# Large Cardinals as Principles of Structural reflection

Joan Bagaria  ICREA (Institució Catalana de Recerca i Estudis Avançats) and
Departament de Matemàtiques i Informàtica, Universitat de Barcelona. Gran Via
de les Corts Catalanes, 585, 08007 Barcelona, Catalonia.
<EMAIL_ADDRESS>
###### Abstract.

After discussing the limitations inherent to all set-theoretic reflection
principles akin to those studied by A. Lévy et. al. in the 1960’s, we
introduce new principles of reflection based on the general notion of
_Structural Reflection_ and argue that they are in strong agreement with the
conception of reflection implicit in Cantor’s original idea of the
unknowability of the _Absolute_ , which was subsequently developed in the
works of Ackermann, Lévy, Gödel, Reinhardt, and others. We then present a
comprehensive survey of results showing that different forms of the new
principles of Structural Reflection are equivalent to well-known large
cardinals axioms covering all regions of the large-cardinal hierarchy, thereby
justifying the naturalness of the latter.

Part of this research was supported by the Spanish Government under grant
MTM2017-86777-P, and by the Generalitat de Catalunya (Catalan Government)
under grant SGR 270-2017.

In the framework of Zermelo-Fraenkel (ZF) set theory111For all undefined set-
theoretic notions see [Jec02] or [Kan03]. the universe $V$ of all sets is
usually represented as a stratified cumulative hierarchy of sets indexed by
the ordinal numbers. Namely, $V=\bigcup_{\alpha\in OR}V_{\alpha}$, where

  1. $V_{0}=\varnothing$

  2. $V_{\alpha+1}=\mathcal{P}(V_{\alpha}),\;\mbox{ namely the set of all subsets of }V_{\alpha},\mbox{ and}$

  3. $V_{\lambda}=\bigcup_{\alpha<\lambda}V_{\alpha},\mbox{ if $\lambda$ is a limit ordinal.}$

This view of the set-theoretic universe justifies _a posteriori_ the ZF
axioms. For not only all ZF axioms are true in $V$, but they are also
necessary to build $V$. Indeed, the axioms of Extensionality, pairing, Union,
Power-set, and Separation are used to define the set-theoretic operation given
by

$G(x)=\bigcup\\{\mathcal{P}(y):\exists z(\langle z,y\rangle\in x)\\}.$

The axiom of Replacement is then needed to prove by transfinite recursion that
the operation $V$ on the ordinals given by $V(\alpha)=G(V\restriction\alpha)$
is well-defined and unique. Then $V(\alpha)=V_{\alpha}$ is as defined above.
The two remaining ZF axioms are easily justified: the axiom of Infinity leads
the ordinal sequence into the transfinite, and is the essence of set theory,
for its negation (namely, ZFC minus Infinity plus the negation of Infinity)
yields a theory mutually interpretable with Peano Arithmetic; and the axiom of
Regularity simply says, in the presence of the other ZF axioms, that there are
no more sets beyond those in $V$.

In the context of ZFC (ZF plus the Axiom of Choice), the _Axiom of Choice_ is
clearly true in $V$, although it is not necessary to build $V$.

In ZF, or ZFC, other representations of $V$ as a cumulative hierarchy are
possible. By “a cumulative hierarchy” we mean a union of sets $X_{\alpha}$,
defined by transfinite recursion on a _club_ , i.e., closed and unbounded,
class $C$ of ordinals $\alpha$, such that:

  1. $\alpha\leq\beta\mbox{ implies }X_{\alpha}\subseteq X_{\beta},$

  2. $X_{\lambda}=\bigcup_{\alpha<\lambda}X_{\alpha},\mbox{ if $\lambda$ is a limit point of $C$, and}$

  3. $V=\bigcup_{\alpha\in C}X_{\alpha}.$

For example, let $H_{\kappa}$, for $\kappa$ an infinite cardinal, be the set
of all sets whose transitive closure has cardinality less than $\kappa$. Then
the $H_{\kappa}$ also form a cumulative hierarchy indexed by the club class
$C\\!ARD$ of infinite cardinal numbers. (Note however that to prove
$\bigcup_{\kappa\in C\\!ARD}H_{\kappa}=V$ one needs the Axiom of Choice to
guarantee that every (transitive) set has a cardinality, and therefore every
set belongs to some $H_{\kappa}$.) Nevertheless, all representations of $V$ as
a cumulative hierarchy are essentially the same, in the following sense:
Suppose $V=\bigcup_{\alpha\in C}X_{\alpha}$ and $V=\bigcup_{\alpha\in
D}Y_{\alpha}$ are two cumulative hierarchies, where $C$ and $D$ are club
classes of ordinals. Then there is a club class of ordinals $E\subseteq C\cap
D$ such that $X_{\alpha}=Y_{\alpha}$, for all $\alpha\in E$.

Every representation of $V$ as a cumulative hierarchy is subject to the
_reflection_ phenomenon. Namely, the fact that every sentence of the first-
order language of set theory that holds in $V$, holds already at some stage of
the hierarchy. Indeed, the _Principle of Reflection_ of Montague and Lévy
([Lév60]), provable in $ZF$, asserts that every formula of the first-order
language of set theory true in $V$ holds in some $V_{\alpha}$. In fact, for
every formula $\varphi(x_{1},\ldots,x_{n})$ of the language of set theory, ZF
proves that there exists an ordinal $\alpha$ such that for every
$a_{1},\ldots,a_{n}\in V_{\alpha}$,

$\varphi(a_{1},\ldots,a_{n})\quad\mbox{ if and only if }\quad
V_{\alpha}\models\varphi(a_{1},\ldots,a_{n}).$

Even more is true: for each natural number $n$, there is a $\Pi_{n}$ definable
club proper class $C^{(n)}$ of ordinals such that ZF proves that for every
$\kappa\in C^{(n)}$, every $\Sigma_{n}$ formula $\varphi(x_{1},\ldots,x_{n})$,
and every $a_{1},\ldots,a_{n}\in V_{\kappa}$,

$\varphi(a_{1},\ldots,a_{n})\quad\mbox{ if and only if }\quad
V_{\kappa}\models\varphi(a_{1},\ldots,a_{n}).$

That is, $(V_{\kappa},\in)$ is a $\Sigma_{n}$-elementary substructure of
$(V,\in)$, henceforth written as $(V_{\kappa},\in)\preceq_{n}(V,\in)$, or
simply as $V_{\kappa}\preceq_{n}V$.

The import of the Principle of Reflection is highlighted by the result of Lévy
[Lév60] showing that the ZF axioms of Extensionality, Separation, and
Regularity, together with the principle of Complete Reflection (CR) imply ZF.
The CR principle is the schema asserting that for every formula
$\varphi(x_{1},\ldots,x_{n})$ of the language of set theory there exists a
transitive set $A$ closed under subsets (i.e, all subsets of elements of $A$
belong to $A$) such that for every $a_{1},\ldots,a_{n}\in A$,

$\varphi(a_{1},\ldots,a_{n})\leftrightarrow
A\models\varphi(a_{1},\ldots,a_{n}).$

Since the $V_{\alpha}$ are transitive and closed under subsets, this shows
that the reflection phenomenon, as expressed by the CR principle or the
Reflection Theorem (given the existence of the $V_{\alpha}$’s), is not only
deeply ingrained in the ZF axioms, but it captures the main content of ZF.

In the series of papers [Lév60, Lév61, Lév62] Lévy considers stronger
principles of reflection, formulated as axiom schemata, and he shows them
equivalent to the existence of inaccessible, Mahlo, and Hyper-Mahlo cardinals.
The unifying idea behind such principles is clearly stated by Lévy at the
beginning of [Lév61]:

> If we start with the idea of the impossibility of distinguishing, by
> specific means, the universe from partial universes we shall be led to the
> following axiom schemata, listed according to increasing strength. These
> axiom schemata will be called _principles of reflection_ since they state
> the existence of standard models (by models we shall mean, for the time
> being, models whose universes are sets) which reflect in some sense the
> state of the universe.

This idea of reflection, namely the impossibility of distinguishing the
universe from its partial universes (such as the $V_{\alpha}$), is also
implicit in earlier work of Ackermann ([Ack56]), but it is Lévy who
demonstrates how it can be used to find new natural theories strengthening ZF.
Indeed, in his review of Levy’s article [Lév62], Feferman [Fef] writes:

> The author’s earlier work demonstrated very well that the diversity of known
> set-theories could be viewed with more uniformity in the light of various
> reflection principles, and that these also provided a natural way to
> “manufacture” new theories. The present paper can only be regarded as a
> beginning of a systematic attempt to compare the results.

More recently, it has been argued by many authors that any intrinsic
justification of new set-theoretic axioms, beyond ZF, and in particular the
axioms of large cardinals, should be based on stronger forms of Lévy’s
reflection principles. In the next section we shall see some of these
arguments, as well as the limitations, clearly exposed by Koellner [Koe09],
inherent to all reflection principles akin to those studied by Lévy. In
Section 2 we will introduce new principles of reflection based on what we call
_Structural Reflection_ and will argue that they are in strong agreement with
the notion of reflection implicit in the original idea of Cantor’s of the
unknowability of the _absolute_ , which was subsequently developed in the
works of Ackermann, Lévy, Gödel, Reinhardt, and others. The rest of the paper,
starting with Section 3, will present a series of results showing that
different forms of the new principles of Structural Reflection are equivalent
to well-known large cardinals axioms covering all regions of the large-
cardinal hierarchy, thereby justifying the naturalness of the latter. 222The
work presented in the following sections started over ten years ago. After a
talk I gave in Barcelona in 2011 on large cardinals as principles of
structural reflection, John Baldwin, who attended the talk and was at the time
editor of the BSL, encouraged me to write a survey article for the Bulletin.
Well, here it is. I’m thankful to John for the invitation and I apologise to
him for the long delay.

##  1\. Set Theoretic axioms as reflection principles

Cantor ([Can83, p. 205, note 2]) emphasizes the unknowability of the
transfinite sequence of all ordinal numbers, which he thinks of as an
“appropriate symbol of the absolute”:

> The absolute can only be acknowledged, but never known, not even
> approximately known.

This principle of the unknowability of the absolute, which in Cantor’s work
seems to have only a metaphysical (non-mathematical) meaning (see [Jan10]),
resurfaces again in the 1950’s in the work of Ackermann and Lévy, taking the
mathematical form of a principle of reflection. Thus, in Ackermann’s set
theory – in fact, a theory of classes – which is formulated in the first-order
language of set theory with an additional constant symbol for the class $V$ of
all sets, the idea of reflection is expressed in the form of an axiom schema
of comprehension:

> _Ackermann’s Reflection:_ Let $\varphi(x,z_{1},\ldots,z_{n})$ be a formula
> which does not contain the constant symbol $V$. Then for every
> $\vec{a}=a_{1},\ldots,a_{n}\in V$,
>
> $\forall x(\varphi(x,\vec{a})\rightarrow x\in V)\rightarrow\exists y(y\in
> V\land\forall x(x\in y\leftrightarrow\varphi(x,\vec{a}))).$

A consequence of Ackermann’s Reflection is that no formula can define $V$, or
the class OR of all ordinal numbers, and is therefore in agreement with
Cantor’s principle of the unknowability of the absolute. However, Ackermann’s
set theory (with Foundation) was shown by Lévy [Lév59] and Reinhardt [Rei70]
to be essentially equivalent to ZF, in the sense that both theories are
equiconsistent and prove the same theorems about sets. Thus Ackermann’s set
theory did not provide any real advantage with respect to the simpler and
intuitively clearer ZF axioms, and so it was eventually forgotten.

Later on, in the context of the wealth of independence results in set theory
that were obtained starting in the mid-1960’s thanks to the forcing technique,
and as a result of the subsequent need for the identification of new set-
theoretic axioms, Gödel (as quoted by Wang [Wan96]), places Ackermann’s
principle (stated in a Cantorian, non-mathematical form) as the main source
for new set-theoretic axioms beyond ZFC:

> All the principles for setting up the axioms of set theory should be
> reducible to a form of Ackermann’s principle: The Absolute is unknowable.
> The strength of this principle increases as we get stronger and stronger
> systems of set theory. The other principles are only heuristic principles.
> Hence, the central principle is the reflection principle, which presumably
> will be understood better as our experience increases.

Thus, according to Gödel, the fundamental guiding principle in setting up new
axioms of set theory is the unknowability of the absolute, and so any new
axiom should be based on such principle. Gödel’s program consisted, therefore,
in formulating stronger and stronger systems of set theory by adding to the
base theory, which presumably could be taken as ZFC, new principles akin to
Ackermann’s.

So the question is how should one understand and formulate the idea of
reflection embodied in Ackermann’s principle. Some light is provided by Gödel
in the following quote from [Wan96, p. 285], where he asserts that the
indefinability of $V$ should be the source of all axioms of infinity, i.e.,
all large-cardinal axioms.

> Generally I believe that, in the last analysis, every axiom of infinity
> should be derivable from the (extremely plausible) principle that $V$ is
> indefinable, where definability is to be taken in [a] more and more
> generalized and idealized sense.

One possible interpretation of Gödel’s principle of the indefinability of $V$
is as an unrestricted version of the Montague-Lévy Principle of Reflection.
Namely: every formula, with parameters, in _any_ formal language with the
membership relation, that holds in $V$, must also hold in some $V_{\alpha}$.
This has been indeed the usual way to interprete Gödel’s view of reflection as
a justification for the axioms of large cardinals. This is made explicit in
Koellner [Koe09, p. 208], where he identifies reflection principles with
generalized forms of the Montague-Lévy Principle of Reflection. Namely,

> Reflection principles aim to articulate the informal idea that the height of
> the universe is “absolute infinite” and hence cannot be “characterized from
> below”. These principles assert that any statement true in $V$ is true in
> some smaller $V_{\alpha}$.

Moreover, he explicitly interpretes Gödel’s view of reflection as a source of
large cardinals in this way ([Koe09, p. 208]):

> Since the most natural way to assert that $V$ is indefinable is via
> reflection principles and since to assert this in a “more and more
> generalized and idealized sense” is to move to languages of higher-order
> with higher-order parameters, Gödel is (arguably) espousing the view that
> higher-order reflection principles imply all large cardinal axioms.

This kind of reflection, namely generalised forms of the Principle of
Reflection of Lévy and Montague allowing for the reflection of second-order
formulas, has been used to justify weak large-cardinal axioms, such as the
existence of inaccessible, Mahlo, or even weakly-compact cardinals. Let us see
briefly some examples to illustrate how the arguments work.

The Principle of Reflection also holds with proper classes as additional
predicates. Namely, for every (definable, with set parameters) proper class
$A$, and every $n$, there is an ordinal $\alpha$ such that

$(V_{\alpha},\in,A\cap V_{\alpha})\preceq_{n}(V,\in,A).$

Thus, if we interpret second-order quantifiers over $V$ as ranging only over
definable classes, then for each $n$, the following second-order sentence,
call it $\varphi_{n}$, is true in $V$

$\forall A\exists\alpha(V_{\alpha},\in,A\cap V_{\alpha})\preceq_{n}(V,\in,A).$

Now, applying reflection, $\varphi_{n}$ must reflect to some $V_{\kappa}$. The
second-order universal quantifier in $\varphi_{n}$ is now interpreted in
$V_{\kappa}$, hence ranging over _all_ subsets of $V_{\kappa}$, which are now
available in $V_{\kappa+1}$. So we have that for each subset $A$ of
$V_{\kappa}$, there is some $\alpha<\kappa$ such that

$(V_{\alpha},\in,A\cap V_{\alpha})\preceq_{n}(V_{\kappa},\in,A).$

If this is so, and even just for $n=1$, then $\kappa$ must be an inaccessible
cardinal, and this property actually characterises inaccessible cardinals
[Lév60]. Thus the existence of an inaccessible cardinal follows rather easily
from the reflection of the single $\Pi^{1}_{1}$ sentence $\varphi_{1}$ to some
$V_{\kappa}$.

Further, the following stronger form of reflection is provable in ZFC (as a
schema): if $C$ is a definable club proper class of ordinals, then for every
proper class $A$, and every $n$, there is a club proper class of $\alpha$ in
$C$ such that

$(V_{\alpha},\in,A\cap V_{\alpha})\preceq_{n}(V,\in,A).$

Hence, for each $n$, the following $\Pi^{1}_{1}$ sentence, with $C$ as a
second-order parameter, holds in $V$ (again, interpreting the universal
second-order quantifier as ranging over definable classes):

$\forall A\forall\beta\exists\alpha>\beta(\alpha\in
C\wedge(V_{\alpha},\in,A\cap V_{\alpha})\preceq_{n}(V,\in,A)).$

Applying reflection, there is $\kappa$ such that for every subset $A$ of
$V_{\kappa}$ there are unboundedly many $\alpha$ in $C\cap\kappa$ such that

$(V_{\alpha},\in,A\cap V_{\alpha})\preceq_{n}(V_{\kappa},\in,A).$

But since $C$ is a club, $\kappa\in C$, and so (in the case of $n\geq 1$)
$\kappa$ is an inaccessible cardinal in $C$. Now consider the following
$\Pi^{1}_{1}$ sentence, which we have just shown (using $\Pi^{1}_{1}$
reflection) to hold in $V$:

$\forall C(C\mbox{ is a club subclass of
ordinals}\rightarrow\exists\kappa(\kappa\mbox{ innaccessible}\wedge\kappa\in
C)).$

Any cardinal that reflects the $\Pi^{1}_{1}$ sentence consisting of the
conjunction of $\varphi_{1}$ above with the last displayed sentence is an
inaccessible cardinal $\lambda$ with the property that every club subset of
$\lambda$ contains an inaccessible cardinal, i.e., $\lambda$ is a Mahlo
cardinal.

Furthermore, if we are willing to assume that all $\Pi^{1}_{1}$ sentences with
second-order parameters reflect, we may as well reflect this property of $V$.
So, consider the following $\Pi^{1}_{2}$ sentence, which says that $V$
reflects all $\Pi^{1}_{1}$ sentences with parameters,

$\forall
A\forall\varphi\in\Pi^{1}_{1}\exists\alpha((V,\in,A)\models\varphi\rightarrow(V_{\alpha},\in,A\cap
V_{\alpha})\models\varphi).$

If $V_{\kappa}$ reflects this sentence, then $\kappa$ is a
$\Pi^{1}_{1}$-indescribable cardinal, i.e., a weakly-compact cardinal (see
[Jec02]). Similar arguments, applied to sentences of order $n$, but only
allowing first-order and second-order parameters, yield the existence of
$\Sigma^{m}_{n}$ and $\Pi^{m}_{n}$ indescribable cardinals.

While the arguments just given may seem reasonable, or even natural, we do not
think they provide a justification for the existence of the large cardinals
obtained in that way. The problem is that the relevant second-order statements
are true in $V$ only when interpreting second-order variables as ranging over
_definable_ classes, yet when the statements are reflected to some
$V_{\alpha}$ the second-order variables are reinterpreted as ranging over the
full power-set of $V_{\alpha}$. It is precisely this transition from definable
classes to the full power-set that yields the large-cardinal strength. Thus,
the fundamental objection is that second-order reflection from $V$ to some
$V_{\alpha}$, or to some set, is always problematic because so is unrestricted
second-order quantification over $V$, as the full _power-class_ of $V$ is not
available.

Nevertheless, even if one is willing to accept the existence of cardinals
$\kappa$ that reflect $n$-th–order sentences, with parameters of order greater
than $2$, one does not obtain large cardinals much stronger than the
indescribable ones. Indeed, to start with, and as first noted in [Rei74], one
cannot even have reflection for $\Pi^{1}_{1}$ sentences with unrestricted
third-order parameters. For suppose, towards a contradiction, that $\kappa$ is
a cardinal that reflects such sentences. Let $A$ be the collection
$\\{V_{\alpha}:\alpha<\kappa\\}$ taken as a third-order parameter, i.e., as a
subset of $\mathcal{P}(V_{\kappa})$. Then the $\Pi^{1}_{1}$ sentence

$\forall X\exists x(X\in A\to X=x)$

where $X$ is a second-order variable and $x$ is first-order, asserts that
every element of $A$ is a set. The sentence is clearly true in
$(V_{\kappa},\in,A)$, but false in any
$(V_{\alpha},\in,A\cap\mathcal{P}(V_{\alpha}))$ with $\alpha<\kappa$, because
$V_{\alpha}$ belongs to $A\cap\mathcal{P}(V_{\alpha})$ but is not an element
of $V_{\alpha}$.

One possible way around the problem of second-order reflection with third-
order parameters is to allow such parameters, or even higher-order parameters,
but to restrict the kind of sentences to be reflected. This is the approach
taken by Tait [Tai01]. He considers the class $\Gamma^{(2)}$ of formulas
which, in normal form, have all universal quantifiers restricted to first-
order and second-order variables and the only atomic formulas allowed to
appear negated are either those of first order or of the form $x\in X$, where
$x$ is a variable of first order and $X$ a variable of second order. Tait
shows that reflection at some $V_{\kappa}$ for the class of $\Gamma^{(2)}$
sentences, allowing parameters of arbitrarily-high finite order, implies that
$\kappa$ is an ineffable cardinal (see [Koe09]), and that $V_{\kappa}$
reflects all such sentences whenever $\kappa$ is a measurable cardinal. A
sharper upper bound on the consistency strength of this kind of reflection is
given by Koellner [Koe09, Theorem 9]. He shows that below the first
$\omega$-Erdös cardinal, denoted by $\kappa(\omega)$, there exists a cardinal
$\kappa$ such that $V_{\kappa}$ reflects all $\Gamma^{(2)}$ formulas. The
existence of $\kappa(\omega)$ is, however, a rather mild large-cardinal
assumption, since it is compatible with $V=L$.

At this point, the question is thus whether reflection can consistently hold
(modulo large cardinals) for a wider class of sentences. But Koellner [Koe09],
building on some results of Tait, shows that no $V_{\kappa}$ can reflect the
class of formulas of the form $\forall X\exists Y\varphi(X,Y,Z)$, where $X$ is
of third-order, $Y$ is of any finite order, $Z$ is of fourth order, and
$\varphi$ has only first-order quantifiers and its only negated atomic
subformulas are either of first order or of the form $x\in X$, where $x$ is of
first order and $X$ is of second order. Other kinds of restrictions on the
class of sentences to be reflected are possible (see [McC13] for the
consistency of some forms of reflection slightly stronger than Tait’s
$\Gamma^{(2)}$), but Koellner [Koe09] convincingly shows that the existence of
a cardinal $\kappa$ such that $V_{\kappa}$ reflects any reasonable expansion
of the class of sentences $\Gamma^{(2)}$, with parameters of order greater
than 2, either follows from the existence of $\kappa(\omega)$ or is outright
inconsistent.

These results seem to put an end to the program of providing an _intrinsic_
333See [Koe09] for a discussion on _intrinsic_ versus _extrinsic_
justification of the axioms of set theory. justification of large-cardinal
axioms, even for axioms as strong as the existence of $\kappa(\omega)$, by
showing that their existence follows from strong higher-order reflection
properties holding at some $V_{\kappa}$. In particular, the program cannot
even provide justification for the existence of measurable cardinals. Thus,
the conclusion is that if one understands reflection principles as asserting
that some sentences (even of higher order, and with parameters) that hold in
$V$ must hold in some $V_{\alpha}$, then reflection principles cannot be used
to justify the existence of large cardinals up to or beyond $\kappa(\omega)$.
Moreover, as we already emphasized, a more fundamental problem with the use of
higher-order reflection principles is that either second-order quantification
over $V$ is interpreted as ranging over definable classes, in which case
second-order reflection does not yield any large cardinals unless one makes
the dubious jump from definable classes to the full power-set, or is ill-
defined, as the full power-class of $V$ does not exist.

###  1.1. A remark on the undefinability of $V$

Before we go on to propose a new kind of reflection principles, let us take a
pause to consider another possible interpretation of Gödel’s principle of the
undefinability of $V$ as a justification for large-cardinal axioms.

The statement that a set $A$ is definable is usually understood in two
different senses:

  1. (1)

There is a formula $\varphi(x)$ that defines $A$. That is, for every set $a$,
$a$ belongs to $A$ if and only if $\varphi(a)$ holds. The formula $\varphi$
may have parameters, provided they are _simpler_ than $A$, e.g., their rank is
less than the rank of $A$.

  2. (2)

$A$ is the unique solution of a formula $\varphi(x)$. Again, $\varphi$ may
have parameters simpler than $A$.

There is, however, no essential difference between (1) and (2). For the
formula $\varphi(x)$ defines a set $A$ in the sense of (1) if and only if the
formula $\forall x(x\in y\leftrightarrow\varphi(x))$ defines $A$ in the sense
of (2). But if $A$ is a proper class, then (1) and (2) are very different,
even if only because (2) needs to be reformulated to make any sense. If we
understand definability as in (1), then there are many formulas that define
$V$, for instance the formula $x=x$. So, the notion of indefinability of $V$
can only be understood in the sense of (2), once properly reformulated. To
express that $V$ is not the unique solution of a formula, possibly with some
sets as parameters, we need to make sense of the fact that a formula is true
_of_ $V$, as opposed to being true _in_ $V$.

Let $\mathcal{L}_{V}$ be the first-order language of set theory expanded with
a constant symbol $\bar{a}$ for every set $a$, and a new constant symbol $v$.
Define the class $\mathcal{T}$ of sentences of $\mathcal{L}_{V}$ recursively
as follows: $\varphi$ belongs to $\mathcal{T}$ if and only if

  1. $\varphi$ is of the form $\bar{a}\in\bar{b}$ or $\bar{a}\in v$, for some sets $a,b$ such that $a\in b$, or

  2. $\varphi$ is of the form $\bar{a}=\bar{a}$ for some set $a$, or $v=v$, or

  3. $\varphi\equiv\neg\psi$, and $\psi$ does not belong to $\mathcal{T}$, or

  4. $\varphi\equiv\psi\wedge\theta$, and both $\psi$ and $\theta$ belong to $\mathcal{T}$, or

  5. $\varphi\equiv\exists y\psi(y)$, and there is a set $a$ such that $\psi(\bar{a})$ belongs to $\mathcal{T}$.

The idea is that if $\varphi$ belongs to $\mathcal{T}$, then $\varphi$ is true
in the structure

$\bar{V}:=\langle V\cup\\{V\\},\in,V,\langle\bar{a}\rangle_{a\in V}\rangle$

where the constant $v$ is interpreted as $V$. And conversely, if $\varphi$ is
a sentence in the language $\mathcal{L}_{V}$ that is true in $\bar{V}$, then
$\varphi\in\mathcal{T}$. Thus one may construe the principle of indefinability
of $V$, as expressed in Gödel’s quote above, as follows

> _Indefinability of $V$:_ Every $\varphi\in\mathcal{T}$ is true in some
>
> $\bar{V}_{\alpha}:=\langle
> V_{\alpha}\cup\\{V_{\alpha}\\},\in,V_{\alpha},\langle\bar{a}\rangle_{a\in
> V_{\alpha}}\rangle.$

Of course, to express this principle in the first-order language of set theory
one needs to do it as a schema. Namely, for each $n$ let $\mathcal{T}_{n}$ be
the class of $\Sigma_{n}$ sentences of $\mathcal{T}$, and let

> _$\Sigma_{n}$ -Indefinability of $V$:_ Every $\varphi\in\mathcal{T}_{n}$ is
> true in some $\bar{V}_{\alpha}$.

Given a $\Sigma_{n}$ sentence $\varphi$ of the language of set theory, where
$n\geq 1$, if it is true in $V$, i.e., if $\models_{n}\varphi$ holds, then the
sentence $\varphi^{v}$ obtained from $\varphi$ by bounding all quantifiers by
$v$ belongs to $\mathcal{T}_{n}$. Hence, by $\Sigma_{n}$_-Indefinability of_
$V$, $\varphi^{v}$ is true in some $\bar{V}_{\alpha}$, and therefore
$V_{\alpha}\models\varphi$. Thus _Indefinability of $V$_ directly implies the
Principle of Reflection of Montague-Lévy (over the theory ZF minus Infinity).
Conversely, by induction on the complexity of $\varphi$, it is easily shown
that ZF proves $\Sigma_{n}$-_Indefinability of $V$_. Thus, to derive stronger
reflection principles based on the indefinability of $V$ one needs to
understand “definability”, following Gödel’s quote above, in a “more and more
generalized and idealized sense”. One could expand the class $\mathcal{T}$ by
adding higher-order sentences that are true of $V$. However this will not lead
us very far. For if $\mathcal{T}^{\prime}$ is any reasonable class of (higher-
order) sentences that are true of $V$, then _Indefinability of $V$_ for the
class $\mathcal{T}^{\prime}$ will imply the reflection, in the sense of
Montague-Lévy, of all sentences in $\mathcal{T}^{\prime}$. Therefore, the
limitations seen above of the extensions of the Montague-Lévy Principle of
Reflection to higher-order formulas apply also to these generalized forms of
_Indefinability of $V$_.

##  2\. Structural Reflection

The main obstacle for the program of finding an intrinsic justification of
large-cardinal axioms via strong principles of reflection lies, we believe, on
a too restrictive interpretation of the notion of reflection, namely the
interpretation investigated by Tait, Koellner, and others, according to which
the reflection properties of $V$ are exhausted by generalized forms of the
Montague-Lévy Principle of Reflection to higher-order logics.

Let us think again about the notion of reflection as derived from the Cantor-
Ackermann principle of the unknowability of the absolute. A different
interpretation of this principle may be extracted from another claim made by
Gödel, as quoted in [Wan96]:

> The universe of sets cannot be uniquely characterized (i.e., distinguished
> from all its initial segments) by any internal structural property of the
> membership relation in it which is expressible in any logic of finite or
> transfinite type, including infinitary logics of any cardinal number.

This quote does not immediately suggest that the uncharacterizability of $V$
should be interpreted in the sense of the Montague-Lévy kind of reflection.
Rather, what it seems to suggest is some sort of reflection, not (only) of
formulas, but of _internal structural properties of the membership relation_.
The quote does also state that the properties should be expressible in some
logic, and any reasonable logic would do. So maybe Gödel is not saying here
anything new, and he is simply advocating for a generalizsation of the
Montague-Lévy type of reflection to formulas belonging to any (reasonable)
kind of logic. But whatever the correct interpretation of Gödel’s quote above
may be, let us consider in some detail the idea of reflection of structural
properties of the membership relation. Thus, what one would want to reflect is
not the theory of $V$, but rather the _structural content_ of $V$.

Whatever one might mean by a “structural property of the membership relation”,
it is clear that such a property should be exemplified in structures of the
form $\langle A,\in,\langle R_{i}\rangle_{i\in I}\rangle$, where $A$ is a set
and $\langle R_{i}\rangle_{i\in I}$ is a family of relations on $A$, and where
$I$ is a set that may be empty. Moreover, any such property should be
expressible by some formula of the language of set theory, maybe involving
some set parameters. Thus, any internal structural property of the membership
relation would be formally given by a formula $\varphi(x)$ of the first-order
language of set theory, possibly with parameters, that defines a class of
structures $\langle A,\in,\langle R_{i}\rangle_{i\in I}\rangle$ of the same
type. As we shall see later on there is no loss of generality in considering
only classes of structures whose members are of the form $\langle
V_{\alpha},\in,\langle R_{i}\rangle_{i\in I}\rangle$. Now, Gödel’s vague
assertion (as quoted above) that $V$ “cannot be uniquely characterized (i.e.,
distinguished from all its initial segments) by any internal structural
property of the membership relation” can be naturally interpreted in the sense
that no formula $\varphi(x)$ characterizes $V$, meaning that some $V_{\alpha}$
reflects _the structural property defined by $\varphi(x)$_. Let us emphasize
that what is reflected is not the formula $\varphi(x)$, but the structural
property defined by $\varphi(x)$, i.e., the class of structures defined by
$\varphi(x)$. This is the crucial difference with the Montague-Lévy type of
reflection. The most natural way to make this precise is to assert that there
exists an ordinal $\alpha$ such that for every structure $A$ in the class
(i.e., for every structure $A$ that satisfies $\varphi(x)$) there exists a
structure $B$ also in the class which belongs to $V_{\alpha}$ and is _like_
$A$. Since, in general, $A$ may be much larger than any $B$ in $V_{\alpha}$,
the closest resemblance of $B$ to $A$ is attained in the case $B$ is
isomorphic to an elementary substructure of $A$, i.e., $B$ can be elementarily
embedded into $A$. We can now formulate (an informal and preliminary version
of) the general principle of Structural Reflection as follows:

>   * ${\rm SR}$:
>
> (_Structural Reflection_) For every definable, in the first-order language
> of set theory, possibly with parameters, class $\mathcal{C}$ of relational
> structures of the same type there exists an ordinal $\alpha$ that _reflects_
> $\mathcal{C}$, i.e., for every $A$ in $\mathcal{C}$ there exists $B$ in
> $\mathcal{C}\cap V_{\alpha}$ and an elementary embedding from $B$ into $A$.
>
>

Observe that when $\mathcal{C}$ is a set the principle becomes trivial.

We do not wish to claim that the ${\rm SR}$ principle is what Gödel had in
mind when talking about reflection of internal structural properties of the
membership relation, but we do claim that ${\rm SR}$ is a form of reflection
that derives naturally from the Cantor-Ackermann principle of unknowability of
the Absolute and is at least compatible with Gödel’s interpretation of this
principle.

In the remaining sections we will survey a collection of results showing the
equivalence of different forms of ${\rm SR}$ with the existence of different
kinds of large cardinals. Our goal is to illustrate the fact that ${\rm SR}$
is a general principle underlying a wide variety of large-cardinal principles.
Many of the results have already been published ([Bag12, BCMR15, BV16, BGS17,
BW20, Lüc21]) or are forthcoming ([BL21]), but some are new (4.1, 4.2, 4.5,
4.7, 4.8, 4.9, 5.1, 5.11, 5.17, 5.18, 5.19, 5.21, and 7.6). Each of these
results should be regarded as a small step towards the ultimate objective of
showing that all large cardinals are in fact different manifestations of a
single general reflection principle.

##  3\. From supercompactness to Vopěnka’s Principle

We shall begin with the ${\rm SR}$ principle, as stated above, which is
properly formulated in the first-order language of set theory as an axiom
schema. Namely, for each natural number $n$ let

>   * $\mathbf{\Sigma_{n}}$-${\rm SR}$:
>
> (_$\mathbf{\Sigma_{n}}$ -Structural Reflection_) For every
> $\Sigma_{n}$-definable, with parameters, class $\mathcal{C}$ of relational
> structures of the same type there is an ordinal $\alpha$ that _reflects_
> $\mathcal{C}$.
>
>

$\mathbf{\Pi_{n}}$-${\rm SR}$ may be formulated analogously. We may also
define the _lightface_ , i.e., parameter-free versions (as customary we use
the lightface types $\Sigma_{n}$ and $\Pi_{n}$ for that). Namely,

>   * $\Sigma_{n}$-${\rm SR}$:
>
> (_$\Sigma_{n}$ -Structural Reflection_) For every $\Sigma_{n}$-definable,
> without parameters, class $\mathcal{C}$ of relational structures of the same
> type there exists an ordinal $\alpha$ that _reflects_ $\mathcal{C}$.
>
>

Similarly for $\Pi_{n}$-${\rm SR}$.

A standard closing-off argument shows that $\mathbf{\Sigma_{n}}$-${\rm SR}$ is
equivalent to the assertion that there exists a proper class of ordinals
$\alpha$ such that $\alpha$ reflects _all_ classes of structures of the same
type that are $\Sigma_{n}$-definable, with parameters in $V_{\alpha}$. Also,
$\Sigma_{n}$-${\rm SR}$ is equivalent to the assertion that there exists an
ordinal $\alpha$ that reflects _all_ classes of structures of the same type
that are $\Sigma_{n}$-definable, without parameters. Similarly for
$\mathbf{\Pi_{n}}$-${\rm SR}$ and $\Pi_{n}$-${\rm SR}$. Thus, for $\Gamma$ a
_definability class_ (i.e., one of $\mathbf{\Sigma_{n}}$, $\mathbf{\Pi_{n}}$,
$\Sigma_{n}$, or $\Pi_{n}$), let us say that an ordinal $\alpha$ _witnesses_
$\Gamma$-${\rm SR}$ if $\alpha$ reflects all classes of structures of the same
type that are $\Gamma$-definable (allowing for parameters in $V_{\alpha}$, in
the case of boldface classes). Then, in the case of boldface classes $\Gamma$,
$\Gamma$-${\rm SR}$ holds if and only if $\Gamma$-${\rm SR}$ is witnessed by a
proper class of ordinals.

The first observation is that, as the next proposition shows,
$\mathbf{\Sigma_{1}}$-${\rm SR}$ is provable in ZFC. Recall444See [Bag12].
that, for every $n>0$, $C^{(n)}$ is the $\Pi_{n}$-definable club proper class
of cardinals $\kappa$ such that $V_{\kappa}\preceq_{\Sigma_{n}}V$, i.e.,
$V_{\kappa}$ is a $\Sigma_{n}$-elementary substructure of $V$. In particular,
every element of $C^{(1)}$ is an uncountable cardinal and a fixed point of the
$\beth$ function.

######  Proposition 3.1.

The following are equivalent for every ordinal $\alpha$:

  1. (1)

$\alpha$ witnesses $\mathbf{\Sigma_{0}}$-${\rm SR}$

  2. (2)

$\alpha$ witnesses $\mathbf{\Sigma_{1}}$-${\rm SR}$

  3. (3)

$\alpha\in C^{(1)}$.

###### Proof.

The implication (2)$\Rightarrow$(1) is trivial. The implication
(3)$\Rightarrow$(2) is proved in [BCMR15], using a Löwenheim-Skolem type of
argument.

To show that (1) implies (3), let $\alpha$ witness $\mathbf{\Sigma_{0}}$-${\rm
SR}$ and suppose $\varphi(a_{1},\ldots,a_{n})$ is a $\Sigma_{1}$ sentence,
with parameters $a_{1},\ldots,a_{n}$ in $V_{\alpha}$, that holds in $V$. Let
${\mathcal{C}}$ be the $\Sigma_{0}$-definable, with $a_{1},\ldots,a_{n}$ as
parameters, class of structures of the form

$\langle M,\in,\\{a_{1},\ldots,a_{n}\\}\rangle$

where $M$ is a transitive set that contains $a_{1},\ldots,a_{n}$. Let $M$ be
any transitive set such that $M\models\varphi(a_{1},\ldots,a_{n})$. By ${\rm
SR}$, there is an elementary embedding

$j:\langle N,\in,\\{a_{1},\ldots,a_{n}\\}\rangle\to\langle
M,\in,\\{a_{1},\ldots,a_{n}\\}\rangle$

where $\langle N,\in\\{a_{1},\ldots,a_{n}\\}\rangle\in{\mathcal{C}}\cap
V_{\alpha}$. Since $j$ fixes $a_{1},\ldots,a_{n}$, by elementarity
$N\models\varphi(a_{1},\ldots,a_{n})$, and by upwards absoluteness for
$\Sigma_{1}$ sentences with respect to transitive sets,
$V_{\alpha}\models\varphi(a_{1},\ldots,a_{n})$. This shows $\alpha\in
C^{(1)}$. ∎

Thus, $\mathbf{\Sigma_{1}}$-${\rm SR}$ is provable in ZFC, and therefore does
not yield any large cardinals. But $\Pi_{1}$-${\rm SR}$ does, and is indeed
very strong. The following theorem hinges on Magidor’s characterization of the
first supercompact cardinal as the first cardinal that reflects the
$\Pi_{1}$-definable class of structures of the form $\langle
V_{\alpha},\in\rangle$, $\alpha$ an ordinal ([Mag71]).

######  Theorem 3.2 ([Bag12, BCMR15]).

The following are equivalent:

  1. (1)

$\Pi_{1}$-${\rm SR}$.

  2. (2)

$\Sigma_{2}$-${\rm SR}$.

  3. (3)

There exists a supercompact cardinal.

The proof of the theorem shows in fact that the following are equivalent for
an ordinal $\kappa$:

  1. (1)

$\kappa$ is the least ordinal that witnesses ${\rm SR}$ for the
$\Pi_{1}$-definable class of structures $\langle V_{\alpha},\in\rangle$,
$\alpha$ an ordinal.

  2. (2)

$\kappa$ is the least cardinal that witnesses $\Pi_{1}$-${\rm SR}$.

  3. (3)

$\kappa$ is the least cardinal that witnesses $\mathbf{\Sigma_{2}}$-${\rm
SR}$.

  4. (4)

$\kappa$ is the least supercompact cardinal.

The following global parametrized version then follows. Namely,

######  Theorem 3.3 ([Bag12, BCMR15]).

The following are equivalent:

  1. (1)

$\mathbf{\Pi_{1}}$-${\rm SR}$.

  2. (2)

$\mathbf{\Sigma_{2}}$-${\rm SR}$.

  3. (3)

There exists a proper class of supercompact cardinals.

The proof of the theorem also shows that if $\kappa$ witnesses
$\mathbf{\Pi_{1}}$-${\rm SR}$, then $\kappa$ is either supercompact or a limit
of supercompact cardinals.

Some remarks are in order. First, the equivalence of $\Pi_{1}$-${\rm SR}$ and
$\Sigma_{2}$-${\rm SR}$, and also of their boldface forms, is due to the
following general fact. Given a $\Sigma_{n+1}$ definable (possibly with
parameters, and with $n>0$) class $\mathcal{C}$ of relational structures of
the same type, let $\mathcal{C}^{\ast}$ be the class of structures of the form
$\langle V_{\alpha},\in,A\rangle$, where $\alpha$ is the least cardinal in
$C^{(n)}$ such that $A\in V_{\alpha}$ and $V_{\alpha}\models\varphi(A)$, where
$\varphi(x)$ is a fixed $\Sigma_{n+1}$ formula that defines ${\mathcal{C}}$.
Then,

$A\in\mathcal{C}\mbox{ if and only if }\langle
V_{\alpha},\in,A\rangle\in\mathcal{C}^{\ast}.$

Now notice that $\mathcal{C}^{\ast}$ is $\Pi_{n}$ definable, with the same
parameters as $\mathcal{C}$ (see [Bag12]). Moreover, if a cardinal $\kappa$
reflects the class ${\mathcal{C}}^{\ast}$, then it also reflects
${\mathcal{C}}$: for if $A\in{\mathcal{C}}$, let $\alpha$ be the least
cardinal in $C^{(n)}$ such that $\langle
V_{\alpha},\in,A\rangle\models\varphi(A)$, where $\varphi(x)$ is a fixed
$\Sigma_{n+1}$ formula that defines ${\mathcal{C}}$. Let $j:\langle
V_{\beta},\in,B\rangle\to\langle V_{\alpha},\in,A\rangle$ be elementary with
$\langle V_{\beta},\in,B\rangle\in{\mathcal{C}}^{\ast}\cap V_{\kappa}$. Then,
since $\beta\in C^{(n)}$ and $V_{\beta}\models\varphi(B)$, we have that
$B\in{\mathcal{C}}$ and the restriction map $j\restriction A:A\to B$ is an
elementary embedding.

For $P$ a set or a proper class and $\Gamma$ a definability class, we shall
write $\Gamma(P)$-${\rm SR}$ for the assertion that ${\rm SR}$ holds for all
$\Gamma$-definable, with parameters in $P$, classes of structure of the same
type. Thus, e.g., $\mathbf{\Sigma_{n}}$-${\rm SR}$ is $\Sigma_{n}(V)$-${\rm
SR}$, and $\Sigma_{n}$-${\rm SR}$ is $\Sigma_{n}(\varnothing)$-${\rm SR}$. Our
remarks above yield now the following:

######  Proposition 3.4.

For $P$ any set or proper class, the assertions $\Pi_{n}(P)$-${\rm SR}$ and
$\Sigma_{n+1}(P)$-${\rm SR}$ are equivalent. In particular $\Pi_{n}$-${\rm
SR}$ and $\Sigma_{n+1}$-${\rm SR}$ are equivalent; and so are
$\mathbf{\Pi_{n}}$-${\rm SR}$ and $\mathbf{\Sigma_{n+1}}$-${\rm SR}$.

Second, the remarks above also show that for principles of Structural
Reflection of the form $\Gamma(P)$-${\rm SR}$ the relevant structures to
consider are those of the form $\langle V_{\alpha},\in,A\rangle$, where $A\in
V_{\alpha}$. Let us say that a structure is _natural_ if it is of this form.
Therefore, we may reformulate $\Gamma$-${\rm SR}$, for $\Gamma$ a lightface
definability class, as follows:

>   * $\Gamma$-${\rm SR}$:
>
> (_$\Gamma$ -Structural Reflection. Second version_) There exists a cardinal
> $\kappa$ that reflects all $\Gamma$-definable classes ${\mathcal{C}}$ of
> natural structures, i.e., for every $A\in{\mathcal{C}}$ there exists
> $B\in{\mathcal{C}}\cap V_{\kappa}$ and an elementary embedding $j:B\to A$.
>
>

The version for $\Gamma$ a boldface definability class being as follows:

>   * There exist a proper class of cardinals $\kappa$ that reflect all
> $\Gamma$-definable, with parameters in $V_{\kappa}$, classes ${\mathcal{C}}$
> of natural structures.
>
>

At the next level of definitional complexity, i.e., $n=2$, we have the
following:

######  Theorem 3.5 ([Bag12, BCMR15]).

The following are equivalent:

  1. (1)

$\Pi_{2}$-${\rm SR}$.

  2. (2)

There exists an extendible cardinal.

The proof of the theorem shows that the first extendible cardinal is precisely
the first cardinal that witnesses SR for one particular $\Pi_{2}$-definable
class of natural structures. The parameterized version also holds:

######  Theorem 3.6 ([Bag12, BCMR15]).

The following are equivalent:

  1. (1)

$\mathbf{\Pi_{2}}$-${\rm SR}$.

  2. (2)

There exists a proper class of extendible cardinals.

Moreover, if $\kappa$ witnesses $\mathbf{\Pi_{2}}$-${\rm SR}$, then $\kappa$
is either extendible or a limit of extendible cardinals.

For the higher levels of definitional complexity we need the notion of
$C^{(n)}$-extendible cardinal from [Bag12, BCMR15]: $\kappa$ is _$C^{(n)}$
-extendible_ if for every $\lambda$ greater than $\kappa$ there exists an
elementary embedding $j:V_{\lambda}\to V_{\mu}$, some $\mu$, with
$crit(j)=\kappa$, $j(\kappa)>\lambda$, and $j(\kappa)\in C^{(n)}$. Note that
the only difference with the notion of extendibility is that we require the
image of the critical point to be in $C^{(n)}$. Also note that every
extendible cardinal is $C^{(1)}$-extendible. We then have the following level-
by-level characterizations of ${\rm SR}$ in terms of the existence of large
cardinals:

######  Theorem 3.7 ([Bag12, BCMR15]).

The following are equivalent for $n\geq 1$:

  1. (1)

$\Pi_{n+1}$-${\rm SR}$.

  2. (2)

There exists a $C^{(n)}$-extendible cardinal.

######  Theorem 3.8 ([Bag12, BCMR15]).

The following are equivalent for $n\geq 1$:

  1. (1)

$\mathbf{\Pi_{n+1}}$-${\rm SR}$.

  2. (2)

There exists a proper class of $C^{(n)}$-extendible cardinals.

Similarly as in the case of supercompact and extendible cardinals, the proofs
of the theorems above actually show that the first $C^{(n)}$-extendible
cardinal is the first cardinal that witnesses SR for one single
$\Pi_{n+1}$-definable class of natural structures. Also, if $\kappa$ witnesses
$\mathbf{\Pi_{n+1}}$-${\rm SR}$, then $\kappa$ is either a
$C^{(n)}$-extendible cardinal or a limit of $C^{(n)}$-extendible cardinals.

Recall that _Vopěnka’s Principle $({\rm VP})$_ is the assertion that for every
proper class $\mathcal{C}$ of relational structures of the same type there
exist $A\neq B$ in $\mathcal{C}$ such that $A$ is elementarily embeddable into
$B$. In the first-order language of set theory ${\rm VP}$ can be formulated as
a schema. The following corollary to the theorems stated above yields a
characterization of ${\rm VP}$ in terms of ${\rm SR}$. Moreover, it shows
that, globally, the lightface and boldface forms of ${\rm SR}$ are equivalent.

######  Theorem 3.9 ([Bag12, BCMR15]).

The following schemata are equivalent:

  1. (1)

${\rm SR}$, i.e., $\mathbf{\Pi_{n}}$-${\rm SR}$ for all $n$.

  2. (2)

$\Pi_{n}$-${\rm SR}$ for all $n$.

  3. (3)

There exists a $C^{(n)}$-extendible cardinal, for every $n$.

  4. (4)

There is a proper class of $C^{(n)}$-extendible cardinals, for every $n$.

  5. (5)

${\rm VP}$.

##  4\. Structural Reflection below supercompactness

We have just seen that a natural hierarchy of large cardinals in the region
between the first supercompact cardinal and ${\rm VP}$ can be characterized in
terms of ${\rm SR}$. Now the question is if the same is true for other well-
known regions of the large cardinal hierarchy. Since $\mathbf{\Sigma_{1}}$-SR
is provable in ZFC and $\Pi_{1}$-SR implies already the existence of a
supercompact cardinal, if large cardinals weaker than supercompact admit a
characterization as principles of structural reflection, then we need to look
either for ${\rm SR}$ restricted to particular (collections of)
$\Pi_{1}$-definable classes of structures, or for classes of structures whose
definitional complexity is between $\Sigma_{1}$ and $\Pi_{1}$ (e.g.,
$\Sigma_{1}$-definability with additional $\Pi_{1}$ predicates), or for weaker
forms of structural reflection. Let us consider first the ${\rm SR}$ principle
restricted to particular definable classes of structures contained in
canonical inner models.

###  4.1. Structural Reflection relative to canonical inner models

There is one single class ${\mathcal{C}}$ of structures in $L$ that is
$\Pi_{1}$-definable in $V$, without parameters, and such that ${\rm
SR}({\mathcal{C}})$ is equivalent to the existence of $0^{\sharp}$. Namely,
let ${\mathcal{C}}$ be the class of structures of the form $\langle
L_{\beta},\in,\gamma\rangle$, with $\gamma<\beta$ uncountable cardinals (in
$V$).

######  Theorem 4.1.

The following are equivalent:

  1. (1)

${\rm SR}({\mathcal{C}})$

  2. (2)

$0^{\sharp}$ exists.

###### Proof.

(1) implies (2): Suppose that $\alpha$ reflects ${\mathcal{C}}$. Pick
$V$-cardinals $\gamma<\beta$ with $\alpha\leq\gamma$. By reflection, there are
$V$-cardinals $\gamma^{\prime}<\beta^{\prime}<\alpha$ and an elementary
embedding

$j:\langle L_{\beta^{\prime}},\in,\gamma^{\prime}\rangle\to\langle
L_{\beta},\in,\gamma\rangle.$

Since $j(\gamma^{\prime})=\gamma$, $j$ is not the identity. Let $\kappa$ be
the critical point of $j$. Thus, $\kappa\leq\gamma^{\prime}<\beta^{\prime}$.
Hence by Kunen’s Theorem ([Kun71]) $0^{\sharp}$ exists.

(2) implies (1): Assume $0^{\sharp}$ exists. Let $\alpha$ be an uncountable
limit cardinal in $V$. We claim that $\alpha$ reflects ${\mathcal{C}}$. For
suppose $\langle L_{\beta},\in,\gamma\rangle\in{\mathcal{C}}$ with
$\alpha\leq\beta$. Let $\gamma^{\prime}<\beta^{\prime}<\alpha$ be uncountable
cardinals in $V$ such that $\gamma^{\prime}\leq\gamma$. Let $I$ denote the
class of Silver indiscernibles. Let
$j:I\cap[\gamma^{\prime},\beta^{\prime}]\to I\cap[\gamma,\beta]$ be order-
preserving and such that $j(\gamma^{\prime})=\gamma$ and
$j(\beta^{\prime})=\beta$. Then $j$ generates an elementary embedding

$j:\langle L_{\beta^{\prime}},\in,\gamma^{\prime}\rangle\to\langle
L_{\beta},\in,\gamma\rangle$

as required. ∎

The existence of $0^{\sharp}$ yields also the ${\rm SR}$ principle restricted
to classes of structures that are definable in $L$.

######  Theorem 4.2.

If $0^{\sharp}$ exists, then ${\rm SR}({\mathcal{C}})$ holds for every class
${\mathcal{C}}$ that is definable in $L$, with parameters.

###### Proof.

Fix $\mathcal{C}$ and a formula $\varphi(x)$, possibly with ordinals
$\alpha_{0}<\ldots<\alpha_{m}$ as parameters, that defines it in $L$. Let
$\kappa$ be a limit of Silver indiscernibles greater than $\alpha_{m}$. We
claim that $\kappa$ reflects $\mathcal{C}$. For suppose $B\in\mathcal{C}$.
Without loss of generality, $B\not\in L_{\kappa}$. Since $0^{\sharp}$ exists,
there is an increasing sequence of Silver indiscernibles
$i_{0},\ldots,i_{n},i_{n+1}$ and a formula $\psi(y,z_{0},\ldots,z_{n})$,
without parameters, such that

$B=\\{y:L_{i_{n+1}}\models\psi(y,i_{0},\ldots,i_{n})\\}.$

Choose indiscernibles $j_{0}<\ldots<j_{n}<j_{n+1}<\kappa$, with
$\alpha_{m}<j_{0}$, and let

$A=\\{y:L_{j_{n+1}}\models\psi(y,j_{0},\ldots,j_{n})\\}.$

Thus $A\in L_{\kappa}$. Since $L\models\varphi(B)$, we have that

$L\models\forall x(\forall y(y\in x\leftrightarrow
L_{i_{n+1}}\models\psi(y,i_{0},\ldots,i_{n}))\to\varphi(x)).$

By indiscernibility,

$L\models\forall x(\forall y(y\in x\leftrightarrow
L_{j_{n+1}}\models\psi(y,j_{0},\ldots,j_{n}))\to\varphi(x))$

which implies $L\models\varphi(A)$, i.e., $A\in\mathcal{C}$.

Let $j:L\to L$ be an elementary embedding that sends $i_{k}$ to $j_{k}$, all
$k\leq n+1$. Then by indiscernibility, the map $j\restriction A:A\to B$ is an
elementary embedding. ∎

However, the ${\rm SR}$ principle restricted to classes of structures that are
definable in $L$ falls very short of yielding $0^{\sharp}$, as we shall next
show. Let us recall the following definition:

######  Definition 4.3 ([BGS17]).

A cardinal $\kappa$ is _$n$ -remarkable_, for $n>0$, if for all
$\lambda>\kappa$ in $C^{(n)}$ and every $a\in V_{\lambda}$, there is
$\bar{\lambda}<\kappa$ also in $C^{(n)}$ such that in
$V^{\rm{Coll}(\omega,<\kappa)}$ there exists an elementary embedding
$j:V_{\bar{\lambda}}\to V_{\lambda}$ with $j({\rm{crit}}(j))=\kappa$ and
$a\in{\rm{range}}(j)$.

A cardinal $\kappa$ is $1$-remarkable if and only if it is remarkable, in the
sense of Schindler (see [BGS17] and definition 7.2 below).

If $0^{\sharp}$ exists, then every Silver indiscernible is completely
remarkable in $L$ (i.e., $n$-remarkable for every $n>0$). Moreover, the
consistency strength of the existence of a $1$-remarkable cardinal is strictly
weaker than the existence of a $2$-iterable cardinal, which in turn is weaker
than the existence of an $\omega$-Erdös cardinal (see [BGS17]).

A weaker notion than $n$-remarkability is obtained by eliminating from its
definition the requirement that $j({\rm crit}(j))=\kappa$. So, let’s define:

######  Definition 4.4.

A cardinal $\kappa$ is _almost $n$-remarkable_, for $n>0$, if for all
$\lambda>\kappa$ in $C^{(n)}$ and every $a\in V_{\lambda}$, there is
$\bar{\lambda}<\kappa$ also in $C^{(n)}$ such that in
$V^{\rm{Coll}(\omega,<\kappa)}$ there exists an elementary embedding
$j:V_{\bar{\lambda}}\to V_{\lambda}$ with $a\in{\rm{range}}(j)$.

We say that $\kappa$ is _almost completely-remarkable_ if it is
almost-$n$-remarkable for every $n$.

######  Theorem 4.5.

A cardinal $\kappa$ is almost $n$-remarkable if and only if in
$V^{\rm{Coll}(\omega,<\kappa)}$ $\kappa$ witnesses ${\rm SR}({\mathcal{C}})$
for every class ${\mathcal{C}}$ that is $\Pi_{n}$-definable in $V$ with
parameters in $V_{\kappa}$.

###### Proof.

Assume $\kappa$ is almost $n$-remarkable. Fix $\mathcal{C}$ and a $\Pi_{n}$
formula $\varphi(x)$ that defines it in $V$, possibly with parameters in
$V_{\kappa}$. Suppose $B\in\mathcal{C}$. In $V$, let $\lambda\in C^{(n)}$ be
greater than the rank of $B$. Thus, $V_{\lambda}\models\varphi(B)$. Since
$\kappa$ is almost $n$-remarkable, there is $\bar{\lambda}<\kappa$ also in
$C^{(n)}$ such that in $V^{\rm{Coll}(\omega,<\kappa)}$ there exists an
elementary embedding $j:V_{\bar{\lambda}}\to V_{\lambda}$ with
$B\in{\rm{range}}(j)$. Let $A$ be the preimage of $B$ under $j$. So $A\in
V_{\kappa}$. By elementarity of $j$, $V_{\bar{\lambda}}\models\varphi(A)$.
Hence, since $\bar{\lambda}\in C^{(n)}$, $A\in\mathcal{C}$. Moreover,
$j\restriction A:A\to B$ is an elementary embedding.

Conversely, assume that in $V^{\rm{Coll}(\omega,<\kappa)}$, $\kappa$ witnesses
${\rm SR}({\mathcal{C}})$ for every class ${\mathcal{C}}$ that is
$\Pi_{n}$-definable in $V$ with parameters in $V_{\kappa}$. Let
${\mathcal{C}}$ be the $\Pi_{n}$-definable class of structures of the form
$\langle V_{\alpha},\in,a\rangle$ where $\alpha\in C^{(n)}$ and $a\in
V_{\alpha}$. Given $\lambda\in C^{(n)}$ and $a\in V_{\lambda}$, in
$V^{\rm{Coll}(\omega,<\kappa)}$ there exists some $\langle
V_{\bar{\lambda}},\in,b\rangle\in{\mathcal{C}}$ together with an elementary
embedding

$j:\langle V_{\bar{\lambda}},\in,b\rangle\to\langle V_{\lambda},\in,a\rangle.$

Since $j(b)=a$, $a\in{\rm{range}}(j)$. This shows $\kappa$ is almost
$n$-remarkable. ∎

######  Corollary 4.6.

If $\kappa$ is an almost completely-remarkable cardinal in $L$, then in
$L^{\rm{Coll}(\omega,<\kappa)}$ $\kappa$ witnesses ${\rm SR}({\mathcal{C}})$
for all classes ${\mathcal{C}}$ of structures that are definable in $L$ with
parameters in $L_{\kappa}$.

Similar arguments yield analogous results for $X^{\sharp}$, for every set of
ordinals $X$. Given a set of ordinals $X$, let $\mathcal{C}_{X}$ be the class
of structures of the form $\langle L_{\beta}[X],\in,\gamma\rangle$, where
$\gamma$ and $\beta$ are cardinals (in $V$) and $sup(X)<\gamma<\beta$.
Clearly, ${\mathcal{C}}_{X}$ is $\Pi_{1}$ definable with $X$ as a parameter.

######  Theorem 4.7.

  1. (1)

${\rm SR}({\mathcal{C}}_{X})$ holds if and only if $X^{\sharp}$ exists.

  2. (2)

If $X^{\sharp}$ exists, then ${\rm SR}(\mathcal{C})$ holds for all classes
$\mathcal{C}$ that are definable in $L[X]$, with parameters.

These results suggest the following forms of ${\rm SR}$ restricted to inner
models. Let $M$ be an inner model. Writing $M_{\alpha}$ for
$(V_{\alpha})^{M}$, consider the following principle for $\Gamma$ a lightface
definability class:

>   * $\Gamma$-${\rm SR}(M)$:
>
> (_$\Gamma$ -Structural Reflection for $M$_) There exists an ordinal $\alpha$
> that reflects every $\Gamma$-definable class ${\mathcal{C}}$ of relational
> structures of the same type such that ${\mathcal{C}}\subseteq M$, i.e., for
> every $A$ in $\mathcal{C}$ there exists $B$ in $\mathcal{C}\cap M_{\alpha}$
> and an elementary embedding $j$ from $B$ into $A$. (_Warning:_ $j$ may not
> be in $M$.)
>
>

The corresponding version for a boldface $\Gamma$ being as follows:

>   * There exist a proper class of ordinals $\alpha$ that reflect every
> $\Gamma$-definable, with parameters in $M_{\alpha}$, class ${\mathcal{C}}$
> of relational structures of the same type such that ${\mathcal{C}}\subseteq
> M$.
>
>

The last theorem shows that, for every set of ordinals $X$, the existence of
$X^{\sharp}$ implies ${\rm SR}(L[X])$, i.e., $\mathbf{\Sigma_{n}}$-${\rm
SR}(L[X])$, for every $n$.

Similar results may be obtained for other canonical inner models, e.g.,
$L[U]$, the canonical inner model for one measurable cardinal $\kappa$, where
$U$ is the (unique) normal measure on $\kappa$ in $L[U]$. Let
${\mathcal{C}}^{U}$ be the class of structures of the form $\langle
L_{\beta}[U],\in,\gamma\rangle$, with $\gamma<\beta$ uncountable cardinals (in
$V$). The class ${\mathcal{C}}^{U}$ is $\Pi_{1}$-definable in $V$, with $U$ as
a parameter. By using well-known facts about $0^{\dagger}$ due to Solovay (see
[Kan03] 21), and arguing similarly as in Theorems 4.1 and 4.2, respectively,
one obtains the following:

######  Theorem 4.8.

The following are equivalent:

  1. (1)

${\rm SR}({\mathcal{C}}^{U})$ holds if and only if $0^{\dagger}$ exists.

  2. (2)

If $0^{\dagger}$ exists, then ${\rm SR}({\mathcal{C}})$ holds for every class
${\mathcal{C}}$ that is definable in $L[U]$, with parameters.

Also, similarly as in Theorem 4.7, one can obtain the analogous result for
$X^{\dagger}$, for every set of ordinals $X$. Namely, given a set of ordinals
$X$, let $\mathcal{C}^{U}_{X}$ be the class of structures of the form $\langle
L_{\beta}[U,X],\in,\gamma\rangle$, where $\gamma$ and $\beta$ are cardinals
(in $V$) and $sup(X)<\gamma<\beta$. Then ${\mathcal{C}}^{U}_{X}$ is
$\Pi_{1}$-definable with $U$ and $X$ as parameters.

######  Theorem 4.9.

  1. (1)

${\rm SR}({\mathcal{C}}^{U}_{X})$ holds if and only if $X^{\dagger}$ exists.

  2. (2)

If $X^{\dagger}$ exists, then ${\rm SR}(\mathcal{C})$ holds for all classes
$\mathcal{C}$ that are definable in $L[U,X]$, with parameters.

Analogous results should also hold for canonical inner models for stronger
large-cardinal notions. E.g., for the canonical inner model $L[\mathcal{E}]$
for a strong cardinal, as in [Koe89] or [Mit10], and its sharp, _zero pistol_
$0^{\P}$. Also for a canonical inner model for a proper class of strong
cardinals, as in [Sch02], and its sharp, _zero hand grenade_. For inner models
for stronger large cardinal notions, e.g., one Woodin cardinal, the situation
is less clear, although analogous results should hold given the appropriate
canonical inner model and its corresponding sharp.

##  5\. Product Structural reflection

Recall that for any set $S$ of relational structures $\mathcal{A}=\langle
A,\ldots\rangle$ of the same type, the set-theoretic product $\prod S$ is the
structure whose universe is the set of all functions $f$ with domain $S$ such
that $f(\mathcal{A})\in A$, for every $\mathcal{A}\in S$, and whose relations
are defined point-wise.

In this section we shall consider the following general product form of
structural reflection, which is a variation of the Product Reflection
Principle $(\rm{PRP})$ introduced in [BW20]:

>   * $\rm{PSR}$:
>
> (_Product Structural Reflection_) For every definable class of relational
> structures $\mathcal{C}$ of the same type, $\tau$, there exists an ordinal
> $\alpha$ that _product-reflects_ $\mathcal{C}$, i.e., for every
> ${\mathcal{A}}$ in $\mathcal{C}$ there exists a set $S$ of structures of
> type $\tau$ (although not necessarily in ${\mathcal{C}}$) with
> ${\mathcal{A}}\in S$ and an elementary embedding $j:\prod({\mathcal{C}}\cap
> V_{\alpha})\to\prod S$.
>
>

Similarly as in the case of ${\rm SR}$ (see section 2), we may formally define
$\rm{PSR}$ as a schema. Thus, we say that an ordinal $\alpha$ witnesses
$\Gamma(P)$-$\rm{PSR}$ (where $\Gamma$ is a definability class and $P$ a set
or a proper class) if $\alpha$ product-reflects all classes ${\mathcal{C}}$
that are $\Gamma$-definable with parameters in $P$. Our remarks in section 2
also apply here. In particular, an ordinal $\alpha$ witnesses
$\mathbf{\Pi_{n}}$-$\rm{PSR}$ if and only if it witnesses
$\mathbf{\Sigma_{n+1}}$-$\rm{PSR}$. Moreover, we obtain equivalent principles
by restricting to classes of natural structures. Thus, for $\Gamma$ a
lightface definability class, we define:

>   * $\Gamma$-$\rm{PSR}$:
>
> (_$\Gamma$ -Product Structural Reflection_) There exists an ordinal $\alpha$
> that product-reflects all $\Gamma$-definable classes ${\mathcal{C}}$ of
> natural structures.
>
>

The corresponding version for $\Gamma$ boldface being as follows:

>   * There exist a proper class of ordinals $\alpha$ that product-reflect all
> $\Gamma$-definable, with parameters in $V_{\alpha}$, classes ${\mathcal{C}}$
> of natural structures.
>
>

As in [BW20, Proposition 3.2] one can show that every cardinal $\kappa$ in
$C^{(1)}$ witnesses $\mathbf{\Sigma_{1}}$-$\rm{PSR}$. The converse also holds,
and in fact we have the following:

######  Proposition 5.1.

For every $n$, if $\kappa$ witnesses $\mathbf{\Pi_{n}}$-$\rm{PSR}$, then
$\kappa\in C^{(n+1)}$.

###### Proof.

We shall prove the case $n=1$. The general case follows by induction, using a
similar argument. The case $n=0$ is similar to the case $n=1$, but simpler, as
it suffices to consider a class of structures with domain a transitive set
(see the proof of Proposition 3.1). So, suppose $\varphi(x,y)$ is a $\Pi_{1}$
formula with $x,y$ as the only free variables, $a\in V_{\kappa}$, and
$V\models\exists x\varphi(x,a)$. Let ${\mathcal{C}}$ be the
$\Pi_{1}$-definable, with $a$ as a parameter, class of structures of the form
${\mathcal{A}}_{\alpha}=\langle
V_{\alpha},\in,a,\\{R^{\alpha}_{\varphi}\\}_{\varphi\in\Pi_{1}}\rangle$ with
$\alpha\in C^{(1)}$, and where
$\\{R^{\alpha}_{\varphi}\\}_{\varphi\in\Pi_{1}}$ is the $\Pi_{1}$ relational
diagram for $\langle V_{\alpha},\in,a\rangle$, i.e., if
$\varphi(x_{1},\ldots,x_{n},\overline{a})$, $\overline{a}$ a constant symbol,
is a $\Pi_{1}$ formula in the language of $\langle V_{\alpha},\in,a\rangle$,
then

$R^{\alpha}_{\varphi}=\\{\langle a_{1},\ldots,a_{n}\rangle:\langle
V_{\alpha},\in,a\rangle\models``\varphi[a_{1},\ldots,a_{n},a]"\\}\,.$

Let $\lambda>\kappa$ be in $C^{(2)}$, so that $V_{\lambda}\models\exists
x\varphi(x,a)$. By $\rm{PSR}$ there exists a set $S$ that contains
${\mathcal{A}}_{\lambda}$ and an elementary embedding

$j:\prod_{\alpha<\kappa}{\mathcal{A}}_{\alpha}\to\prod S.$

Since ${\mathcal{A}}_{\lambda}\models\exists x\varphi(x,a)$, we have
$R^{\lambda}_{\varphi}\neq\varnothing$. Hence, since
${\mathcal{A}}_{\lambda}\in S$,

$\prod S\not\models\langle
R^{\alpha}_{\varphi}\rangle_{{\mathcal{A}}_{\alpha}\in S}=\varnothing$

and therefore, by elementarity of $j$,

$\prod_{\alpha<\kappa}{\mathcal{A}}_{\alpha}\not\models\langle
R^{\alpha}_{\varphi}\rangle_{\alpha<\kappa}=\varnothing$

which implies that ${\mathcal{A}}_{\alpha}\models
R^{\alpha}_{\varphi}\neq\varnothing$, for some $\alpha<\kappa$. Hence,
${\mathcal{A}}_{\alpha}\models\exists x\varphi(x,a)$.

Note that, if $\varphi(x,y)$ had been a bounded formula, instead of $\Pi_{1}$,
then we would have, by upward absoluteness, that
${\mathcal{A}}_{\kappa}\models\exists x\varphi(x,y)$, thus showing that
$\kappa\in C^{(1)}$. Thus, since $\alpha,\kappa\in C^{(1)}$, we have that
$V_{\alpha}\preceq_{\Sigma_{1}}V_{\kappa}$, and therefore
$V_{\kappa}\models\exists x\varphi(x,a)$.

Now suppose $V_{\kappa}\models\exists x\varphi(x,a)$. Since $\kappa\in
C^{(1)}$, by upward absoluteness, $V\models\exists x\varphi(x,a)$. ∎

Recall that a cardinal $\kappa$ is _$\lambda$ -strong_, where
$\lambda>\kappa$, if there exists an elementary embedding $j:V\to M$, with $M$
transitive, ${\rm crit}(j)=\kappa$, $j(\kappa)>\lambda$, and with
$V_{\lambda}$ contained in $M$. A cardinal $\kappa$ is _strong_ if it is
$\lambda$-strong for every cardinal $\lambda>\kappa$.

The following proposition is proved similarly as in [BW20, 3.3].

######  Proposition 5.2.

If $\kappa$ is a strong cardinal, then $\kappa$ witnesses
$\mathbf{\Pi_{1}}$-$\rm{PSR}$.

###### Proof.

Let $\kappa$ be a strong cardinal and let ${\mathcal{C}}$ be a
$\Pi_{1}$-definable, with parameters in $V_{\kappa}$, proper class of
structures in a fixed relational language $\tau\in V_{\kappa}$. Let
$\varphi(x)$ be a $\Pi_{1}$ formula defining it.

Given any $\mathcal{A}\in{\mathcal{C}}$, let $\lambda\in C^{(1)}$ be greater
than or equal to $\kappa$ and with $\mathcal{A}\in V_{\lambda}$.

Let $j:V\to M$ be an elementary embedding, with $crit(j)=\kappa$,
$V_{\lambda}\subseteq M$, and $j(\kappa)>\lambda$.

By elementarity, the restriction of $j$ to ${\mathcal{C}}\cap V_{\kappa}$
yields an elementary embedding

$h:\prod({\mathcal{C}}\cap V_{\kappa})\to\prod(\\{X:M\models\varphi(X)\\}\cap
V^{M}_{j(\kappa)}).$

Let $S:=\\{X:M\models\varphi(X)\\}\cap V^{M}_{j(\kappa)}$. Since
$\mathcal{A}\in V_{\lambda}$ and $\varphi(x)$ is $\Pi_{1}$, by downward
absoluteness $V_{\lambda}\models\varphi(\mathcal{A})$. Hence, since the fact
that $\lambda\in C^{(1)}$ is $\Pi_{1}$-expressible and therefore downwards
absolute for transitive classes, and since $V_{\lambda}\subseteq M$, it
follows that $V_{\lambda}\preceq_{\Sigma_{1}}M$ and therefore
$M\models\varphi(\mathcal{A})$. Moreover $\mathcal{A}\in V_{\lambda}\subseteq
V^{M}_{j(\kappa)}$. Thus, ${\mathcal{A}}\in S$. ∎

Since the Product Reflection Principle ($\rm{PRP}$) introduced in [BW20] when
restricted to $\Pi_{1}$-definable classes is an easy consequence of
$\Pi_{1}$-$\rm{PSR}$, from [BW20] we obtain the following:

######  Theorem 5.3.

The following are equivalent:

  1. (1)

$\Pi_{1}$-$\rm{PSR}$

  2. (2)

There exists a strong cardinal.

as well as its boldface version:

######  Theorem 5.4.

The following are equivalent:

  1. (1)

$\mathbf{\Pi_{1}}$-$\rm{PSR}$

  2. (2)

There exists a proper class of strong cardinals.

This shows that strong cardinals are related to $\rm{PSR}$ as supercompact
cardinals are to ${\rm SR}$ (Theorems 3.2 and 3.3). For the higher levels of
definability, i.e., $n>1$, the large cardinal notion that corresponds to
$\Pi_{n}$-$\rm{PSR}$, analogous to the notion of $C^{(n)}$-extendible cardinal
in the case of ${\rm SR}$, is the following:

######  Definition 5.5.

[BW20] For $\Gamma$ a definability class, a cardinal $\kappa$ is _$\lambda$
-$\Gamma$-strong_, $\lambda$ an ordinal, if for every $\Gamma$-definable
(without parameters) class $A$ there is an elementary embedding $j:V\to M$,
with $M$ transitive, ${\rm crit}(j)=\kappa$, $V_{\lambda}\subseteq M$, and
$A\cap V_{\lambda}\subseteq j(A)$.

A cardinal $\kappa$ is _$\Gamma$ -strong_ if it is $\lambda$-$\Gamma$-strong
for every ordinal $\lambda$.

As with the case of strong cardinals, standard arguments show (cf. [Kan03]
26.7(b)) that $\kappa$ is _$\lambda$ -$\Gamma$-strong_ if and only if for
every $\Gamma$-definable (without parameters) class $A$ there is an elementary
embedding $j:V\to M$, with $M$ transitive, ${\rm crit}(j)=\kappa$,
$V_{\lambda}\subseteq M$, $j(\kappa)>\lambda$, and $A\cap V_{\lambda}\subseteq
j(A)$. As shown in [BW20], every strong cardinal is $\Sigma_{2}$-strong. Also,
a cardinal is $\Pi_{n}$-strong if and only if is $\Sigma_{n+1}$-strong.
Moreover, if $n\geq 1$ and $\lambda\in C^{(n+1)}$, then the following are
equivalent for a cardinal $\kappa<\lambda$:

  1. (1)

$\kappa$ is $\lambda$-$\Pi_{n}$-strong.

  2. (2)

There is an elementary embedding $j:V\to M$, with $M$ transitive, ${\rm
crit}(j)=\kappa$, $V_{\lambda}\subseteq M$, and $M\models``\lambda\in
C^{(n)}"$.

Similarly as in Proposition 5.2, one can prove the following (see [BW20] for
details):

######  Proposition 5.6.

If $\kappa$ is a $\Pi_{n}$-strong cardinal, then $\kappa$ witnesses
$\mathbf{\Pi_{n}}$-$\rm{PSR}$.

The following theorem then follows from the main result in [BW20]:

######  Theorem 5.7.

The following are equivalent for $n\geq 1$:

  1. (1)

$\Pi_{n}$-$\rm{PSR}$

  2. (2)

There exists a $\Pi_{n}$-strong cardinal.

The corresponding boldface version also holds. Namely,

######  Theorem 5.8.

The following are equivalent for $n\geq 1$:

  1. (1)

$\mathbf{\Pi_{n}}$-$\rm{PSR}$

  2. (2)

There exists a proper class of $\Pi_{n}$-strong cardinals.

Finally, for the $\rm{PSR}$ principle, the statement analogous to Vopěnka’s
Principle in the case of ${\rm SR}$ is the following:

######  Definition 5.9.

[BW20] _${\rm Ord}$ is Woodin_ if for every definable $A\subseteq V$ there
exists some $\alpha$ which is $A$-strong, i.e., for every $\gamma$ there is an
elementary embedding $j:V\to M$ with ${\rm crit}(j)=\alpha$,
$\gamma<j(\alpha)$, $V_{\gamma}\subseteq M$, and $A\cap V_{\gamma}=j(A)\cap
V_{\gamma}$.

Note that if $\delta$ is a Woodin cardinal (see [Kan03] for the definition of
Woodin cardinal and its equivalent formulation in terms of $A$-strength), then
$V_{\delta}$ satisfies ${\rm Ord}$ is Woodin. The following equivalences then
follow (cf. Theorem 3.9):

######  Theorem 5.10 ([BW20]).

The following schemata are equivalent:

  1. (1)

$\rm{PSR}$, i.e., $\mathbf{\Pi_{n}}$-$\rm{PSR}$ for all $n$.

  2. (2)

$\Pi_{n}$-$\rm{PSR}$ for all $n$.

  3. (3)

There exists a $\Pi_{n}$-strong cardinal, for every $n$.

  4. (4)

There is a proper class of $\Pi_{n}$-strong cardinals, for every $n$.

  5. (5)

${\rm Ord}$ is Woodin.

A close inspection of the proofs of Propositions 5.2 and 5.6 reveals that, for
$n>0$, if $\kappa$ is a $\Pi_{n}$-strong cardinal, then for every
$\Pi_{n}$-definable (with parameters in $V_{\kappa}$) class ${\mathcal{C}}$ of
relational structures of the same type $\tau$, and for every
$\beta\geq\kappa$, there exists a set $S$ of structures of type $\tau$
(although possibly not in ${\mathcal{C}}$) that contains ${\mathcal{C}}\cap
V_{\beta}$ and there exists an elementary embedding $h:\prod({\mathcal{C}}\cap
V_{\kappa})\to\prod S$ with the following properties:

  1. (1)

_Faithful_ : For every $f\in\prod({\mathcal{C}}\cap V_{\kappa})$,
$h(f)\restriction({\mathcal{C}}\cap V_{\kappa})=f$.

  2. (2)

_$\subseteq$ -chain-preserving_: If $f\in\prod({\mathcal{C}}\cap V_{\kappa})$
is so that $f({\mathcal{A}})\subseteq f({\mathcal{A}}^{\prime})$ whenever
$A\subseteq A^{\prime}$, then so is $h(f)$.

Moreover, if $\kappa$ witnesses $\Pi_{n}$-$\rm{PSR}$, then some cardinal les
than or equal to $\kappa$ is $\Pi_{n}$-strong. Thus, the following is an
equivalent reformulation of $\Gamma$-$\rm{PSR}$, for $\Gamma=\Gamma_{n}$ a
lightface definability class with $n>0$:

>   * $\Gamma$-$\rm{PSR}$:
>
> (_$\Gamma$ -Product Structural Reflection. Second version_) There exists a
> cardinal $\kappa$ that _product-reflects_ all $\Gamma$-definable proper
> classes ${\mathcal{C}}$ of relational structures of the same type $\tau$,
> i.e., for every $\beta$ there exists a set $S$ of structures of type $\tau$
> that contains ${\mathcal{C}}\cap V_{\beta}$, and there exists a faithful and
> $\subseteq$-chain-preserving elementary embedding $h:\prod({\mathcal{C}}\cap
> V_{\kappa})\to\prod S$.
>
>

The corresponding version for a boldface $\Gamma$ being as follows:

>   * There exist a proper class of cardinals $\kappa$ that _product-reflect_
> all $\Gamma$-definable, with parameters in $V_{\kappa}$, proper class
> ${\mathcal{C}}$ of relational structures of the same type.
>
>

The next theorem implies that strong cardinals can be characterized in terms
of $\Pi_{1}$-$\rm{PSR}$. The proof follows closely [BW20, Theorem 5.1], with
the properties of faithfulness and $\subseteq$-chain preservation ((1) and (2)
above) playing now a key role.

######  Theorem 5.11.

There is a $\Pi_{1}$-definable, without parameters, class ${\mathcal{C}}$ of
natural structures such that if a cardinal $\kappa$ product-reflects
${\mathcal{C}}$ (second version), then $\kappa$ is a strong cardinal.

###### Proof.

Let $C$ be the class of all ordinals $\alpha<\kappa$ of uncountable cofinality
such that $\alpha$ is the $\alpha$-th element of $C^{(1)}$. Let $\mathcal{C}$
be the $\Pi_{1}$-definable class of all structures

${\mathcal{A}}_{\alpha}:=\langle V_{\lambda_{\alpha}},\in,\alpha\rangle$

where $\alpha\in C$, and $\lambda_{\alpha}$ is the least cardinal in $C^{(1)}$
greater than $\alpha$

Note that, since by proposition 5.1, $\kappa\in C^{(2)}$, $\kappa$ is a limit
point of $C$.

Pick any $\gamma$ in $C$ greater than $\kappa$. We will show that $\kappa$ is
$\gamma$-strong. By $\rm{PSR}$ there is a faithful $\subseteq$-chain-
preserving elementary embedding $j:\prod({\mathcal{C}}\cap V_{\kappa})\to\prod
S$, where $S$ is some set with $({\mathcal{C}}\cap
V_{\kappa})\cup\\{{\mathcal{A}}_{\gamma}\\}\subseteq S$.

Now pick any ${\mathcal{A}}_{\beta}\in{\mathcal{C}}\cap S$ and let

$h_{\beta}:\prod S\to{\mathcal{A}}_{\beta}$

be the projection map. Let $I:=C\cap\kappa$ and define

$k_{\beta}:V_{\kappa+1}\to V_{\beta+1}$

by:

$k_{\beta}(X)=h_{\beta}(j(\\{X\cap V_{\alpha}\\}_{\alpha\in I})).$

Since $j$ is elementary, for all formulas $\varphi(x_{1},\ldots,x_{n})$ and
all $a_{1},\ldots,a_{n}\in\prod({\mathcal{C}}\cap V_{\kappa})$, if
$\prod({\mathcal{C}}\cap V_{\kappa})\models\varphi[a_{1},\ldots,a_{n}]$, then
$\prod S\models\varphi[j(a_{1})\ldots,j(a_{n})].$ It easily follows that
$k_{\beta}$ preserves Boolean operations, the subset relation, and is the
identity on $\omega+1$.

Note that $k_{\beta}(\kappa)=h_{\beta}(j(\\{\alpha\\}_{\alpha\in I}))=\beta.$

For each $a\in[\beta]^{<\omega}$, define $E^{\beta}_{a}$ by

$X\in E^{\beta}_{a}\quad\mbox{ iff }\quad X\subseteq[\kappa]^{|a|}\mbox{ and
}a\in k_{\beta}(X)\,.$

Since $k_{\beta}(\kappa)=\beta$ and $k_{\beta}(|a|)=|a|$, we also have
$k_{\beta}([\kappa]^{|a|})=[\beta]^{|a|}$, hence $[\kappa]^{|a|}\in
E^{\beta}_{a}$. Since $k_{\beta}$ preserves Boolean operations and the
$\subseteq$ relation, $E^{\beta}_{a}$ is a proper ultrafilter over
$[\kappa]^{|a|}$. Moreover, since $k_{\beta}(\omega)=\omega$, a simple
argument shows that $E^{\beta}_{a}$ is $\omega_{1}$-complete, hence the
ultrapower ${\rm{Ult}}(V,E^{\beta}_{a})$ is well-founded. Furthermore, since
$j$ is faithful, if $\beta<\kappa$, then $E^{\beta}_{a}$ is the principal
ultrafilter generated by $\\{a\\}$. Let

$j^{\beta}_{a}:V\to M^{\beta}_{a}\cong{\rm{Ult}}(V,E^{\beta}_{a})$

with $M_{a}^{\beta}$ transitive, be the corresponding ultrapower embedding.
Note that if $\beta<\kappa$, then $M^{\beta}_{a}=V$ and $j^{\beta}_{a}$ is the
identity.

Let $\mathcal{E}_{\beta}:=\\{E^{\beta}_{a}:a\in[\beta]^{<\omega}\\}$. As in
[BW20], one can show that $\mathcal{E}_{\beta}$ is normal and coherent. Thus,
for each $a\subseteq b$ in $[\beta]^{<\omega}$ the maps
$i^{\beta}_{ab}:M^{\beta}_{a}\to M^{\beta}_{b}$ given by

$i^{\beta}_{ab}([f]_{E^{\beta}_{a}})=[f\circ\pi_{ba}]_{E^{\beta}_{b}}$

for all $f:[\kappa]^{|a|}\to V$, are well-defined and commute with the
ultrapower embeddings $j^{\beta}_{a}$ (see [Kan03] 26).

Let $M_{{\mathcal{E}}_{\beta}}$ be the direct limit of the directed system

$\langle\langle M^{\beta}_{a}:a\in[\beta]^{<\omega}\rangle,\langle
i^{\beta}_{ab}:a\subseteq b\rangle\rangle$

and let $j_{{\mathcal{E}}_{\beta}}:V\to M_{{\mathcal{E}}_{\beta}}$ be the
corresponding direct limit elementary embedding, i.e.,

$j_{{\mathcal{E}}_{\beta}}(x)=[a,[c^{a}_{x}]_{E^{\beta}_{a}}]_{{\mathcal{E}}_{\beta}}$

for some (any) $a\in[\beta]^{<\omega}$, and where
$c^{a}_{x}:[\kappa]^{|a|}\to\\{x\\}$.

As in [BW20] one can also show that $M_{{\mathcal{E}}_{\beta}}$ is well-
founded. So, let $\pi_{\beta}:M_{{\mathcal{E}}_{\beta}}\to N_{\beta}$ be the
transitive collapse, and let $j_{N_{\beta}}:V\to N_{\beta}$ be the
corresponding elementary embedding, i.e., $j_{N_{\beta}}=\pi\circ
j_{{\mathcal{E}}_{\beta}}$. Then, as in [BW20] we can show that
$V_{\beta}\subseteq N_{\beta}$ and $j_{N_{\beta}}(\kappa)\geq\beta$. If
$\beta>\kappa$, this implies that ${\rm crit}(j_{N_{\beta}})\leq\kappa$. (If
$\beta<\kappa$, then $j_{N_{\beta}}:V\to V$ is the identity.)

Let $I_{S}:=\\{\beta:{\mathcal{A}}_{\beta}\in{\mathcal{C}}\cap S\\}$.

######  Claim 5.12.

If $\beta\leq\beta^{\prime}$ are in $I_{S}$, then
$E^{\beta}_{a}=E^{\beta^{\prime}}_{a}$, for every $a\in[\beta]^{<\omega}$.

###### Proof of claim.

Since $E^{\beta}_{a},E^{\beta^{\prime}}_{a}$ are proper ultrafilters over
$[\kappa]^{|a|}$, it is sufficient to see that $E^{\beta}_{a}\subseteq
E^{\beta^{\prime}}_{a}$. So, suppose $X\in E^{\beta}_{a}$. Then $a\in
k_{\beta}(X)=h_{\beta}(j(\\{X\cap V_{\alpha}\\}_{\alpha\in I}))$. Since
$\\{X\cap V_{\alpha}\\}_{\alpha\in I}$ forms an $\subseteq$-chain and $j$ is
$\subseteq$-chain-preserving, so does $j(\\{X\cap V_{\alpha}\\}_{\alpha\in
I})$. Hence, $k_{\beta}(X)\subseteq k_{\beta^{\prime}}(X)$, and therefore
$a\in k_{\beta^{\prime}}(X)$, which yields $X\in E^{\beta^{\prime}}_{a}$. ∎

By the claim above, for every $\beta<\beta^{\prime}$ in $I_{S}$ the map

$k_{\beta,\beta^{\prime}}:M_{{\mathcal{E}}_{\beta}}\to
M_{{\mathcal{E}}_{\beta^{\prime}}}$

given by

$k_{\beta,\beta^{\prime}}([a,[f]_{E^{\beta}_{a}}]_{{\mathcal{E}}_{\beta}})=[a,[f]_{E^{\beta^{\prime}}_{a}}]_{{\mathcal{E}}_{\beta^{\prime}}}$

is well-defined and elementary. Moreover, it commutes with the embeddings
$j_{{\mathcal{E}}_{\beta}}:V\to M_{{\mathcal{E}}_{\beta}}$ and
$j_{{\mathcal{E}}_{\beta^{\prime}}}:V\to M_{{\mathcal{E}}_{\beta^{\prime}}}$.
Let $M$ be the direct limit of

$\langle\langle M_{{\mathcal{E}}_{\beta}}:\beta\in I^{\prime}\rangle,\langle
k_{\beta,\beta^{\prime}}:\beta<\beta^{\prime}\mbox{ in }I_{S}\rangle\rangle$

and let $j_{M}:V\to M$ be the corresponding direct limit elementary embedding,
which is given by

$j_{M}(x)=[\beta,[a,[c^{a}_{x}]_{E^{\beta}_{a}}]_{{\mathcal{E}}_{\beta}}]$

for some (any) $a\in[\beta]^{<\omega}$. Let $\pi^{M}:M\to N$ be the transitive
collapse, and let $j_{N}=\pi^{M}\circ j_{M}:V\to N$.

Let $\xi=\mbox{sup}(I_{S})$. Note that, as $\gamma\in I_{S}$, $\xi>\kappa$.

######  Claim 5.13.

$j_{N}(\kappa)=\xi$

###### Proof of claim.

As in [BW20], we can show that $j_{N_{\beta}}(\kappa)\geq\beta$, for every
$\beta\in I_{S}\setminus\kappa$. So, for such a $\beta$, letting
$\ell_{\beta,N}$ be the unique elementary embedding such that
$j_{N}=\ell_{\beta,N}\circ j_{N_{\beta}}$, we have:

$j_{N}(\kappa)=\ell_{\beta,N}(j_{N_{\beta}}(\kappa))\geq\ell_{\beta,N}(\beta)\geq\beta.$

Hence, $j_{N}(\kappa)\geq\xi$. Also, $j_{E^{\beta}_{a}}(\kappa)$ can be
computed in $V_{\beta}$, for all $a\in[\beta]^{<\omega}$, and therefore
$j_{N_{\beta}}(\kappa)\leq\beta$. Hence, $j_{N}(\kappa)\leq\xi$. ∎

Since $\kappa<\xi$, it follows from the claim above that ${\rm
crit}(j_{N})\leq\kappa$. But since for $\beta<\kappa$ the map $j_{N_{\beta}}$
is the identity, we must have ${\rm crit}(j_{N})=\kappa$. Also, since
$\gamma\in I_{S}$, $V_{\gamma}\subseteq N_{\gamma}$, hence
$V_{\gamma}\subseteq N$. This shows that $\kappa$ is $\gamma$-strong, as
wanted. ∎

From proposition 5.2 and theorem 5.11 we obtain now the following
characterization of strong cardinals.

######  Corollary 5.14.

A cardinal $\kappa$ is strong if and only if it witnesses $\Pi_{1}$-$\rm{PSR}$
(second version).

Similar results can be proven for $\Gamma$-strong cardinals (definition 5.5).
On the one hand, proposition 5.6 shows that if $\kappa$ is $\Pi_{n}$-strong,
then $\kappa$ witnesses $\mathbf{\Pi_{n+1}}$-$\rm{PSR}$. On the other hand,
similarly as in 5.11, we can prove that if $\kappa$ witnesses
$\Pi_{n}$-$\rm{PSR}$, then $\kappa$ is $\Pi_{n}$-strong. This yields the
following characterization of $\Pi_{n}$-strong cardinals:

######  Theorem 5.15.

For every $n>0$, a cardinal $\kappa$ is $\Pi_{n}$-strong if and only it it
witnesses $\Pi_{n}$-$\rm{PSR}$ (second version).

###  5.1. Strong Product Structural reflection

Let us consider next the following, arguably more natural, strengthening of
$\rm{PSR}$:

>   * $\rm{SPSR}$:
>
> (_Strong Product Structural Reflection_) For every definable class of
> relational structures $\mathcal{C}$ of the same type, $\tau$, there exists
> an ordinal $\alpha$ that _strongly product-reflects_ $\mathcal{C}$, i.e.,
> for every ${\mathcal{A}}$ in $\mathcal{C}$ there exists an ordinal $\beta$
> with ${\mathcal{A}}\in V_{\beta}$ and an elementary embedding
> $j:\prod({\mathcal{C}}\cap V_{\alpha})\to\prod({\mathcal{C}}\cap
> V_{\beta})$.
>
>

Similarly as in the case of $\rm{PSR}$, let us say that a cardinal $\kappa$
witnesses $\Gamma(P)$-$\rm{SPSR}$ if $\kappa$ strongly product-reflects all
classes ${\mathcal{C}}$ that are $\Gamma$-definable (with parameters in $P$).
Also, a cardinal $\kappa$ witnesses $\mathbf{\Pi_{n}}$-$\rm{SPSR}$ if and only
if it witnesses $\mathbf{\Sigma_{n+1}}$-$\rm{SPSR}$, and similarly for the
lightface definability classes. Moreover, we obtain equivalent principles by
restricting to classes of natural structures. Thus, we may formally define
$\Gamma$-$\rm{SPSR}$, for $\Gamma$ a lightface definability class, as follows:

>   * $\Gamma$-$\rm{SPSR}$:
>
> (_$\Gamma$ -Strong Product Structural Reflection_) There exists a cardinal
> $\kappa$ that strongly product-reflects all $\Gamma$-definable class
> ${\mathcal{C}}$ of natural structures.
>
>

The boldface version being as follows:

>   * There exist a proper class of cardinals $\kappa$ that strongly product-
> reflect all $\Gamma$-definable, with parameters in $V_{\kappa}$, class
> ${\mathcal{C}}$ of natural structures.
>
>

Note that $\Gamma$-$\rm{SPSR}$ implies $\Gamma$-$\rm{PSR}$, for any
definability class $\Gamma$.

We shall see next that the large cardinal notions that correspond to the
$\rm{SPSR}$ principle are those of superstrong, globally superstrong, and
$C^{(n)}$-globally superstrong cardinals.

######  Definition 5.16.

[CT19] A cardinal $\kappa$ is _superstrong above $\lambda$_, for some
$\lambda\geq\kappa$, if there exists an elementary embedding $j:V\to M$, with
$M$ transitive, ${\rm crit}(j)=\kappa$, $j(\kappa)>\lambda$, and
$V_{j(\kappa)}\subseteq M$.

A cardinal $\kappa$ is _globally superstrong_ if it is superstrong above
$\lambda$, for every $\lambda\geq\kappa$.

More generally, a cardinal $\kappa$ is _$C^{(n)}$ -superstrong above
$\lambda$_, for some $\lambda\geq\kappa$, if there exists an elementary
embedding $j:V\to M$, with $M$ transitive, ${\rm crit}(j)=\kappa$,
$j(\kappa)>\lambda$, $V_{j(\kappa)}\subseteq M$, and $j(\kappa)\in C^{(n)}$.

A cardinal $\kappa$ is _$C^{(n)}$ -globally superstrong_ if it is
$C^{(n)}$-superstrong above $\lambda$, for every $\lambda\geq\kappa$.

Note that every globally superstrong cardinal is $C^{(1)}$-globally
superstrong. Also, every globally superstrong cardinal is superstrong, and
every $C^{(n)}$-globally superstrong cardinal belongs to $C^{(n+2)}$ ([CT19]).
As shown in [CT19], on the one hand, if $\kappa$ is $C^{(n)}$-gobally
superstrong, then there are many $C^{(n)}$-superstrong cardinals below
$\kappa$. On the other hand, if $\kappa$ is $\kappa+1$-extendible, then
$V_{\kappa}$ satisfies that there is a proper class of $C^{(n)}$-globally
superstrong cardinals, for every $n$. Moreover, if $\kappa$ is
$C^{(n)}$-extendible, then there are many $C^{(n)}$-globally superstrong
cardinals below $\kappa$.

Similarly as in Proposition 5.2 we can prove the following:

######  Proposition 5.17.

If $\kappa$ is $C^{(n)}$-globally superstrong, then it witnesses
$\mathbf{\Pi_{n}}$-$\rm{SPSR}$.

###### Proof.

Let $\kappa$ be $C^{(n)}$-globally superstrong and let ${\mathcal{C}}$ be a
class of relational structures of the same type that is definable by a
$\Pi_{n}$ formula $\varphi(x)$, with parameters in $V_{\kappa}$.

Given any $\mathcal{A}\in{\mathcal{C}}$, let $\lambda\in C^{(n)}$ be greater
than or equal to $\kappa$ and with $\mathcal{A}\in V_{\lambda}$.

Let $j:V\to M$ be an elementary embedding witnessing that $\kappa$ is
$C^{(n)}$-superstrong above $\lambda$. Then the restriction of $j$ to
${\mathcal{C}}\cap V_{\kappa}$ yields an elementary embedding

$h:\prod({\mathcal{C}}\cap V_{\kappa})\to\prod(\\{X:M\models\varphi(X)\\}\cap
V^{M}_{j(\kappa)}).$

Let $S:=\\{X:M\models\varphi(X)\\}\cap V^{M}_{j(\kappa)}$. Since
$j(\kappa)>\lambda$ and $j(\kappa)\in C^{(n)}$, we have
$V_{j(\kappa)}\models\varphi({\mathcal{A}})$. Hence, since $\kappa\in C^{(n)}$
(in fact $\kappa\in C^{(n+2)}$ ([CT19]), by elementarity
$V_{j(\kappa)}=V_{j(\kappa)}^{M}\preceq_{\Sigma_{n}}M$, and thus
$M\models\varphi({\mathcal{A}})$. It follows that ${\mathcal{A}}\in S$.
Moreover, if ${\mathcal{B}}\in S$, then $M\models\varphi({\mathcal{B}})$.
Hence, since $V_{j(\kappa)}^{M}\preceq_{\Sigma_{n}}M$, we have that
$V_{j(\kappa)}^{M}=V_{j(\kappa)}\models\varphi({\mathcal{B}})$. Since
$j(\kappa)\in C^{(n)}$, $\varphi({\mathcal{B}})$ holds in $V$, and therefore
${\mathcal{B}}\in{\mathcal{C}}$. This shows that $S={\mathcal{C}}\cap
V_{j(\kappa)}$, and so

$h:\prod({\mathcal{C}}\cap V_{\kappa})\to\prod({\mathcal{C}}\cap
V_{j(\kappa)}).$

Hence, $h$ witnesses $\rm{SPSR}$ for ${\mathcal{A}}$. ∎

Observe that since the function $h$ in the proof of the last proposition is
the restriction of $j$ to $\prod({\mathcal{C}}\cap V_{\kappa})$, and $j$ is
elementary, it preserves all first-order properties. In particular, it is
$\subseteq$-chain-preserving; and since $\kappa={\rm crit}(j)$, $h$ is
faithful. Thus, taking into consideration our remarks from section 2, as well
as those made in the previous section before we stated the second version of
the $\rm{PSR}$ schema, we may reformulate $\Gamma$-$\rm{SPSR}$ for $\Gamma$ a
lightface definability class as follows:

>   * $\Gamma$-$\rm{SPSR}$:
>
> (_$\Gamma$ -Strong Product Structural Reflection. Second version_) There
> exists a cardinal $\kappa$ that _strongly-product-reflects_ all
> $\Gamma$-definable classes ${\mathcal{C}}$ of natural structures, i.e., for
> every ${\mathcal{A}}\in{\mathcal{C}}$ there exists an ordinal $\beta$ with
> ${\mathcal{A}}\in V_{\beta}$ and a faithful and $\subseteq$-chain-preserving
> elementary embedding $h:\prod({\mathcal{C}}\cap
> V_{\kappa})\to\prod({\mathcal{C}}\cap V_{\beta})$.
>
>

The corresponding version for boldface $\Gamma$ being:

>   * There exist a proper class of cardinals $\kappa$ that _strongly-product-
> reflect_ all $\Gamma$-definable, with parameters in $V_{\kappa}$, proper
> classes ${\mathcal{C}}$ of natural structures.
>
>

Arguing similarly as in the proof of Theorem 5.11 (and [BW20]), we can now
prove the following:

######  Theorem 5.18.

For every $n>0$, there is a $\Pi_{n}$-definable, without parameters, class
${\mathcal{C}}$ of natural structures such that if a cardinal $\kappa$
strongly-product-reflects ${\mathcal{C}}$, then $\kappa$ is a
$C^{(n)}$-globally superstrong cardinal.

###### Proof.

Let $\mathcal{C}$ be the $\Pi_{n}$-definable class of all structures

${\mathcal{A}}_{\alpha}:=\langle V_{\lambda_{\alpha}},\in,\alpha\rangle$

where $\alpha$ has uncountable cofinality and is the $\alpha$-th element of
$C^{(n)}$, and $\lambda_{\alpha}$ is the least cardinal in $C^{(n)}$ greater
than $\alpha$.

Let $\kappa$ witness $\rm{SPSR}$ for ${\mathcal{C}}$. Let
$I:=\\{\alpha:{\mathcal{A}}_{\alpha}\in V_{\kappa}\\}$. Since $\kappa\in
C^{(n+1)}$ (proposition 5.1), ${\rm{sup}}(I)=\kappa$.

Pick any ordinal $\lambda\geq\kappa$ and let us show that $\kappa$ is
$\lambda$-$C^{(n)}$-superstrong. Let ${\mathcal{A}}_{\beta}$ in
${\mathcal{C}}$ with $\lambda<\beta$. Let $\kappa^{\prime}$ be such that
${\mathcal{A}}_{\beta}\in V_{\kappa^{\prime}}$ and there is a faithful
$\subseteq$-chain-preserving elementary embedding

$j:\prod({\mathcal{C}}\cap V_{\kappa})\to\prod({\mathcal{C}}\cap
V_{\kappa^{\prime}}).$

Let

$h_{\beta}:\prod({\mathcal{C}}\cap
V_{\kappa^{\prime}})\to{\mathcal{A}}_{\beta}$

be the projection map and define $k_{\beta}:V_{\kappa+1}\to V_{\beta+1}$ by:

$k_{\beta}(X)=h_{\beta}(j(\\{X\cap V_{\alpha}\\}_{\alpha\in I})).$

As in 5.11, for each $a\in[\beta]^{<\omega}$, define $E^{\beta}_{a}$ by

$X\in E^{\beta}_{a}\quad\mbox{ iff }\quad X\subseteq[\kappa]^{|a|}\mbox{ and
}a\in k_{\beta}(X)\,.$

Then $E^{\beta}_{a}$ is an $\omega_{1}$-complete proper ultrafilter over
$[\kappa]^{|a|}$, and so the ultrapower ${\rm{Ult}}(V,E^{\beta}_{a})$ is well-
founded. Furthermore, since $j$ is faithful, if $\beta\in I$, then
$E^{\beta}_{a}$ is the principal ultrafilter generated by $\\{a\\}$. Let

$j^{\beta}_{a}:V\to M^{\beta}_{a}\cong{\rm{Ult}}(V,E^{\beta}_{a})$

and let $\mathcal{E}_{\beta}:=\\{E^{\beta}_{a}:a\in[\beta]^{<\omega}\\}$. As
in [BW20], $\mathcal{E}_{\beta}$ is normal and coherent. Let
$M_{{\mathcal{E}}_{\beta}}$ be the direct limit of

$\langle\langle M^{\beta}_{a}:a\in[\beta]^{<\omega}\rangle,\langle
i^{\beta}_{ab}:a\subseteq b\rangle\rangle$

where the $i_{ab}^{\beta}$ are the standard projection maps, and let
$j_{{\mathcal{E}}_{\beta}}:V\to M_{{\mathcal{E}}_{\beta}}$ be the
corresponding limit elementary embedding. As in [BW20],
$M_{{\mathcal{E}}_{\beta}}$ is well-founded. So, let
$\pi_{\beta}:M_{{\mathcal{E}}_{\beta}}\to N_{\beta}$ be the transitive
collapse, and let $j_{N_{\beta}}=\pi\circ j_{{\mathcal{E}}_{\beta}}:V\to
N_{\beta}$. We have that $V_{\beta}\subseteq N_{\beta}$ and
$j_{N_{\beta}}(\kappa)\geq\beta$ (see [BW20]).

By claim 5.12, if $\beta\leq\beta^{\prime}$ are in
$I^{\prime}:=\\{\alpha:{\mathcal{A}}_{\alpha}\in V_{\kappa^{\prime}}\\}$, then
$E^{\beta}_{a}=E^{\beta^{\prime}}_{a}$, for every $a\in[\beta]^{<\omega}$.
Hence, for every $\beta<\beta^{\prime}$ in $I^{\prime}$, the map

$k_{\beta,\beta^{\prime}}:M_{{\mathcal{E}}_{\beta}}\to
M_{{\mathcal{E}}_{\beta^{\prime}}}$

given by

$k_{\beta,\beta^{\prime}}([a,[f]_{E_{a}}]_{{\mathcal{E}}_{\beta}})=[a,[f]_{E_{a}}]_{{\mathcal{E}}_{\beta^{\prime}}}$

is well-defined, elementary, and commutes with the embeddings
$j_{{\mathcal{E}}_{\beta}}:V\to M_{{\mathcal{E}}_{\beta}}$ and
$j_{{\mathcal{E}}_{\beta^{\prime}}}:V\to M_{{\mathcal{E}}_{\beta^{\prime}}}$.
Let $M$ be the direct limit of

$\langle\langle M_{{\mathcal{E}}_{\beta}}:\beta\in I^{\prime}\rangle,\langle
k_{\beta,\beta^{\prime}}:\beta<\beta^{\prime}\mbox{ in
}I^{\prime}\rangle\rangle$

and let $j_{M}:V\to M$ be the corresponding limit elementary embedding. Let
$\pi^{M}:M\to N$ be the transitive collapse, and let $j_{N}=\pi^{M}\circ
j_{M}:V\to N$.

Let $\xi=\mbox{sup}(I^{\prime})$. Note that $\xi\in C^{(n)}$ and $\xi>\kappa$.
As in claim 5.13, $j_{N}(\kappa)=\xi$, hence ${\rm crit}(j_{N})\leq\kappa$.
But since for $\beta\in I$ the map $j_{N_{\beta}}$ is the identity, ${\rm
crit}(j_{N})=\kappa$. Also, since $V_{\xi}=\bigcup_{\beta\in I}V_{\beta}$, and
$V_{\beta}\subseteq N_{\beta}$ for all $\beta\in I$, it follows that
$V_{\xi}\subseteq N$. This shows that $\kappa$ is $\xi$-superstrong, hence
also $\lambda$-$C^{(n)}$-superstrong, as wanted. ∎

We have thus proved the following:

######  Theorem 5.19.

For every $n\geq 1$, the following are equivalent for any cardinal $\kappa$:

  1. (1)

$\kappa$ witnesses $\Pi_{n}$-$\rm{SPSR}$

  2. (2)

$\kappa$ is a $C^{(n)}$-globally superstrong cardinal.

######  Corollary 5.20.

The following are equivalent:

  1. (1)

$\rm{SPSR}$, i.e., $\mathbf{\Pi_{n}}$-$\rm{SPSR}$ for every $n$.

  2. (2)

$\Pi_{n}$-$\rm{SPSR}$ for every $n$.

  3. (3)

There exists a $C^{(n)}$-globally superstrong cardinal, for every $n$.

  4. (4)

There exist a proper class of $C^{(n)}$-globally superstrong cardinals.

###  5.2. Bounded Product Structural Reflection

Let us consider next some _bounded_ forms of $\rm{PSR}$. Namely, for $\Gamma$
a lightface definability class and any ordinal $\beta$ let:

>   * $\Gamma$-$\rm{PSR}_{\beta}$:
>
> There exists a cardinal $\kappa$ that $\beta$-_product-reflects_ every
> $\Gamma$-definable proper class ${\mathcal{C}}$ of natural structures, i.e.,
> for every ${\mathcal{A}}$ in ${\mathcal{C}}$ of rank $\leq\kappa+\beta$
> there exists a set $S$ with ${\mathcal{A}}$ in $S$ and an elementary
> embedding $h:\prod({\mathcal{C}}\cap V_{\kappa})\to\prod S$.
>
>

Thus $\Gamma$-$\rm{PSR}$ holds if and only if there exists a cardinal $\kappa$
that witnesses $\Gamma$-$\rm{PSR}_{\beta}$ for all (equivalently, a proper
class of) ordinals $\beta$.

The following theorem shows that measurable cardinals can be characterized in
terms of bounded $\rm{PSR}$.

######  Theorem 5.21.

The following are equivalent:

  1. (1)

$\Pi_{1}$-$\rm{PSR}_{1}$

  2. (2)

There exists a measurable cardinal.

###### Proof.

(1) implies (2): Let ${\mathcal{C}}$ be the $\Pi_{1}$-definable class of
$\langle V_{\gamma},\in\rangle$, $\gamma\geq\omega$. Let $\kappa$ witness
$\rm{PSR}_{1}$ for ${\mathcal{C}}$, and let $S$ be a set that contains
$V_{\kappa+1}$ such that there exists an elementary embedding

$h:\prod({\mathcal{C}}\cap V_{\kappa})\to\prod S.$

Define $k:V_{\kappa+1}\to V_{\kappa+2}$ by:

$k(X)=h_{\kappa+1}(h(\\{X\cap V_{\gamma}\\}_{\gamma<\kappa}))$

where $h_{\kappa+1}$ is the projection on $V_{\kappa+1}$. Then $k$ preserves
Boolean operations and the subset relation, and is the identity on $\omega+1$.
Moreover, $k(\kappa)=\kappa+1$. Now for each $a\in V_{\kappa+1}$, define
${\mathcal{U}}_{a}$ by

$X\in{\mathcal{U}}_{a}\quad\mbox{ iff }\quad X\subseteq\kappa\mbox{ and }a\in
k(X)\,.$

Clearly, $\kappa\in{\mathcal{U}}_{a}$. Also, since $k$ preserves Boolean
operations and the $\subseteq$ relation, ${\mathcal{U}}_{a}$ is a proper
ultrafilter over $\kappa$. Moreover, since $k(\omega)=\omega$,
${\mathcal{U}}_{a}$ is $\omega_{1}$-complete. Furthermore, since
$|V_{\kappa+1}|=2^{|V_{\kappa}|}>|\kappa|$, some ${\mathcal{U}}_{a}$ is non-
principal. So, some cardinal less than or equal to $\kappa$ is measurable.

(2) implies (1): Let $\kappa$ be a measurable cardinal, and let $\varphi(x)$
be a $\Pi_{1}$ formula (we may allow parameters in $V_{\kappa}$) that defines
a proper class ${\mathcal{C}}$ of natural structures. We claim that $\kappa$
$1$-product-reflects ${\mathcal{C}}$.

Let $j:V\to M$ be an ultrapower elementary embedding, given by some
$\kappa$-complete normal measure over $\kappa$. Thus, ${\rm crit}(j)=\kappa$
and $V_{\kappa+1}^{M}=V_{\kappa+1}$. By elementarity, the restriction of $j$
to ${\mathcal{C}}\cap V_{\kappa}$ yields an elementary embedding

$h:\prod({\mathcal{C}}\cap V_{\kappa})\to\prod(\\{X:M\models\varphi(X)\\}\cap
V^{M}_{j(\kappa)}).$

Let $S:=\\{X:M\models\varphi(X)\\}\cap V^{M}_{j(\kappa)}$. If
$\mathcal{A}=\langle V_{\gamma},\in\rangle\in{\mathcal{C}}$, with
$\gamma\leq\kappa+1$, then ${\mathcal{A}}\in M$, and since
$\varphi({\mathcal{A}})$ holds in $V$, by $\Pi_{1}$ downward absoluteness for
transitive classes it also holds in $M$. Moreover, since $j(\kappa)>\kappa+1$,
${\mathcal{A}}\in V^{M}_{j(\kappa)}$, hence ${\mathcal{A}}\in S$. ∎

Let us note that the proof of the theorem above also shows that the least
measurable cardinal is precisely the least cardinal $\kappa$ that witnesses
$\rm{PSR}_{1}$ for all classes ${\mathcal{C}}$ that are $\Pi_{1}$-definable
with parameters in $V_{\kappa}$.

##  6\. Large cardinals below measurability

We shall next consider Structural Reflection for classes of relational
structures that are $\Sigma_{1}$-definable in the language of set theory
extended with additional $\Pi_{1}$ predicates. That is, classes of structures
of complexity between $\Sigma_{1}$ and $\Sigma_{2}$.

Let ${\mathcal{R}}$ be a set of $\Pi_{1}$ predicates or relations. A class
${\mathcal{C}}$ of structures in a fixed countable relational type is said to
be $\Sigma_{1}({\mathcal{R}})$-definable if it is definable by means of a
$\Sigma_{1}$ formula of the first-order language of set theory with additional
predicate symbols for the predicates in ${\mathcal{R}}$, without parameters.
We define the following form of ${\rm SR}$:

>   * $\Sigma_{1}({\mathcal{R}})$-${\rm SR}$:
>
> For every $\Sigma_{1}({\mathcal{R}})$-definable class ${\mathcal{C}}$ of
> structures of the same type there exists a cardinal $\kappa$ that _reflects_
> $\mathcal{C}$, i.e., for every $A$ in $\mathcal{C}$ there exists $B$ in
> $\mathcal{C}\cap V_{\kappa}$ and an elementary embedding from $B$ into $A$.
>
>

For the rest of this section we shall write ${\rm SR}_{\mathcal{R}}$ instead
of the more cumbersome $\Sigma_{1}({\mathcal{R}})$-${\rm SR}$. Also, if
${\mathcal{R}}=\\{R_{1},\ldots,R_{n}\\}$, then we may write ${\rm
SR}_{R_{1},\ldots,R_{n}}$ for ${\rm SR}_{\mathcal{R}}$.

We have that ${\rm SR}_{\varnothing}$, i.e., $\Sigma_{1}$-${\rm SR}$, is
provable in ZFC (Proposition 3.1). However, if $R$ is the $\Pi_{1}$ relation
“$x$ is an ordinal and $y=V_{x}$”, then ${\rm SR}_{R}$ holds if and only if
there exists a supercompact cardinal ([Bag12, BCMR15]; see also Theorem 3.2).
Moreover, if $\kappa$ is supercompact, then ${\rm SR}_{\mathcal{R}}$ holds for
$\kappa$, for any set ${\mathcal{R}}$ of $\Pi_{1}$ predicates (cf. Theorem
3.2).

###  6.1. The principle ${\rm SR}^{-}_{\mathcal{R}}$

For $\Gamma$ any lightface definability class, the following is a natural
restricted form of $\Gamma$-${\rm SR}$:

>   * $\Gamma$-${\rm SR}^{-}:$
>
> There exists a cardinal $\kappa$ such that for every $\Gamma$-definable
> class ${\mathcal{C}}$ of structures of the same type and every
> $A\in{\mathcal{C}}$ of cardinality $\kappa$ there exists
> $B\in{\mathcal{C}}\cap H_{\kappa}$ and an elementary embedding from $B$ into
> $A$. We say that the cardinal $\kappa$ $\kappa$-_reflects_ $\mathcal{C}$.
>
>

The restriction of ${\rm SR}^{-}$ to $\Sigma_{1}(\mathcal{R})$-definable
classes of structures was first introduced in [BV16]. Namely, for
${\mathcal{R}}$ a finite set of $\Pi_{1}$ predicates or relations, let

>   * ${\rm SR}^{-}_{\mathcal{R}}:$
>
> There exists a cardinal $\kappa$ that $\kappa$-reflects every
> $\Sigma_{1}({\mathcal{R}})$-definable (with parameters in $H_{\kappa}$)
> class ${\mathcal{C}}$ of structures of the same type.
>
>

####  6.1.1. The Cardinality predicate

Let $\mathop{Cd}$ be the $\Pi_{1}$ predicate $``x$ is a cardinal”. Magidor and
Väänänen [MV11] show that the principle ${\rm SR}_{\mathop{Cd}}$ implies
$0^{\sharp}$, and much more, e.g., there are no good scales. The principle
${\rm SR}^{-}_{\mathop{Cd}}$ is much weaker, but it does have some large-
cardinal strength, as the next theorem shows.

######  Theorem 6.1 ([BV16]).

${\rm SR}^{-}_{\mathop{Cd}}$ holds, witnessed by $\kappa$, then there exists a
weakly inaccessible cardinal $\lambda\leq\kappa$.

It is shown in [MV11] that, starting form a supercompact cardinal, one can
produce a model of ZFC in which ${\rm SR}_{\mathop{Cd}}$ holds for the first
weakly inaccessible cardinal. Thus, no large-cardinal properties beyond weak
inaccessibility may be proved in ZFC to hold for the least cardinal witnessing
${\rm SR}_{\mathop{Cd}}$.

####  6.1.2. The Regularity predicate

Let $\mathop{Rg}$ be the $\Pi_{1}$ predicate $``x$ is a regular ordinal”.

######  Theorem 6.2 ([BV16]).

If ${\rm SR}^{-}_{\mathop{Rg}}$ holds, witnessed by $\kappa$, then there
exists a weakly Mahlo cardinal $\lambda\leq\kappa$.

It follows from [MV11] that one cannot hope to get from ${\rm
SR}_{\mathop{Rg}}$ more than a weakly Mahlo cardinal $\leq\kappa$, for
starting from a weakly Mahlo cardinal one can obtain a model in which ${\rm
SR}_{\mathop{Rg}}$ is witnessed by the least weakly Mahlo cardinal. One cannot
hope either to show that the least $\kappa$ witnessing ${\rm
SR}_{\mathop{Rg}}$ is strongly inaccessible, for in [SV02] it is shown that
one can have ${\rm SR}_{\mathop{Rg}}$ witnessed by $\kappa=2^{\aleph_{0}}$.

Let us note that, since the predicate $\mathop{Cd}$ is
$\Sigma_{1}(\mathop{Rg})$-definable (see [BV16]), the principle ${\rm
SR}_{\mathop{Cd},\mathop{Rg}}$ is equivalent to ${\rm SR}_{\mathop{Rg}}$.

####  6.1.3. The Weakly Inaccessible predicate

There is a principle between ${\rm SR}^{-}_{\mathop{Cd}}$ and ${\rm
SR}^{-}_{\mathop{Rg}}$, namely ${\rm SR}^{-}_{\mathop{Cd},\mathop{W\\!I}}$,
where $\mathop{W\\!I}$ is the $\Pi_{1}$ predicate $``x$ is weakly
inaccessible”.

######  Proposition 6.3 ([BV16]).

If ${\rm SR}^{-}_{\mathop{Cd},\mathop{W\\!I}}$ holds, witnessed by $\kappa$,
then there exists a $2$-weakly inaccessible cardinal $\lambda\leq\kappa$.

We may also consider predicates $\alpha$-$\mathop{W\\!I}$, for $\alpha$ an
ordinal. That is, the predicate “$x$ is $\alpha$-weakly inaccessible”. Then,
similar arguments as in [BV16] would show that the principle ${\rm
SR}^{-}_{\mathop{Cd},\,\alpha\,\mbox{-}\mathop{W\\!I}}$ holding for $\kappa$
implies that there is an $(\alpha+1)$-weakly inaccessible cardinal
$\lambda\leq\kappa$.

####  6.1.4. Weak compactness

Let $\mathop{W\\!C}(x,\alpha)$ be the $\Pi_{1}$ relation $``\alpha$ is a limit
ordinal and $x$ is a partial ordering with no chain of order-type $\alpha$”.

######  Theorem 6.4 ([BV16]).

If ${\rm SR}^{-}_{\mathop{Cd},\mathop{W\\!C}}$ holds, witnessed by some
$\kappa$ such that if $\gamma\leq\kappa$ is weakly inaccessible, then
$2^{\delta}\leq\gamma$ for all cardinals $\delta<\gamma$, then there exists a
weakly compact cardinal $\lambda\leq\kappa$.555In [BV16] it is only assumed
that $\kappa$ witnesses ${\rm SR}^{-}_{\mathop{Cd},\mathop{W\\!C}}$. However,
Philipp Lücke [Lüc21] has shown that some additional assumption on $\kappa$ is
needed. Note that our assumption on $\kappa$ in the current statement of the
Theorem follows form the GCH.

Since the first weakly Mahlo cardinal may satisfy ${\rm SR}_{\mathop{Rg}}$
([MV11]), we cannot prove the existence of a weakly compact cardinal
$\leq\kappa$ just from ${\rm SR}_{\mathop{Rg}}$. Hence, ${\rm
SR}_{\mathop{Cd},\mathop{W\\!C}}$ is stronger than ${\rm SR}_{\mathop{Rg}}$.

Let $\mathop{PwSet}$ be the $\Pi_{1}$ relation $\\{(x,y):y=\mathcal{P}(x)\\}$.
Then we have the following:

######  Theorem 6.5 ([Mag71]).

$\kappa$ is the least cardinal witnessing ${\rm SR}_{\mathop{PwSet}}$ if and
only if $\kappa$ is the first supercompact cardinal.

It follows from Theorem 3.2 that ${\rm SR}_{\mathop{PwSet}}$ is in fact
equivalent to $\Pi_{1}$-${\rm SR}$, and also equivalent to $\Sigma_{2}$-${\rm
SR}$. However, Lücke [Lüc21] has established that ${\rm
SR}^{-}_{\mathop{PwSet}}$ is much weaker than ${\rm SR}_{\mathop{PwSet}}$.
Indeed, he shows that ${\rm SR}^{-}_{\mathop{PwSet}}$ is equivalent to the
existence of a weakly shrewd cardinal, a large cardinal notion obtained by
weakening the definition of shrewd cardinal studied by M. Rathjen in [Rat95]
and whose consistency strength is strictly between the large cardinal notions
of total indescribability and subtleness. However, as shown in [Lüc21], shrewd
and weakly shrewd cardinal are equiconsistent.

######  Definition 6.6 ([Lüc21]).

A cardinal $\kappa$ is _weakly shrewd_ if for every formula $\varphi(x,y)$ of
the language of set theory, every cardinal $\theta>\kappa$, and every
$A\subseteq\kappa$ such that $\varphi(A,\kappa)$ holds in $H_{\theta}$, there
exist cardinals $\bar{\kappa}<\bar{\theta}$ such that $\bar{\kappa}<\kappa$
and $\varphi(A\cap\bar{\kappa},\bar{\kappa})$ holds in $H_{\bar{\theta}}$.

######  Theorem 6.7 ([Lüc21]).

The following are equivalent:

  1. (1)

$\kappa$ is the least weakly shrewd cardinal.

  2. (2)

$\kappa$ is the least cardinal witnessing ${\rm SR}^{-}_{\mathop{PwSet}}$.

  3. (3)

$\kappa$ is the least cardinal witnessing $\Sigma_{2}$-${\rm SR}^{-}$.

Since, as shown in [Lüc21], weakly shrewd cardinals may be smaller than
$2^{\aleph_{0}}$, the principle $\Sigma_{2}$-${\rm SR}^{-}$, and therefore
also ${\rm SR}^{-}_{\mathcal{R}}$, for any set $\mathcal{R}$ of $\Pi_{1}$
predicates, does not imply the existence of a strongly inaccessible cardinal.
Moreover, [Lüc21] shows that it is consistent, modulo the existence of a
weakly shrewd cardinal that is not shrewd (a large cardinal notion
consistency-wise weaker than subtleness), that there exists a cardinal less
than $2^{\aleph_{0}}$ witnessing the principle ${\rm SR}^{-}$ for all
definable classes of structures of the same type, taken as a schema, i.e.,
$\Sigma_{n}$-${\rm SR}^{-}$, for all $n<\omega$. Thus, even ${\rm SR}^{-}$
cannot imply the existence of a strongly inaccessible cardinal.

###  6.2. Strong $\Sigma_{1}(\mathcal{R})$-definability

Notice that a class ${\mathcal{C}}$ is $\Sigma_{1}$-definable iff there is a
$\Sigma_{1}$ formula $\varphi$ such that for every $A$, $A\in{\mathcal{C}}$ if
and only if some transitive structure $\langle M,\in\rangle$ that contains $A$
satisfies $\varphi(A)$. Now let $\mathcal{L}_{\dot{R}}$ be the language of set
theory expanded with an additional predicate symbol $\dot{R}$, and suppose $R$
is a predicate. Naturally, one may define a class ${\mathcal{C}}$ to be
$\Sigma_{1}(R)$ if it is $\Sigma_{1}$-definable in the language
$\mathcal{L}_{\dot{R}}$ with $\dot{R}$ being interpreted as $R$. However,
unlike the case of $\Sigma_{1}$-definability, this is not equivalent to saying
that there is a $\Sigma_{1}$ formula $\varphi$ of the language
$\mathcal{L}_{\dot{R}}$ such that for every $A$, $A\in{\mathcal{C}}$ if and
only if some transitive structure $\langle M,\in,R^{M}\rangle$ that contains
$A$ satisfies $\varphi(A)$. For the equivalence to hold we need to require
that $R^{M}$ is precisely $R\cap M$. Namely,

######  Proposition 6.8.

The following are equivalent for all classes ${\mathcal{C}}$ and predicates
$R$:

  1. (1)

${\mathcal{C}}$ is $\Sigma_{1}(R)$, i.e., there exists a $\Sigma_{1}$ formula
$\varphi(x)$ of $\mathcal{L}_{\dot{R}}$ such that

${\mathcal{C}}=\\{A:\varphi(A),\mbox{with $\dot{R}$ interpreted as $R$}\\}$

  2. (2)

There is a $\Sigma_{1}$ formula $\varphi(x)$ of the language
$\mathcal{L}_{\dot{R}}$ such that for every $A$, $A\in{\mathcal{C}}$ if and
only if

$\langle M,\in,R\cap M\rangle\models\varphi(A)$

for some transitive structure $\langle M,\in\rangle$ that contains $A$.

Notice also that if ${\mathcal{C}}$ is a $\Sigma_{1}$-definable class of
structures of the same type, then the closure of ${\mathcal{C}}$ under
isomorphisms is also $\Sigma_{1}$-definable, and we have the following
equivalences:

######  Proposition 6.9.

The following are equivalent for any class ${\mathcal{C}}$ of structures of
the same type that is closed under isomorphisms:

  1. (1)

${\mathcal{C}}$ is $\Sigma_{1}$.

  2. (2)

There is a $\Sigma_{1}$ formula $\varphi(x)$ of the language of set theory
such that for every $A$, $A\in{\mathcal{C}}$ if and only if

$\langle M,\in\rangle\models\varphi(B)$

for some transitive structure $\langle M,\in\rangle$ of size $|A|$ that
contains $B$, where $B$ is isomorphic to $A$.

Based on the considerations above, the following is therefore a natural
definition for a class of structures closed under isomorphisms to be
$\Sigma_{1}$-definable with an additional predicate $R$. This is a
reformulation, for the case $n=1$, of Lücke’s [Lüc21] definition of _local
$\Sigma_{n}(R)$-class_:

######  Definition 6.10.

A class ${\mathcal{C}}$ of structures of the same type and closed under
isomorphisms is $\Sigma_{1}(R)^{\ast}$ if there is a $\Sigma_{1}$ formula
$\varphi(x)$ of the language $\mathcal{L}_{\dot{R}}$ such that for every $A$,
$A\in{\mathcal{C}}$ if and only if

$\langle M,\in,R\cap M\rangle\models\varphi(B)$

for some transitive structure $\langle M,\in\rangle$ of size $|A|$ that
contains $B$, where $B$ is isomorphic to $A$.

Observe that although every $\Sigma_{1}(R)^{\ast}$ class ${\mathcal{C}}$ is
$\Sigma_{1}(R)$, the converse is not true, even assuming closure under
isomorphisms. An example is the class ${\mathcal{C}}$ of all structures
isomorphic to some transitive $\langle M,\in,{\rm{Cd}}\cap M\rangle$, where
${\rm{Cd}}$ is the class of cardinals.

The closure under isomorphisms of the $\Sigma_{1}(\mathcal{R})$-definable
classes of structures that are used in the proofs of 6.1, 6.2, 6.3, and 6.4
(as given in [BV16]) are easily seen to be $\Sigma_{1}(\mathcal{R})^{\ast}$,
for the corresponding $\mathcal{R}$. Thus, the results follow from the weaker
$\Sigma_{1}(\mathcal{R})^{\ast}$-${\rm SR}^{-}$ corresponding assumptions.
Also, the argument in the proof of Theorem 5.5 from [BV16] can be adapted to
show that if $\mathcal{L}^{\ast}$ and $\mathcal{R}$ are symbiotic, then the
$SLST(\mathcal{L}^{\ast})$ property implies
$\Sigma_{1}(\mathcal{R})^{\ast}$-${\rm SR}^{-}$ (see [BV16]).666However, as
shown in [Lüc21], it does not imply $({\rm SR})^{-}_{\mathcal{R}}$, as claimed
in [BV16]. Thus, from the results in section 8 of [BV16] one may obtain the
following equivalences:

######  Theorem 6.11.

  1. (1)

$($[Lüc21]$)$ $\kappa$ is the least weakly inaccessible cardinal iff $\kappa$
is the least cardinal witnessing $\Sigma_{1}(\mathop{Cd})^{\ast}$-${\rm
SR}^{-}$.

  2. (2)

$($[Lüc21]$)$ $\kappa$ is the least weakly Mahlo cardinal iff $\kappa$ is the
least cardinal witnessing $\Sigma_{1}(\mathop{Rg})^{\ast}$-${\rm SR}^{-}$.

  3. (3)

$\kappa$ is the least $\alpha$-weakly inaccessible cardinal iff $\kappa$ is
the least cardinal witnessing
$\Sigma_{1}(\mathop{Cd},\alpha$-$\mathop{W\\!I})^{\ast}$-${\rm SR}^{-}$.

  4. (4)

Suppose $\kappa$ is such that if $\gamma\leq\kappa$ is weakly inaccessible,
then $2^{\delta}\leq\gamma$ for all cardinals $\delta<\gamma$. Then $\kappa$
is the least weakly compact cardinal iff $\kappa$ is the least cardinal
witnessing $\Sigma_{1}(\mathop{Cd},\mathop{W\\!C})^{\ast}$-${\rm SR}^{-}$.

Under the assumption of GCH, or just assuming that every weakly inaccessible
cardinal is inaccessible, the theorem above yields exact characterizations in
terms of ${\rm SR}$ for the first inaccessible, Mahlo, $\alpha$-inaccessible,
and weakly-compact cardinals.

######  Corollary 6.12 (GCH).

  1. (1)

$\kappa$ is the least inaccessible cardinal iff $\kappa$ is the least cardinal
witnessing $\Sigma_{1}(Cd)^{\ast}$-${\rm SR}^{-}$.

  2. (2)

$\kappa$ is the least Mahlo cardinal iff $\kappa$ is the least cardinal
witnessing $\Sigma_{1}(Rg)^{\ast}$-${\rm SR}^{-}$.

  3. (3)

$\kappa$ is the least $\alpha$-inaccessible cardinal iff $\kappa$ is the least
cardinal witnessing $\Sigma_{1}(Cd,\alpha$-$WI)^{\ast}$-${\rm SR}^{-}$.

  4. (4)

$\kappa$ is the least weakly compact cardinal iff $\kappa$ is the least
cardinal witnessing $\Sigma_{1}(Cd,WC)^{\ast}$-${\rm SR}^{-}$.

In items (1)-(4) above one may, equivalently, strengthen
$\Sigma_{1}(\mathcal{R})^{\ast}$-${\rm SR}^{-}$ by allowing
$\kappa$-reflection for classes of structures ${\mathcal{C}}$ that are
$\Sigma_{1}(\mathcal{R})^{\ast}$-definable with parameters in $H_{\kappa}$.

##  7\. Generic Structural Reflection

If $A$ and $B$ are structures of the same type, we say that an elementary
embedding $j:A\to B$ is _generic_ if it exists in some forcing extension of
$V$. We shall next consider the following _generic_ version of ${\rm SR}$:

>   * ${\rm GSR}$:
>
> (_Generic Structural Reflection_) For every definable (with parameters)
> class $\mathcal{C}$ of relational structures of the same type there exists
> an ordinal $\alpha$ that _generically-reflects_ $\mathcal{C}$, i.e., for
> every $A$ in $\mathcal{C}$ there exists $B$ in $\mathcal{C}\cap V_{\alpha}$
> and a generic elementary embedding from $B$ into $A$.
>
>

Thus, ${\rm GSR}$ is just like ${\rm SR}$, but the elementary embeddings may
not exist in $V$ but in some forcing extension of $V$. The next proposition
shows that this is equivalent to requiring that the elementary embedding
exists in any forcing extension resulting from collapsing the structure $B$ to
make it countable.

######  Proposition 7.1 (([BGS17]).

The following are equivalent for structures $B$ and $A$ of the same type.

  1. (1)

$V^{{\rm Coll}(\omega,B)}\models``\text{There is an elementary embedding
}j:B\to A."$

  2. (2)

For some forcing notion ${\mathbb{P}}$,
$V^{{\mathbb{P}}}\models``\text{There is an elementary embedding }j:B\to A."$

Taking into account similar considerations as in the case of ${\rm SR}$ and
$\rm{PSR}$, we may properly formulate ${\rm GSR}$ as a schema. Namely, for
$\Gamma$ a lightface definability class, let:

>   * $\Gamma$-${\rm GSR}$:
>
> (_$\Gamma$ -Generic Structural Reflection_) There exists a cardinal $\kappa$
> that _generically-reflects_ all $\Gamma$-definable, with parameters in
> $V_{\kappa}$, classes $\mathcal{C}$ of natural structures, i.e., for every
> $A$ in $\mathcal{C}$ there exists $B$ in $\mathcal{C}\cap V_{\alpha}$ such
> that in $V^{{\rm Coll}(\omega,B)}$ there is an elementary embedding from $B$
> into $A$.
>
>

The boldface version being:

>   * There exist a proper class of cardinals $\kappa$ that _generically-
> reflect_ all $\Gamma$-definable, with parameters in $V_{\kappa}$, classes
> $\mathcal{C}$ of natural structures.
>
>

The assertion that $\kappa$ witnesses $\Gamma$-${\rm GSR}$, for $\Gamma$ a
boldface definability class, is equivalent to the _Generic Vopěnka Principle_
${\rm{gVP}}(\kappa,\Gamma)$ introduced in [BGS17].

Similar considerations as in the case of ${\rm SR}$ (see the remarks before
and after proposition 3.4) show that $\mathbf{\Pi_{n}}$-${\rm GSR}$ and
$\mathbf{\Sigma_{n+1}}$-${\rm GSR}$ are equivalent; and also $\Pi_{n}$-${\rm
GSR}$ and $\Sigma_{n+1}$-${\rm GSR}$ are equivalent.

We shall see next that some large cardinals, such as Schindler’s _remarkable_
cardinals, can be characterized in terms of ${\rm GSR}$.

######  Definition 7.2 ([Sch00, Sch14]).

A cardinal $\kappa$ is _remarkable_ if for every regular cardinal
$\lambda>\kappa$, there is a regular cardinal $\bar{\lambda}<\kappa$ such that
in $V^{{\rm Coll}(\omega,{\mathrel{\mathchoice{\raise
2.0pt\hbox{$\scriptstyle<$}}{\raise 1.0pt\hbox{$\scriptstyle<$}}{\raise
0.0pt\hbox{$\scriptscriptstyle<$}}{\scriptscriptstyle<}}}\kappa)}$ there is an
elementary embedding $j:H_{\bar{\lambda}}^{V}\to H_{\lambda}^{V}$ with $j({\rm
crit}(j))=\kappa$.

A cardinal is remarkable if and only if it is $1$-remarkable (Definition 4.3).
Remarkable cardinals are downward absolute to $L$ and their consistency
strength is strictly below a $2$-iterable cardinal. Remarkable cardinals are
in $C^{(2)}$, and they are totally indescribable and ineffable, hence limits
of totally indescribable cardinals (see [BGS17]).

######  Theorem 7.3 ([BGS17]).

The following are equiconsistent:

  1. (1)

$\Pi_{1}$-${\rm GSR}$

  2. (2)

There exists a cardinal $\kappa$ that witnesses $\mathbf{\Pi_{1}}$-${\rm GSR}$

  3. (3)

There exists a remarkable cardinal.

Let us say that a cardinal $\kappa$ is _almost remarkable_ if it is
almost-$1$-remarkable (Definition 4.4), namely: for all $\lambda>\kappa$ in
$C^{(1)}$ and every $a\in V_{\lambda}$, there is $\bar{\lambda}<\kappa$ also
in $C^{(1)}$ such that in $V^{\rm{Coll}(\omega,<\kappa)}$ there exists an
elementary embedding $j:V_{\bar{\lambda}}\to V_{\lambda}$ with
$a\in{\rm{range}}(j)$. Then theorem 4.5 yields the following:

######  Theorem 7.4.

A cardinal $\kappa$ witnesses $\mathbf{\Pi_{1}}$-${\rm GSR}$ if and only if
$\kappa$ is almost remarkable.

It follows that the notions of remarkable cardinal and of almost-remarkable
cardinal are equiconsistent.

Magidor [Mag71] shows that a cardinal $\kappa$ is supercompact if and only if
for every regular cardinal $\lambda>\kappa$ there is a regular cardinal
$\bar{\lambda}<\kappa$ and an elementary embedding $j:H_{\bar{\lambda}}\to
H_{\lambda}$ with $j({\rm crit}(j))=\kappa$. One can thus view a remarkable
cardinal as a _virtually supercompact_ 777We choose to call it _virtually
supercompact_ , as in [BGS17], instead of the perhaps more natural _generic
supercompact_ , for the latter notion already exists in the literature with a
different meaning. cardinal. In analogy with theorem 3.2 one might therefore
expect $\Pi_{1}$-${\rm GSR}$ to be not just equiconsistent, but actually
equivalent with the existence of a remarkable cardinal. Even more, since the
first supercompact cardinal is precisely the first cardinal that witnesses
$\mathbf{\Pi_{1}}$-${\rm SR}$, one might conjecture that the first remarkable
cardinal is the first cardinal that witnesses $\mathbf{\Pi_{1}}$-${\rm GSR}$.
This is almost true, but not exactly. On the one hand, if $\kappa$ is a
remarkable cardinal, then $\kappa$ witnesses $\mathbf{\Pi_{1}}$-${\rm GSR}$
([BGS17]). On the other hand, as shown in theorem 7.6 below, if there is no
$\omega$-Erdös cardinal in $L$, then the least cardinal witnessing
$\mathbf{\Pi_{1}}$-${\rm GSR}$ is also the first remarkable cardinal.

The following equivalent definition of remarkability was given in [BGS17]: a
cardinal $\kappa$ is remarkable if and only if for every $\lambda>\kappa$
there exists some $\bar{\lambda}<\kappa$ and a generic elementary embedding
$j:V_{\bar{\lambda}}\to V_{\lambda}$ with $j({\rm crit}(j))=\kappa$.

Wilson [Wil19] defines the notion of weakly remarkable cardinal by not
requiring that $\bar{\lambda}$ is strictly below $\kappa$. Namely,

######  Definition 7.5 ([Wil19]).

$\kappa$ is _weakly remarkable_ if and only if for every $\lambda>\kappa$
there exists some $\bar{\lambda}$ and a generic elementary embedding
$j:V_{\bar{\lambda}}\to V_{\lambda}$ with $j({\rm crit}(j))=\kappa$.

Wilson shows that if there exists a weakly remarkable non-remarkable, cardinal
$\kappa$, then some ordinal greater than $\kappa$ is an $\omega$-Erdös
cardinal in $L$. Moreover, the statements “There exists an $\omega$-Erdös
cardinal” and “There exists a weakly remarkable non-remarkable cardinal” are
equiconsistent modulo ZFC, and equivalent assuming $V=L$.

Observe that if $\kappa$ is cardinal witnessing $\mathbf{\Pi_{1}}$-${\rm
GSR}$, then $\kappa$ also witnesses $\mathbf{\Pi_{1}}$-${\rm GSR}$ in $L$.
Thus, if $\kappa$ witnesses $\mathbf{\Pi_{1}}$-${\rm GSR}$ and in $L$
$\lambda$ is the least inaccessible cardinal above $\kappa$, then
$L_{\lambda}$ is a model of ZFC in which $\kappa$ satisfies $\Pi_{1}$-${\rm
GSR}$ and there is no $\omega$-Erdös cardinal above $\kappa$. By combining
arguments from [BGS17] and [Wil19] we have the following:

######  Theorem 7.6.

Assume there is no $\omega$-Erdös cardinal in $L$. Then, the least cardinal
that satisfies $\Pi_{1}$-${\rm GSR}$, if it exists, is remarkable.

###### Proof.

Let $\kappa$ be the least cardinal witnessing $\Pi_{1}$-${\rm GSR}$. Let
${\mathcal{C}}$ be the $\Pi_{1}$-definable class of structures of the form
$\langle V_{\lambda+1},\in\rangle$ with $\lambda\in C^{(1)}$. Pick a singular
cardinal $\lambda\in C^{(2)}$ greater than $\kappa$. By $\Pi_{1}$-${\rm GSR}$,
let $j:V_{\bar{\lambda}+1}\to V_{\lambda+1}$ be a generic elementary embedding
with $\bar{\lambda}<\kappa$. Let $\bar{\alpha}={\rm crit}(j)$. Note that
$\bar{\alpha}<\bar{\lambda}$, because $\bar{\alpha}$ is regular and
$\bar{\lambda}$ is not.

We claim that $\bar{\alpha}$ is weakly remarkable up to $\bar{\lambda}$. So,
fix some $\delta>\bar{\alpha}$ smaller than $\bar{\lambda}$. Consider the
restriction $j:V_{\delta}\to V_{j(\delta)}$, which has $j({\rm
crit}(j))=j(\bar{\alpha})$. Then $V_{\lambda+1}$ satisfies that for some
$\bar{\delta}$ there exists a generic elementary embedding
$j^{\ast}:V_{\bar{\delta}}\to V_{j(\delta)}$ such that $j^{\ast}({\rm
crit}(j^{\ast}))=j(\bar{\alpha})$. Hence, by elementarity
$V_{\bar{\lambda}+1}$ satisfies that for some $\bar{\delta}$ there exists a
generic elementary embedding $j^{\ast}:V_{\bar{\delta}}\to V_{\delta}$ with
$j^{\ast}({\rm crit}(j^{\ast}))=\bar{\alpha}$.

By elementarity, $\alpha:=j(\bar{\alpha})$ is weakly remarkable up to
$\lambda$; and since $\lambda\in C^{(2)}$, $\alpha$ is weakly remarkable.
Since the existence of a weakly remarkable non-remarkable cardinal implies the
existence of an $\omega$-Erdös cardinal in $L$ ([Wil19]), by our assumption we
have that $\alpha$ is in fact remarkable. Hence, since every remarkable
cardinal witnesses $\Pi_{1}$-${\rm GSR}$ ([BGS17]), we have that
$\kappa\leq\alpha$.

The theorem will be proved by showing that $\alpha=\kappa$. For suppose,
aiming for a contradiction, that $\kappa<\alpha$. Since $\alpha$ is remarkable
and therefore belongs to $C^{(2)}$, we have

$V_{\alpha}\models``\kappa\mbox{ witnesses }\Pi_{1}\mbox{-}{\rm GSR}".$

By elementarity, there is some $\gamma<\bar{\alpha}$ such that

$V_{\bar{\alpha}}\models``\gamma\mbox{ witnesses }\Pi_{1}\mbox{-}{\rm GSR}".$

Hence, since $j(\gamma)=\gamma$, again by elementarity,

$V_{\alpha}\models``\gamma\mbox{ witnesses }\Pi_{1}\mbox{-}{\rm GSR}"$

and therefore $\gamma$ witnesses $\Pi_{1}$-${\rm GSR}$, thus contradicting the
minimality of $\kappa$. ∎

######  Corollary 7.7.

Assume there is no $\omega$-Erdös cardinal in $L$. Then, the following are
equivalent for a cardinal $\kappa$:

  1. (1)

$\kappa$ is the least cardinal witnessing $\Pi_{1}$-${\rm GSR}$.

  2. (2)

$\kappa$ is the least cardinal witnessing $\mathbf{\Pi_{1}}$-${\rm GSR}$.

  3. (3)

$\kappa$ is the least almost remarkable cardinal.

  4. (4)

$\kappa$ is the least weakly remarkable cardinal.

  5. (5)

$\kappa$ is the least remarkable cardinal.

We don’t know if the assumption that there is no $\omega$-Erdös cardinal in
$L$ is necessary for the equivalence above to hold. However, we have the
following:

######  Proposition 7.8.

For every $n>0$, if $\kappa$ witnesses $\mathbf{\Pi_{n}}$-${\rm GSR}$, then
$\kappa\in C^{(n+1)}$. In particular, if $\kappa$ witnesses
$\mathbf{\Pi_{1}}$-${\rm GSR}$, then $\kappa\in C^{(2)}$.

###### Proof.

Let us prove the case $n=1$. The general case follows by induction, using a
similar argument. So, suppose $a\in V_{\kappa}$, $\varphi(x,y)$ is a $\Pi_{1}$
formula with $x,y$ as the only free variables, and $V\models\exists
x\varphi(x,a)$. Pick $\lambda\in C^{(2)}$ greater than $\kappa$, so that
$V_{\lambda}\models\exists x\varphi(x,a)$. Since the class of structures of
the form $\langle V_{\alpha},\in,a\rangle$ is $\Pi_{1}$-definable with $a$ as
a parameter, there exists a generic elementary embedding $j:\langle
V_{\bar{\lambda}},\in,a\rangle\to\langle V_{\lambda},\in,a\rangle$ with
$\bar{\lambda}<\kappa$. Note that, on the one hand, since $\lambda$ belongs to
$C^{(1)}$ so does $\bar{\lambda}$, hence by downward absoluteness for
$\Pi_{1}$ sentences, $V_{\kappa}\models``\bar{\lambda}\in C^{(1)}"$. On the
other hand, by elementarity of $j$, $V_{\bar{\lambda}}\models\exists
x\varphi(x,a)$. Hence by upwards absoluteness, $V_{\kappa}\models\exists
x\varphi(x,a)$.

A simpler similar argument, using the fact that $\Sigma_{1}$ sentences are
absolute for transitive sets, shows that $\kappa\in C^{(1)}$. Hence, if
$\varphi(x,y)$ and $a$ are as above, and $V_{\kappa}\models\exists
x\varphi(x,a)$, then by upwards absoluteness, $V\models\exists x\varphi(x,a)$.

For the general case $n>1$, assume $\kappa\in C^{(n)}$, and consider the
$\Pi_{n}$-definable (with $a$ as a parameter) class of of structures of the
form $\langle V_{\alpha},\in,a\rangle$ with $\alpha\in C^{(n)}$. ∎

As Wilson [Wil19] shows that a cardinal is remarkable if and only if it is
weakly remarkable and belongs to $C^{(2)}$, if the least cardinal $\kappa$
that witnesses $\mathbf{\Pi_{1}}$-${\rm GSR}$ is not remarkable, then is not
weakly remarkable. Thus, the question is if it is provable in ZFC that the
least cardinal $\kappa$ witnessing $\mathbf{\Pi_{1}}$-${\rm GSR}$, if it
exists, is weakly remarkable. Notice, however, that the proof of theorem 7.6
does show that if $\kappa$ is the least cardinal witnessing $\Pi_{1}$-${\rm
GSR}$, then either $\kappa$ is remarkable or there is a weakly remarkable
cardinal below $\kappa$. Also, if $\kappa$ is the least cardinal witnessing
$\mathbf{\Pi_{1}}$-${\rm GSR}$, then either $\kappa$ is remarkable or there
are unboundedly-many weakly remarkable cardinals below $\kappa$.

More generally, recall (Definition 4.3 above) that a cardinal $\kappa$ is _$n$
-remarkable_, for $n>0$, if for every $\lambda>\kappa$ in $C^{(n)}$, there is
$\bar{\lambda}<\kappa$ also in $C^{(n)}$ such that in $V^{{\rm
Coll}(\omega,{\mathrel{\mathchoice{\raise 2.0pt\hbox{$\scriptstyle<$}}{\raise
1.0pt\hbox{$\scriptstyle<$}}{\raise
0.0pt\hbox{$\scriptscriptstyle<$}}{\scriptscriptstyle<}}}\kappa)}$, there is
an elementary embedding $j:V_{\bar{\lambda}}\to V_{\lambda}$ with $j({\rm
crit}(j))=\kappa$. Equivalently, we may additionally require that for any
given $a\in V_{\lambda}$, $a$ is in the range of $j$. A cardinal $\kappa$ is
_completely remarkable_ if it is $n$-remarkable for every $n>0$. Remarkable
cardinals are precisely the $1$-remarkable cardinals.

As shown in [BGS17], if $0^{\sharp}$ exists, then every Silver indiscernible
is completely remarkable in $L$. Moreover, if $\kappa$ is $2$-iterable, then
$V_{\kappa}$ is a model of ZFC in which there exist a proper class of
completely remarkable cardinals.

Theorem 7.3 also holds for $n$-remarkable cardinals. Namely,

######  Theorem 7.9.

The following are equiconsistent for $n>0$:

  1. (1)

$\Pi_{n}$-${\rm GSR}$

  2. (2)

There exists a cardinal $\kappa$ that witnesses $\mathbf{\Pi_{n}}$-${\rm GSR}$

  3. (3)

There exists an $n$-remarkable cardinal.

As it turns out, $n+1$-remarkable cardinals correspond precisely to the
virtual form of $C^{(n)}$-extendible cardinals. Namely,

######  Definition 7.10 ([BGS17]).

A cardinal $\kappa$ is _virtually extendible_ if for every $\alpha>\kappa$
there is a generic elementary embedding $j:V_{\alpha}\to V_{\beta}$ such that
${\rm crit}(j)=\kappa$ and $j(\kappa)>\alpha$.

A cardinal $\kappa$ is _virtually $C^{(n)}$-extendible_ if additionally
$j(\kappa)\in C^{(n)}$.

Note that virtually extendible cardinals are virtually $C^{(1)}$-extendible
because $j(\kappa)$ must be inaccessible in $V$.

In contrast with the definition of extendible cardinal, in which the
requirement that $j(\kappa)>\alpha$ is superfluous, in the definition of
virtually extendible cardinal it is necessary. The reason is that while there
is no non-trivial elementary embedding $j:V_{\lambda+2}\to V_{\lambda+2}$,
such an embedding may exist generically (see [BGS17]).

######  Theorem 7.11 ([BGS17]).

A cardinal $\kappa$ is virtually extendible if and only if it is
$2$-remarkable. More generally, $\kappa$ is virtually $C^{(n)}$-extendible if
and only if it is $n+1$-remarkable.

The requirement that $j(\kappa)>\alpha$ in the definition of virtually
extendible cardinals suggests the following strengthening of ${\rm GSR}$. Let
us say that an elementary embedding $j:V_{\alpha}\to V_{\beta}$ is
_overspilling_ if $j$ has a critical point and $j({\rm crit}(j))>\alpha$. For
$\Gamma$ a lightface definability class, let:

>   * $\Gamma$-${\rm SGSR}$:
>
> (_$\Gamma$ -Strong Generic Structural Reflection_) There exists a cardinal
> $\kappa$ that _strongly generically-reflects_ all $\Gamma$-definable classes
> $\mathcal{C}$ of natural structures, i.e., for every $A$ in $\mathcal{C}$
> there exists $B$ in $\mathcal{C}\cap V_{\alpha}$ such that in $V^{{\rm
> Coll}(\omega,B)}$ there is an overspilling elementary embedding from $B$
> into $A$.
>
>

With the boldface version being:

>   * There exist a proper class of cardinals $\kappa$ that _strongly
> generically-reflect_ all $\Gamma$-definable, with parameters in
> $V_{\kappa}$, classes $\mathcal{C}$ of natural structures.
>
>

Then we have the following:

######  Theorem 7.12.

The following are equivalent for every $n\geq 1$:

  1. (1)

$\Pi_{n}$-${\rm SGSR}$

  2. (2)

There exists an $n+1$-remarkable cardinal

  3. (3)

There exists a virtually $C^{(n)}$-extendible cardinal.

##  8\. Beyond ${\rm VP}$

We have seen that a variety of large cardinal notions, ranging from weakly
inaccessible to Vopěnka’s Principle, can be characterised as some form of
Structural Reflection for classes of relational structures of some degree of
complexity. The question is now if the same is true for large-cardinal notions
stronger than ${\rm VP}$, up to rank-into-rank embeddings, or even for large
cardinals that contradict the Axiom of Choice (see [BKW19]). This is largely a