. Its proof is based on the _$\Sigma_{n}$ -Product Reflection Principle_
introduced by Bagaria and Wilson in [5].

Remember that for any set $S$ of relational structures of the same type, the
_set-theoretic product $\prod S$_ is defined to be the structure whose domain
is the set of all functions $f$ with domain $S$ and the property that
$f(\mathcal{A})\in A$ holds for every structure $\mathcal{A}=\langle
A,\ldots\rangle$ in $S$, and whose interpretation of relation symbols is
defined pointwise, i.e., for every $(n+1)$-ary relation symbol $R$ in the
given language, we have $\dot{R}^{\prod S}(f_{0},\ldots,f_{n})$ if and only if
$R^{\mathcal{A}}(f_{0}(\mathcal{A}),\ldots,f_{n}(\mathcal{A}))$ holds for all
$\mathcal{A}\in S$. For a class $\mathcal{C}$ of relational structures of the
same type, we then let $\mathrm{PRP}_{\mathcal{C}}$ denote the statement that
there exists a non-empty subset $S$ of $\mathcal{C}$ with the property that
for every $\mathcal{A}$ in $\mathcal{C}$, there exists a homomorphism121212As
defined in [10, Section 1.2]. of $\prod S$ into $\mathcal{A}$. Given a natural
number $n>0$, Bagaria and Wilson then defined the _$\Sigma_{n}$ -Product
Reflection Principle_ to be the statement that $\mathrm{PRP}_{\mathcal{C}}$
holds for every non-empty class $\mathcal{C}$ of graphs that is definable by a
$\Sigma_{n}$-formula without parameters (see [5, Definition 3.1]). The results
of [5, Section 5] show that $\operatorname{Ord}$ is Woodin if and only if the
$\Sigma_{n}$-Product Reflection Principle holds for all $n>0$.

######  Lemma 5.5.

The following schemes are equivalent over $\operatorname{ZFC}$:

  1. (1)

$\operatorname{Ord}$ is Woodin.

  2. (2)

For every class $A$, there is a proper class of cardinals that are
$\Psi_{str}$-large for $A$.

  3. (3)

For every parameter-free formula $\varphi(v)$ in the language of set theory,
there is a cardinal that is $\Psi_{str}$-large for the class
$\\{x~{}|~{}\varphi(x)\\}$.

###### Proof.

First, assume that (1) holds. Fix an ordinal $\xi$ and pick a formula
$\varphi(v_{0},\ldots,v_{m})$ and parameters $y_{0},\ldots,y_{m-1}$ with
$A=\\{x~{}|~{}\varphi(x,y_{0},\ldots,y_{m-1})\\}$. Define

$A_{*}~{}=~{}\\{\langle
x_{0},\ldots,x_{m}\rangle~{}|~{}\varphi(x_{0},\ldots,x_{m})\\}.$

Using [5, Theorem 5.13], we find a cardinal $\kappa>\xi$ that satisfies
$y_{0},\ldots,y_{m-1}\in V_{\kappa}$ and is strong for $A_{*}$. Fix a cardinal
$\lambda>\kappa$. We can then find an inner model $M$ with
$V_{\lambda}\subseteq M$ and an elementary embedding
${j}:{V}\longrightarrow{M}$ with $\operatorname{crit}(j)=\kappa$,
$j(\kappa)>\lambda$ and $A_{*}\cap V_{\lambda}=j(A_{*}\cap V_{\kappa})\cap
V_{\lambda}$. Given $x\in V_{\lambda}\subseteq M$, we then know that

$\begin{split}x\in A~{}&\Longleftrightarrow~{}\langle
x,y_{0},\ldots,y_{m-1}\rangle\in A_{*}~{}\Longleftrightarrow~{}\langle
x,y_{0},\ldots,y_{m-1}\rangle\in j(A_{*}\cap V_{\kappa})\\\
&\Longleftrightarrow~{}M\models\varphi(x,y_{0},\ldots,y_{m-1})~{}\Longleftrightarrow~{}x\in
j(A\cap V_{\kappa}).\end{split}$

These equivalences show that $j(A\cap V_{\kappa})\cap V_{\lambda}=A\cap
V_{\lambda}$ holds. Moreover, it follows that
$\Psi_{str}(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds
for all limit ordinals $\kappa<\eta\leq\lambda$ and all limit ordinals
$\vartheta>j(\eta)$. By Lemma 5.3, this shows that the cardinal $\kappa>\xi$
is $\Psi_{str}$-large for $A$. Hence, we have shown that (2) holds in this
case.

The implication from (2) to (3) is immediate.

Now, assume that (3) holds. Let $\mathcal{C}$ denote a non-empty class of
graphs that is defined by a formula $\varphi(v)$ without parameters. By our
assumptions, there is a cardinal $\kappa$ that is $\Psi_{str}$-large for
$\mathcal{C}$. We then know that $\mathcal{C}\cap V_{\kappa}\neq\emptyset$.
Fix a graph $\mathcal{G}$ in $\mathcal{C}$ and pick a cardinal $\eta>\kappa$
with $\mathcal{G}\in H_{\eta}=V_{\eta}$. Using Lemma 5.3, we now find a
cardinal $\lambda>\eta$, a transitive class $M$ with $V_{\lambda}\subseteq M$
and an elementary embedding ${j}:{V}\longrightarrow{M}$ satisfying
$j(\kappa)>\lambda$ and $j(\mathcal{C}\cap V_{\kappa})\cap
V_{\lambda}=\mathcal{C}\cap V_{\lambda}$. In this situation, we know that
$\mathcal{G}\in j(\mathcal{C}\cap V_{\kappa})$. Since the graph
$j(\prod(\mathcal{C}\cap V_{\kappa}))$ is a subgraph of the product graph
$\prod(j(\mathcal{C}\cap V_{\kappa}))$ and therefore the embedding $j$ induces
a homomorphism of $\prod(\mathcal{C}\cap V_{\kappa})$ into
$\prod(j(\mathcal{C}\cap V_{\kappa}))$, we can combine this homomorphism with
the projection from $\prod(j(\mathcal{C}\cap V_{\kappa}))$ to $\mathcal{G}$ to
find a homomorphism from $\prod(\mathcal{C}\cap V_{\kappa})$ to $\mathcal{G}$.
These computations show that the $\Sigma_{n}$-Product Reflection Principle
holds for every natural number $n>0$ and therefore the results of [5, Section
5] allow us to conclude that (1) holds in this case. ∎

##  6\. Abstract logics

In order to connect the existence of cardinals that are $\Psi$-strong for
certain classes to compactness properties of strong logics, we now fix a
notion of _abstract logic_. This notion, as introduced below, is taken from
[8]. As argued in [8, Section 4], results of Makowsky in [15] show that
Vopěnka’s Principle is equivalent to the assumption that every abstract logic
has a strong compactness cardinal (see Theorem 6.3 below), i.e., a direct
analogue of Theorem 1.1 that replaces second-order logic with arbitrary
abstract logics and the existence of an extendible cardinal with the validity
of Vopěnka’s Principle. The goal of this section is to formulate and verify
similar analogues of Theorem 3.5 and Corollary 3.6 for abstract logics.

######  Definition 6.1.

  1. (1)

A _language_ is a tuple

$\tau~{}=~{}\langle\mathfrak{F}_{\tau},\mathfrak{R}_{\tau},\mathfrak{C}_{\tau},a_{\tau}\rangle$

consisting of disjoint sets $\mathfrak{F}_{\tau}$ (the _function symbols_ in
$\tau$), $\mathfrak{R}_{\tau}$ (the _relation symbols_ in $\tau$) and
$\mathfrak{C}_{\tau}$ (the _constant symbols_ in $\tau$) together with a
function
${a_{\tau}}:{\mathfrak{F}_{\tau}\cup\mathfrak{R}_{\tau}}\longrightarrow{\omega}$
(the _arity function_ for $\tau$).

  2. (2)

Given a language $\tau$, a _$\tau$ -structure_ is a tuple

$M~{}=~{}\langle|M|,\langle{f^{M}}~{}|~{}{f\in\mathfrak{F}_{\tau}}\rangle,\langle{r^{M}}~{}|~{}{r\in\mathfrak{R}_{\tau}}\rangle,\langle{c^{M}}~{}|~{}{c\in\mathfrak{C}_{\tau}}\rangle\rangle,$

where $|M|$ is a non-empty set (the _universe of $M$_), each $f^{M}$ is an
$a_{\tau}(f)$-ary function on $|M|$, each $r^{M}$ is an $a_{\tau}(r)$-ary
relation on $|M|$, and each $c^{M}$ is an element of $|M|$. We let
$Str_{\tau}$ denote the collection of all $\tau$-structures.

  3. (3)

A _renaming_ of a language $\sigma$ into a language $\tau$ is a bijection

${r}:{\mathfrak{F}_{\sigma}\cup\mathfrak{R}_{\sigma}\cup\mathfrak{C}_{\sigma}}\longrightarrow{\mathfrak{F}_{\tau}\cup\mathfrak{R}_{\tau}\cup\mathfrak{C}_{\tau}}$

satisfying $r[\mathfrak{F}_{\sigma}]=\mathfrak{F}_{\tau}$,
$r[\mathfrak{R}_{\sigma}]=\mathfrak{R}_{\tau}$,
$r[\mathfrak{C}_{\sigma}]=\mathfrak{C}_{\tau}$ and
$a_{\sigma}(s)=a_{\tau}(r(s))$ for all
$s\in\mathfrak{F}_{\sigma}\cup\mathfrak{R}_{\sigma}$. In this setting, we let
${r^{*}}:{Str_{\sigma}}\longrightarrow{Str_{\tau}}$ denote the canonical
bijective class function induced by $r$ that fixes the universes of the given
$\sigma$-structures.

Given the above definitions, there is a canonical notion of a _sublanguage_ of
a given language. Moreover, given a sublanguage $\sigma$ of a language $\tau$,
there is a canonical notion of a $\sigma$-reduct $M\restriction\sigma\in
Str_{\sigma}$ of a $\tau$-structure $M$. Finally, given a language $\tau$,
there is a canonical notion of an isomorphism of $\tau$-structures.

######  Definition 6.2.

An _abstract logic_ $\langle\mathcal{L},\models_{\mathcal{L}}\rangle$ consists
of a class function $\mathcal{L}$ and a binary class relation
$\models_{\mathcal{L}}$ that satisfy the following conditions:

  1. (1)

The domain of $\mathcal{L}$ is the class of all languages.

  2. (2)

If $M\models_{\mathcal{L}}\phi$, then there is a language $\tau$ with $M\in
Str_{\tau}$ and $\phi\in\mathcal{L}(\tau)$.

  3. (3)

(Monotonicity) If $\sigma$ is a sublanguage of $\tau$, then
$\mathcal{L}(\sigma)\subseteq\mathcal{L}(\tau)$.

  4. (4)

(Expansion) If $\sigma$ is a sublanguage of $\tau$, $M\in Str_{\tau}$ and
$\phi\in\mathcal{L}(\sigma)$, then

$M\models_{\mathcal{L}}\phi~{}\Longleftrightarrow~{}{M\restriction\sigma}\models_{\mathcal{L}}\phi.$

  5. (5)

(Isomorphism) Given a language $\tau$ and isomorphic $\tau$-structures $M$ and
$N$, we have

$M\models_{\mathcal{L}}\phi~{}\Longleftrightarrow~{}N\models_{\mathcal{L}}\phi$

for all $\phi\in\mathcal{L}(\tau)$.

  6. (6)

(Renaming) Every renaming $r$ of a language $\sigma$ into a language $\tau$
induces a unique bijection
${r_{*}}:{\mathcal{L}(\sigma)}\longrightarrow{\mathcal{L}(\tau)}$ with the
property that

$M\models_{\mathcal{L}}\phi~{}\Longleftrightarrow~{}r^{*}(M)\models_{\mathcal{L}}r_{*}(\phi)$

holds for every $M\in Str_{\sigma}$ and every $\phi\in\mathcal{L}(\sigma)$.

  7. (7)

There exists a cardinal $\mathscr{o}$ with the property that for every
language $\tau$ and every $\phi\in\mathcal{L}(\tau)$, there exists a
sublanguage $\sigma$ of $\tau$ with
$|\mathfrak{F}_{\sigma}\cup\mathfrak{R}_{\sigma}\cup\mathfrak{C}_{\sigma}|<\mathscr{o}$
and $\phi\in\mathcal{L}(\sigma)$.

In the remainder of this section, we will usually write $\mathcal{L}$ instead
of $\langle\mathcal{L},\models_{\mathcal{L}}\rangle$ to denote an abstract
logic. Given an abstract logic $\mathcal{L}$, we call $\models_{\mathcal{L}}$
the _satisfaction relation_ of $\mathcal{L}$ and, given a language $\tau$, we
say that $\mathcal{L}(\tau)$ is the set of _$\tau$ -sentences_. In addition,
we call the least cardinal $\mathscr{o}$ witnessing (7) in Definition 6.2 the
_occurrence number_ of $\mathcal{L}$. We say that a set $T$ is an
_$\mathcal{L}$ -theory_ if there is a language $\tau$ with
$T\subseteq\mathcal{L}(\tau)$. We say that an $\mathcal{L}$-theory $T$ is
_consistent_ if there is a language $\tau$ with $T\subseteq\mathcal{L}(\tau)$
and a $\tau$-structure $M$ with $M\models_{\mathcal{L}}\phi$ for all $\phi\in
T$. Given an infinite cardinal $\kappa$, we say that an $\mathcal{L}$-theory
$T$ is _${ <}\kappa$-consistent_ if every subset of $T$ of cardinality less
than $\kappa$ is consistent. Finally, we say that an infinite cardinal
$\kappa$ is a _strong compactness cardinal for $\mathcal{L}$_ if every
${<}\kappa$-consistent $\mathcal{L}$-theory is consistent. Makowsky’s result
from [15] can now be formulated in the following way:

######  Theorem 6.3 (Makowsky [15]).

The following schemes are equivalent over $\operatorname{ZFC}$:

  1. (1)

Vopěnka’s Principle.

  2. (2)

Every abstract logic has a strong compactness cardinal.

We now generalize the notions introduced in Section 3 to obtain
characterizations of the validity of fragments of Vopěnka’s Principle through
compactness properties of abstract logics. In the following, we say that an
abstract logic $\mathcal{L}$ is defined by formulas
$\varphi_{0}(v_{0},v_{1},v_{2})$ and $\varphi_{1}(v_{0},v_{1},v_{2})$ together
with a parameter $z$ if $\mathcal{L}=\\{\langle
x,y\rangle~{}|~{}\varphi_{0}(x,y,z)\\}$ and $\models_{\mathcal{L}}=\\{\langle
x,y\rangle~{}|~{}\varphi_{1}(x,y,z)\\}$.

######  Definition 6.4.

Let $F$ be an $\in$-theory, let $\Psi(v_{0},\ldots,v_{4})$ be a formula in the
language of set theory, let $\kappa$ be an infinite cardinal and let
$\mathcal{L}$ be an abstract logic that is defined by formulas $\varphi_{0}$
and $\varphi_{1}$ together with a parameter $z$.

  1. (1)

An $\mathcal{L}$-theory $T$ is _$F$ -$\Psi$-outward consistent at $\kappa$_ if
there is a limit ordinal $\eta>\kappa$ such that for all cardinals
$\lambda<\kappa$ and all cardinals $\vartheta>\eta$ with $T,z\in
H_{\vartheta}$, the partial order $\mathrm{Col}(\omega,\vartheta)$ forces that
$T$ is a ${<}\lambda$-consistent $\mathcal{L}^{\prime}$-theory in $N$ whenever
$N$ is an outer $F$-model of $V_{\vartheta}^{V}$, $\mathcal{L}^{\prime}$ is an
abstract logic in $N$ and the following statements hold:

    1. (a)

$\Psi(N,V_{\vartheta}^{V},\lambda,\kappa,\eta)$ holds.

    2. (b)

The formulas $\varphi_{0}$ and $\varphi_{1}$ are absolute from $V$ to $N$ with
respect to parameters contained in $V_{\lambda}^{V}$.131313I.e., the statement
$\varphi_{i}(x_{0},x_{1},x_{2})$ holds in $N$ for all $i<2$ and
$x_{0},x_{1},x_{2}\in V_{\lambda}^{V}$ with the property that
$\varphi_{i}(x_{0},x_{1},x_{2})$ holds in $V$.

    3. (c)

The formulas $\varphi_{0}$ and $\varphi_{1}$ together with the parameter $z$
define $\mathcal{L}^{\prime}$ in $N$.

    4. (d)

$T$ is an $\mathcal{L}^{\prime}$-theory in $N$.

  2. (2)

A cardinal $\kappa$ is an _$F$ -$\Psi$-outward compactness cardinal for
$\mathcal{L}$_ if all $\mathcal{L}$-theories that are $F$-$\Psi$-outward
consistent at $\kappa$ are consistent.

In order to motivate the above concept, we now discuss its relation to the
compactness properties of $\mathcal{L}^{2}$ introduced in Section 3. For this
purpose, we make the following definition, which will also be relevant later
in this section.

######  Definition 6.5.

A measure of closeness $\Psi(v_{0},\ldots,v_{4})$ is _strong_ if it is a
$\Delta_{1}^{\operatorname{ZFC}^{-}}$-formula with

$\operatorname{ZFC}\vdash\forall
M,N,\mu,\nu,\rho~{}[\Psi(N,M,\mu,\nu,\rho)~{}\longrightarrow~{}\Psi_{str}(N,M,\mu,\nu,\rho)].$

######  Proposition 6.6.

Given a strong measure of closeness $\Psi$, an $\mathcal{L}^{2}$-theory $T$ is
$\Psi$-outward consistent at a cardinal $\kappa$ if and only if it is
$\operatorname{ZFC}^{*}$-$\Psi$-outward consistent at $\kappa$. In particular,
a cardinal $\kappa$ is a $\Psi$-outward compactness cardinal for
$\mathcal{L}^{2}$ if and only if it is a
$\operatorname{ZFC}^{*}$-$\Psi$-outward compactness cardinal for
$\mathcal{L}^{2}$.

###### Proof.

Our definitions directly ensure that every $\mathcal{L}^{2}$-theory that is
$\Psi$-outward consistent at a cardinal $\kappa$ is also
$\operatorname{ZFC}^{*}$-$\Psi$-outward consistent at the given cardinal
$\kappa$. In the other direction, assume that a limit ordinal $\eta$ witnesses
that an $\mathcal{L}^{2}$-theory $T$ is
$\operatorname{ZFC}^{*}$-$\Psi$-outward consistent at a cardinal $\kappa$. Fix
a cardinal $\lambda<\kappa$ and a cardinal $\vartheta>\eta$ with $T\in
H_{\vartheta}$. In addition, let $G$ be
$\mathrm{Col}(\omega,\vartheta)$-generic over $V$ and pick an outer
$\operatorname{ZFC}^{*}$-model $N$ of $V_{\vartheta}^{V}$ in $V[G]$ with the
property that $\Psi(N,V_{\vartheta}^{V},\lambda,\kappa,\eta)$ holds in $V[G]$.
We then know that the canonical formulas defining $\mathcal{L}^{2}$ define an
abstract logic $\mathcal{L}^{\prime}$ in $N$ and $T$ is an
$\mathcal{L}^{\prime}$-theory in $N$. Moreover, these formulas are absolute
between $V$ and $V_{\lambda}$ as well as between $N$ and $V_{\lambda}^{N}$.
Since our setup ensures that $V_{\lambda}^{N}=V_{\lambda}^{V}$, we can now
conclude that $T$ is ${<}\lambda$-consistent in $N$. ∎

As a further motivation for the results proven below, we now derive an
analogue of Proposition 3.4 for the above compactness property.

######  Proposition 6.7.

Let $\Psi$ be a $\Delta_{1}^{\operatorname{ZFC}}$-formula that is a measure of
closeness and let $n>1$ be a natural number. If $\mathcal{L}$ is an abstract
logic and $\kappa$ is a limit cardinal, then every $\mathcal{L}$-theory that
is $\operatorname{ZFC}_{n}$-$\Psi$-outward consistent at $\kappa$ is
${<}\kappa$-consistent. In particular, if Vopěnka’s Principle holds, then
every abstract logic has a $\operatorname{ZFC}_{n}$-$\Psi$-outward compactness
cardinal.

###### Proof.

Fix formulas $\varphi_{0}(v_{0},v_{1},v_{2})$ and
$\varphi_{1}(v_{0},v_{1},v_{2})$ and a parameter $z$ defining an abstract
logic $\mathcal{L}$ and let $\eta$ be a limit ordinal witnessing that an
$\mathcal{L}$-theory $T$ is $\operatorname{ZFC}_{n}$-$\Psi$-outward consistent
at a cardinal $\kappa$. Assume, towards a contradiction, that there is a
subset $T_{0}$ of $T$ of cardinality less than $\kappa$ that is an
inconsistent $\mathcal{L}$-theory. Pick a cardinal $\lambda<\kappa$ with
$|T_{0}|<\lambda$ and a cardinal $\vartheta>\eta$ with $T,z\in
H_{\vartheta}=V_{\vartheta}$ and $V_{\vartheta}$ sufficiently elementary in
$V$. Then 2(B) in Definition 2.1.2 implies that
$\Psi(V_{\vartheta},V_{\vartheta},\lambda,\kappa,\eta)$ holds. In addition,
our choice of $\vartheta$ ensured that the formulas $\varphi_{0}$ and
$\varphi_{1}$ are absolute between $V$ and $V_{\vartheta}$. Finally, we also
know that, in $V_{\vartheta}$, the formulas $\varphi_{0}$ and $\varphi_{1}$
together with the parameter $z$ define an abstract logic
$\mathcal{L}^{\prime}$, and that $T$ is an $\mathcal{L}^{\prime}$-theory that
is not ${<}\lambda$-consistent. If $G$ is
$\mathrm{Col}(\omega,\vartheta)$-generic over $V$, then $V_{\vartheta}^{V}$ is
an outer $\operatorname{ZFC}_{n}$-model of $V_{\vartheta}^{V}$ in $V[G]$, and
the fact that $\Psi$ is a $\Delta_{1}^{\operatorname{ZFC}}$-formula ensures
that $\Psi(V_{\vartheta}^{V},V_{\vartheta}^{V},\lambda,\kappa,\eta)$ holds in
$V[G]$. However, this contradicts that $T$ is
$\operatorname{ZFC}_{n}$-$\Psi$-outward consistent at $\kappa$ in $V$. ∎

In the following, we will restrict ourselves to abstract logics with the
property that both its formulas and the translations of formulas induced by
renamings are produced from the given signatures and renamings by some simple
recursive procedure. It is easy to see that all _finitely generated_ logics
(see [15, §1]) possess the property introduced below. In particular, we can
apply the results below to second-order logic.

######  Definition 6.8.

An abstract logic $\mathcal{L}$ with occurrence number $\mathscr{o}$ has
_simple formulas_ if there exists a $\Sigma_{1}$-formula
$\varphi(v_{0},\ldots,v_{3})$ and a set $z$ with the property that whenever
$r$ is a renaming of a language $\sigma$ with less than $\mathscr{o}$-many
symbols into a language $\tau$ and
${r_{*}}:{\mathcal{L}(\sigma)}\longrightarrow{\mathcal{L}(\tau)}$ is the
bijection induced by $r$, then

$\\{\langle\phi,r_{*}(\phi)\rangle~{}|~{}\phi\in\mathcal{L}(\sigma)\\}~{}=~{}\\{\langle
x,y\rangle~{}|~{}\varphi(r,x,y,z)\\}.$

Note that if a $\Sigma_{1}$-formula $\varphi(v_{0},\ldots,v_{3})$ and a set
$z$ witness that an abstract logic $\mathcal{L}$ with occurrence number
$\mathscr{o}$ has simple formulas, then

$\mathcal{L}(\sigma)~{}=~{}\\{x~{}|~{}\varphi(\operatorname{id}_{\sigma},x,x,z)\\}$

holds for every language $\sigma$ with less than $\mathscr{o}$-many symbols,
where $\operatorname{id}_{\sigma}$ denotes the trivial renaming of $\sigma$
into itself. This shows that the restriction of the class function
$\mathcal{L}$ to languages with less than $\mathscr{o}$-many symbols is also
definable by a $\Sigma_{1}$-formula with parameter $z$ in this case. The next
theorem generalizes the forward direction of Theorem 3.5 to the setting of
this section.

######  Theorem 6.9.

Let $\Psi$ be a strong measure of closeness. Assume that for every class $A$,
there is a proper class of cardinals that are $\Psi$-strong for $A$. Then for
every natural number $n$, every abstract logic with simple formulas has a
$\operatorname{ZFC}_{n}$-$\Psi$-outward compactness cardinal.

###### Proof.

Let $\mathcal{L}$ be an abstract logic with simple formulas that is defined by
formulas $\varphi_{0}$ and $\varphi_{1}$ together with a parameter $z_{0}$. In
the following, let $\mathscr{o}$ denote the occurrence number of
$\mathcal{L}$. Fix a $\Sigma_{1}$-formula $\varphi(v_{0},\ldots,v_{3})$ and a
parameter $z_{1}$ witnessing that $\mathcal{L}$ has simple formulas. Next,
note that for every language $\tau$, the identity
$\operatorname{id}_{\mathcal{L}(\tau)}$ on $\mathcal{L}(\tau)$ is the unique
(as in (6) of Definition 6.2) permutation $\pi$ of $\mathcal{L}(\tau)$
corresponding to the trivial renaming of $\tau$ into itself. In particular, we
know that for every language $\tau$ and every non-trivial141414I.e., not equal
to the identity on $\mathcal{L}(\tau)$. permutation $\pi$ of
$\mathcal{L}(\tau)$, there exists $\chi\in\mathcal{L}(\tau)$ and a
$\tau$-structure $O$ with

$O\models_{\mathcal{L}}\chi~{}\Longleftrightarrow~{}O\mathbin{{\not\models}_{\mathcal{L}}}\pi(\chi).$
(1)

We can now pick a cardinal $\rho$ above $\mathscr{o}$ that satisfies
$z_{0},z_{1}\in H_{\rho}=V_{\rho}$ and has the property that for every
language $\tau$ in $H_{\mathscr{o}}$ and every non-trivial permutation $\pi$
of $\mathcal{L}(\tau)$, we have $\mathcal{L}(\tau)\in H_{\rho}$ and there
exists $\chi\in\mathcal{L}(\tau)$ and a $\tau$-structure $O$ in $H_{\rho}$
with (1). Let $A$ be a class that codes the classes $\\{\rho\\}$,
$\\{z_{0}\\}$, $\\{z_{1}\\}$, $\\{\langle
y_{0},y_{1},y_{2}\rangle~{}|~{}\varphi_{0}(y_{0},y_{1},y_{2})\\}$ and
$\\{\langle y_{0},y_{1},y_{2}\rangle~{}|~{}\varphi_{1}(y_{0},y_{1},y_{2})\\}$
in some canonical way. Pick a cardinal $\kappa>\rho$ that is $\Psi$-large for
$A$. Fix a natural number $n>1$. Pick an $\mathcal{L}$-theory $T$ that is
$\operatorname{ZFC}_{n}$-$\Psi$-outward consistent at $\kappa$, and let
$\eta>\kappa$ be a limit ordinal witnessing this. Let $\sigma$ be a language
with $T\subseteq\mathcal{L}(\sigma)$. In this situation, we can apply Lemma
5.3 to find a cardinal $\lambda>\eta$ with $\sigma,\mathcal{L}(\sigma)\in
H_{\lambda}$, an inner model $M$ and an elementary embedding
${j}:{V}\longrightarrow{M}$ such that $j(\kappa)>\lambda$, $j(A\cap
V_{\kappa})\cap V_{\lambda}=A\cap M\cap V_{\lambda}$ and
$\Psi(V_{\vartheta},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds for
all limit ordinals $\vartheta>j(\eta)$. Our assumptions on $\Psi$ then ensure
that $V_{\lambda}\subseteq M$. Moreover, by our choice of $A$, we know that
$j(\rho)=\rho$, $j(z_{0})=z_{0}$ and $j(z_{1})=z_{1}$. Given $i<2$, we define

$B_{i}~{}=~{}\\{\langle y_{0},y_{1},y_{2}\rangle\in
V_{\lambda}^{3}~{}|~{}\varphi_{i}(y_{0},y_{1},y_{2})\\}.$

Our setup then ensures that

$B_{i}~{}=~{}j(B_{i}\cap V_{\kappa})\cap V_{\lambda}~{}=~{}\\{\langle
y_{0},y_{1},y_{2}\rangle\in
V_{\lambda}^{3}~{}|~{}M\models\varphi_{i}(y_{0},y_{1},y_{2})\\}$

holds for all $i<2$. Finally, we set $\mu=\operatorname{crit}(j)$.

######  Claim.

$\mu>\rho$ and $z_{0},z_{1}\in V_{\mu}$.

###### Proof of the Claim.

Assume that one of the statements of the claim fails. Then there is a limit
ordinal $\mu<\zeta<\lambda$ with $j(\zeta)=\zeta$. But this implies that the
map ${j\restriction V_{\zeta+2}}:{V_{\zeta+2}}\longrightarrow{V_{\zeta+2}}$ is
a non-trivial elementary embedding, contradicting the _Kunen Inconsistency_. ∎

Next, note that $j(\sigma)$ is a language in $V$.

######  Claim.

$j(T)\subseteq\mathcal{L}(j(\sigma))$.

###### Proof of the Claim.

Fix $\phi\in T$. We can find a sublanguage $\bar{\sigma}$ of $\sigma$ with
less than $\mathscr{o}$-many symbols that satisfies
$\phi\in\mathcal{L}(\bar{\sigma})$. By our assumptions, we know that
$\varphi(\operatorname{id}_{\bar{\sigma}},\phi,\phi,z_{1})$ holds in $V$ and
hence elementarity implies that
$\varphi(\operatorname{id}_{j(\bar{\sigma})},j(\phi),j(\phi),z_{1})$ holds in
$M$. But then $\Sigma_{1}$-upwards absoluteness ensures that this statement
also holds in $V$. Since our first claim shows that $j(\bar{\sigma})$ is a
sublanguage of $j(\sigma)$ with less than $\mathscr{o}$-many symbols,
$j(\phi)\in\mathcal{L}(j(\bar{\sigma}))\subseteq\mathcal{L}(j(\sigma))$. ∎

Notice that the elementarity of $j$ ensures that, in $M$, the formulas
$\varphi_{0}$ and $\varphi_{1}$ together with the parameter $z_{0}$ define an
abstract logic $\mathcal{L}_{M}$ and the limit ordinal $j(\eta)$ witnesses
that the $\mathcal{L}_{M}$-theory $j(T)$ is
$\operatorname{ZFC}_{n}$-$\Psi$-outward consistent at $j(\kappa)$.

######  Claim.

The $\mathcal{L}$-theory $j[T]$ is consistent in $V$.

###### Proof of the Claim.

Let $\vartheta>j(\lambda)$ be a cardinal such that $j(\vartheta)=\vartheta$,
the set $V_{\vartheta}$ is sufficiently elementary in $V$ and the set
$V_{\vartheta}^{M}$ is sufficiently elementary in $M$. Let $G$ be
$\mathrm{Col}(\omega,\vartheta)$-generic over $V$. Then $V_{\vartheta}^{M}$ is
countable in $M[G]$ and we know that, in $M[G]$, $j(T)$ is a
${<}\lambda$-consistent $\mathcal{L}^{\prime}$-theory in $N$ whenever $N$ is a
countable outer $\operatorname{ZFC}_{n}$-model of $V_{\vartheta}^{M}$,
$\mathcal{L}^{\prime}$ is an abstract logic in $N$ and the following
statements hold:

  1. (a)

$\Psi(N,V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds.

  2. (b)

If $i<2$ and $\langle y_{0},y_{1},y_{2}\rangle\in B_{i}$, then
$\varphi_{i}(y_{0},y_{1},y_{2})$ holds in $N$.

  3. (c)

The formulas $\varphi_{0}$ and $\varphi_{1}$ together with the parameter
$z_{0}$ define $\mathcal{L}^{\prime}$ in $N$.

  4. (d)

$j(T)$ is an $\mathcal{L}^{\prime}$-theory in $N$.

Since the given statement can be expressed by a $\Pi_{1}$-formula with
parameters in $H_{\aleph_{1}}^{M[G]}$, _Shoenfield Absoluteness_ ensures that
it also holds in $V[G]$. Thus, $\Sigma_{1}$-upwards absoluteness implies that
$\Psi(V_{\vartheta}^{V},V_{\vartheta}^{M},\lambda,j(\kappa),j(\eta))$ holds in
$V[G]$. Moreover, since $V_{\vartheta}^{V}$ was chosen sufficiently elementary
in $V$, we know that for all $i<2$ and all $\langle
y_{0},y_{1},y_{2}\rangle\in B_{i}$, the statement
$\varphi_{i}(y_{0},y_{1},y_{2})$ holds in $V_{\vartheta}^{V}$. In addition,
since the previous claim shows that $j(T)$ is an $\mathcal{L}$-theory in $V$,
our choice of $\vartheta$ also ensures that the formulas $\varphi_{0}$ and
$\varphi_{1}$ together with the parameter $z_{0}$ define an abstract logic
$\mathcal{L}^{\prime}$ in $V_{\vartheta}^{V}$ and $j(T)$ is an
$\mathcal{L}^{\prime}$-theory in $V_{\vartheta}^{V}$. This allows us to
conclude that $j(T)$ is a ${<}\lambda$-consistent
$\mathcal{L}^{\prime}$-theory in $V_{\vartheta}^{V}$. Since $j[T]$ is an
element of $V_{\vartheta}^{V}$ and this set has cardinality less than
$\lambda$ in $V_{\vartheta}^{V}$, we now know that $j[T]$ is a consistent
$\mathcal{L}^{\prime}$-theory in $V_{\vartheta}^{V}$. In this situation, the
fact that $V_{\vartheta}^{V}$ was chosen to be sufficiently elementary in $V$
allows us to conclude that $j[T]$ is a consistent $\mathcal{L}$-theory in $V$.
∎

Let $\sigma_{j}$ denote the sublanguage of $j(\sigma)$ that is given by the
pointwise image of the symbols in $\sigma$ under $j$, and let $r$ denote the
canonical renaming from $\sigma$ into $\sigma_{j}$ induced by $j$. Since the
occurrence number $\mathscr{o}$ of $\mathcal{L}$ is smaller than the critical
point $\mu$ of $j$, we know that $j[T]\subseteq\mathcal{L}(\sigma_{j})$.
Moreover, the previous claim shows that there is a $\sigma$-structure $N$ with
the property that the $\sigma_{j}$-structure $r^{*}(N)$ obtained from $N$
using the renaming $r$ is a model of $j[T]$. Assume, towards a contradiction,
that $N$ is not a model of $T$ and fix $\phi$ in $T$ such that
$N\mathbin{{\not\models}_{\mathcal{L}}}\phi$. Pick a sublanguage
$\bar{\sigma}$ of $\sigma$ that contains less than $\mathscr{o}$-many symbols
and has the property that $\phi$ is an element of $\mathcal{L}(\bar{\sigma})$.
Then $j$ induces a renaming $\bar{r}$ of $\bar{\sigma}$ into the sublanguage
$j(\bar{\sigma})$ of $\sigma_{j}$, and this function is equal to the
restriction of $r$ to $\bar{\sigma}$. Let $\bar{N}$ denote the
$\bar{\sigma}$-reduct of $N$. Then
$\bar{N}\mathbin{{\not\models}_{\mathcal{L}}}\phi$. Moreover, the
$j(\bar{\sigma})$-structure $\bar{r}^{*}(\bar{N})$ obtained from $\bar{N}$
using $\bar{r}$ is equal to the $j(\bar{\sigma})$-reduct of $r^{*}(N)$ and
therefore we know that $\bar{r}^{*}(\bar{N})\models_{\mathcal{L}}j(\phi)$ and
$\bar{r}^{*}(\bar{N})\mathbin{{\not\models}_{\mathcal{L}}}\bar{r}_{*}(\phi)$.
In particular, it follows that $\bar{r}_{*}(\phi)\neq j(\phi)$.

Now, pick a renaming $s$ of a language $\tau\in H_{\mathscr{o}}$ into
$\bar{\sigma}$. Then $j(s)$ is a renaming of $\tau$ into $j(\bar{\sigma})$ and
our setup ensures that $j(s)=\bar{r}\circ s$, i.e., we know that the following
diagram commutes:

$\textstyle{\bar{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{r}}$$\textstyle{j(\bar{\sigma})}$$\textstyle{\tau\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{\operatorname{id}_{\tau}}$$\textstyle{\tau\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j(s)}$

Since $V_{\lambda}\subseteq M$, we know that the sets $\bar{\sigma}$,
$j(\bar{\sigma})$ and $\tau$ as well as the maps $s$ and $j(s)$ are all
contained in $M$, and this implies that the renaming $\bar{r}$ is also
contained in $M$. In addition, our setup ensures that
$\mathcal{L}(\bar{\sigma})=\mathcal{L}_{M}(\bar{\sigma})$,
$\mathcal{L}(j(\bar{\sigma}))=\mathcal{L}_{M}(j(\bar{\sigma}))$ and
$\mathcal{L}(\tau)=\mathcal{L}_{M}(\tau)$. Let
${s_{*}}:{\mathcal{L}(\tau)}\longrightarrow{\mathcal{L}(\bar{\sigma})}$ denote
the bijection induced by $s$ in $V$. Then $s_{*}$ is contained in $M$ and
$\Sigma_{1}$-upwards absoluteness implies that $s_{*}$ is also the bijection
between $\mathcal{L}_{M}(\bar{\sigma})$ and $\mathcal{L}_{M}(\tau)$ that is
induced by $s$ in $M$. Since $j(s_{*})$ is the bijection between
$\mathcal{L}_{M}(\tau)$ and $\mathcal{L}_{M}(j(\bar{\sigma}))$ induced by
$j(s)$ in $M$, we know that $j(s_{*})\circ s_{*}^{{-}1}$ is the bijection
between $\mathcal{L}_{M}(\bar{\sigma})$ and $\mathcal{L}_{M}(j(\bar{\sigma}))$
induced by ${\bar{r}}$ in $M$.

######  Claim.

$j(s_{*})\circ s_{*}^{{-}1}=j\restriction\mathcal{L}(\bar{\sigma})$.

###### Proof of the Claim.

Assume, towards a contradiction, that the statement of the claim fails. Since
$j\restriction\mathcal{L}(\bar{\sigma})$ is a bijection between
$\mathcal{L}(\bar{\sigma})$ and $\mathcal{L}(j(\bar{\sigma}))$, we then know
that the map $j(s_{*})^{{-}1}\circ j\circ s_{*}$ is a non-trivial permutation
of $\mathcal{L}(\tau)$. In this situation, our setup yields
$\chi\in\mathcal{L}(\tau)$ and a $\tau$-structure $O\in H_{\rho}$ with

$O\models_{\mathcal{L}}\chi~{}\Longleftrightarrow~{}O\mathbin{{\not\models}_{\mathcal{L}}}(j(s_{*})^{{-}1}\circ
j\circ s_{*})(\chi).$

Note that $\mathcal{L}(\tau)=\mathcal{L}_{M}(\tau)$ and, since
$\mathcal{L}(\tau)\subseteq V_{\lambda}$, our setup ensures that

$O\models_{\mathcal{L}}\upsilon~{}\Longleftrightarrow~{}O\models_{\mathcal{L}_{M}}\upsilon$

holds for every $\upsilon\in\mathcal{L}(\tau)$. Moreover, if $s^{*}(O)$
denotes the $\bar{\sigma}$-structure obtained from $O$ using $s$, then, in
$M$, the $j(\bar{\sigma})$-structure $j(s^{*}(O))$ is obtained from $O$ using
the renaming $j(s)$. Since $j(s_{*})$ is the bijection between
$\mathcal{L}_{M}(\tau)$ and $\mathcal{L}_{M}(j(\bar{\sigma}))$ induced by
$j(s)$ in $M$, we then know that

$O\models_{\mathcal{L}_{M}}j(s_{*})(\upsilon)~{}\Longleftrightarrow~{}j(s^{*}(O))\models_{\mathcal{L}_{M}}\upsilon$

holds for all $\upsilon\in\mathcal{L}_{M}(j(\bar{\sigma}))$. By combining
these equivalences with the elementarity of $j$, we can now conclude that

$\begin{split}O\models_{\mathcal{L}}\chi~{}&\Longleftrightarrow~{}s^{*}(O)\models_{\mathcal{L}}s_{*}(\chi)~{}\Longleftrightarrow~{}j(s^{*}(O))\models_{\mathcal{L}_{M}}j(s_{*}(\chi))\\\
&\Longleftrightarrow~{}O\models_{\mathcal{L}_{M}}(j(s_{*})^{{-}1}\circ j\circ
s_{*})(\chi)~{}\Longleftrightarrow~{}O\models_{\mathcal{L}}(j(s_{*})^{{-}1}\circ
j\circ s_{*})(\chi),\end{split}$

a contradiction. ∎

The above claim shows that, in $M$, the map
$j\restriction\mathcal{L}(\bar{\sigma})$ is the bijection between
$\mathcal{L}_{M}(\bar{\sigma})$ and $\mathcal{L}_{M}(j(\bar{\sigma}))$ induced
by $\bar{r}$. This directly implies that $\varphi(\bar{r},\phi,j(\phi),z_{1})$
holds in $M$. By $\Sigma_{1}$-upwards absoluteness, we know that this
statement holds in $V$ and we can conclude that $\bar{r}_{*}(\phi)=j(\phi)$, a
contradiction. ∎

The following result provides a generalization of the backward direction of
Theorem 3.5 that will turn out to be suitable for our characterizations of
fragments of Vopěnka’s Principle.

######  Theorem 6.10.

Let $\Psi$ be an $\mathcal{L}^{2}$-measure of closeness. Assume that for every
natural number $n$, every abstract logic that extends second-order logic and
has simple formulas has a $\operatorname{ZFC}_{n}$-$\Psi$-outward compactness