1 Set-Theoretic axioms as reflection principles

Cantor [Reference Cantor11, 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”:

This principle of the unknowability of the absolute, which in Cantor’s work seems to have only a metaphysical (non-mathematical) meaning (see [Reference Jané13]), resurfaces again in the 1950s 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 $\vec {a}=a_1,\ldots ,a_n \in V$ ,

$$ \begin{align*} \forall x (\varphi (x, \vec{a}) \rightarrow x \in V) \rightarrow \exists y (y \in V \land \forall x (x \in y \leftrightarrow \varphi (x, \vec{a}))). \end{align*} $$

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 [Reference Lévy19] and Reinhardt [Reference Reinhardt30] 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-1960s 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 [Reference Wang36]), places Ackermann’s principle (stated in a Cantorian, non-mathematical form) as the main source for new set-theoretic axioms beyond ZFC:

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 [Reference Wang36, p. 285], where he asserts that the undefinability of V should be the source of all axioms of infinity, i.e., all large-cardinal axioms.

One possible interpretation of Gödel’s principle of the undefinability 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 interpret Gödel’s view of reflection as a justification for the axioms of large cardinals. This is made explicit in Koellner [Reference Koellner16, p. 208], where he identifies reflection principles with generalized forms of the Montague–Lévy Principle of Reflection. Namely,

Moreover, he explicitly interprets Gödel’s view of reflection as a source of large cardinals in this way [Reference Koellner16, p. 208]:

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

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:

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

If this is so, and even just for $n=1$ , then $\kappa $ must be an inaccessible cardinal, and this property actually characterises inaccessible cardinals [Reference Lévy20]. Thus the existence of an inaccessible cardinal follows rather easily from the reflection of the single

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

Hence, for each n, the following

sentence, with C as a second-order parameter, holds in V (again, interpreting the universal second-order quantifier as ranging over definable classes):

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

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

sentence, which we have just shown (using

reflection) to hold in V:

$$ \begin{align*} \forall C(C\mbox{ is a club subclass of ordinals}\rightarrow \exists \kappa (\kappa \mbox{ inaccessible}\wedge \kappa \in C)). \end{align*} $$

Any cardinal that reflects the

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

sentences with second-order parameters reflect, we may as well reflect this property of V. So, consider the following

sentence, which says that V reflects all

sentences with parameters,

If $V_{\kappa }$ reflects this sentence, then $\kappa $ is a

-indescribable cardinal, i.e., a weakly compact cardinal (see [Reference Jech14]). 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

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 [Reference Reinhardt29], one cannot even have reflection for 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 sentence

$$ \begin{align*}\forall X\exists x(X\in A\to X=x),\end{align*} $$

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 [Reference Tait35]. 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 [Reference Koellner16]), 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 [Reference Koellner16, 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 [Reference Koellner16], 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 [Reference McCallum26] for the consistency of some forms of reflection slightly stronger than Tait’s $\Gamma ^{(2)}$ ), but Koellner [Reference Koellner16] 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 Footnote ³ 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 principle, 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 . Again, 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 undefinability 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:

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

$$ \begin{align*} \bar{V}:= \langle V\cup \{ V\}, \in , V, \langle \bar{a}\rangle_{a\in V}\rangle, \end{align*} $$

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 undefinability of V, as expressed in Gödel’s quote above, as follows:

Undefinability of V: Every $\varphi \in \mathcal {T}$ is true in some

$$ \begin{align*} \bar{V}_{\alpha} : = \langle V_{\alpha} \cup \{ V_{\alpha} \}, \in , V_{\alpha} , \langle \bar{a}\rangle_{a\in V_{\alpha}}\rangle.\end{align*} $$

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

$$ \begin{align*} \Sigma_n\text{-Undefinability of } V: \text{ Every } \varphi \in \mathcal{T}_n \text{ is true in some } \bar{V}_{\alpha}. \end{align*} $$

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 $\Sigma _n$ -Undefinability of V, $\varphi ^v$ is true in some $\bar {V}_{\alpha }$ , and therefore $V_{\alpha } \models \varphi $ . Thus Undefinability 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 $\Sigma _n$ -Undefinability of V. Thus, to derive stronger reflection principles based on the undefinability 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 $\mathcal {T}'$ is any reasonable class of (higher-order) sentences that are true of V, then Undefinability of V for the class $\mathcal {T}'$ will imply the reflection, in the sense of Montague–Lévy, of all sentences in $\mathcal {T}'$ . 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 Undefinability 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 [Reference Wang36]:

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 generalization 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:

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

We do not wish to claim that the $\mathrm {SR}$ principle is what Gödel had in mind when talking about reflection of internal structural properties of the membership relation, but we do claim that $\mathrm {SR}$ 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 $\mathrm {SR}$ with the existence of different kinds of large cardinals. Our goal is to illustrate the fact that $\mathrm {SR}$ is a general principle underlying a wide variety of large-cardinal principles. Many of the results have already been published [Reference Bagaria2–Reference Bagaria, Gitman and Schindler4, Reference Bagaria and Lücke7–Reference Bagaria and Wilson9, Reference Lücke23] or are forthcoming [Reference Bagaria and Lücke6], but some are new (Theorems 4.1, 4.2, 4.5, and 4.7–4.9, Proposition 5.1, Theorem 5.11, Proposition 5.17, and Theorems 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 $\mathrm {SR}$ 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

- $\mathrm {SR}$ 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 for that). Namely,

Similarly for - $\mathrm {SR}$ .

A standard closing-off argument shows that $\mathbf {\Sigma _n}$ - $\mathrm {SR}$ 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, $\Sigma _n$ - $\mathrm {SR}$ 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 - $\mathrm {SR}$ and - $\mathrm {SR}$ . Thus, for $\Gamma $ a definability class (i.e., one of $\mathbf {\Sigma _n}$ , , $\Sigma _n$ , or ), let us say that an ordinal $\alpha $ witnesses $\Gamma $ - $\mathrm {SR}$ 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 $ - $\mathrm {SR}$ holds if and only if $\Gamma $ - $\mathrm {SR}$ is witnessed by a proper class of ordinals.

The first observation is that, as the next proposition shows, $\mathbf {\Sigma _1}$ - $\mathrm {SR}$ is provable in ZFC. RecallFootnote ⁴ that, for every $n>0$ , $C^{(n)}$ is the -definable club proper class of cardinals $\kappa $ such that , 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.

Proof The implication (2) $\Rightarrow $ (1) is trivial. The implication (3) $\Rightarrow $ (2) is proved in [Reference Bagaria, Casacuberta, Mathias and Rosický3], using a Löwenheim–Skolem type of argument.

To show that (1) implies (3), let $\alpha $ witness $\mathbf {\Sigma _0}$ - $\mathrm {SR}$ 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

$$ \begin{align*}\langle M,\in, \{a_1,\ldots,a_n\} \rangle,\end{align*} $$

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 $\mathrm {SR}$ , there is an elementary embedding

$$ \begin{align*}j:\langle N,\in, \{a_1,\ldots,a_n\} \rangle\to \langle M,\in ,\{a_1,\ldots,a_n\}\rangle,\end{align*} $$

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, $\mathbf {\Sigma _1}$ - $\mathrm {SR}$ is provable in ZFC, and therefore does not yield any large cardinals. But - $\mathrm {SR}$ 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 -definable class of structures of the form $\langle V_{\alpha }, \in \rangle $ , $\alpha $ an ordinal [Reference Magidor24].

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 $\mathrm {SR}$ for the -definable class of structures $\langle V_{\alpha } ,\in \rangle $ , $\alpha $ an ordinal.

2. (2) $\kappa $ is the least cardinal that witnesses - $\mathrm {SR}$ .

3. (3) $\kappa $ is the least cardinal that witnesses $\mathbf {\Sigma _2}$ - $\mathrm {SR}$ .

4. (4) $\kappa $ is the least supercompact cardinal.

The following global parametrized version then follows. Namely,

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

Some remarks are in order. First, the equivalence of - $\mathrm {SR}$ and $\Sigma _2$ - $\mathrm {SR}$ , 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,

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

Now notice that $\mathcal {C}^{\ast }$ is definable, with the same parameters as $\mathcal {C}$ (see [Reference Bagaria2]). 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 $\Gamma (P)$ - $\mathrm {SR}$ for the assertion that $\mathrm {SR}$ holds for all $\Gamma $ -definable, with parameters in P, classes of structures of the same type. Thus, e.g., $\mathbf {\Sigma _n}$ - $\mathrm {SR}$ is $\Sigma _n(V)$ - $\mathrm {SR}$ , and $\Sigma _n$ - $\mathrm {SR}$ is $\Sigma _n(\varnothing )$ - $\mathrm {SR}$ . Our remarks above yield now the following:

Second, the remarks above also show that for principles of Structural Reflection of the form $\Gamma (P)$ - $\mathrm {SR}$ 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 $ - $\mathrm {SR}$ , for $\Gamma $ a lightface definability class, as follows:

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

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

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

Moreover, if $\kappa $ witnesses - $\mathrm {SR}$ , 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 [Reference Bagaria2, Reference Bagaria, Casacuberta, Mathias and Rosický3]: $\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 $\mathrm {SR}$ in terms of the existence of large 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 -definable class of natural structures. Also, if $\kappa $ witnesses - $\mathrm {SR}$ , then $\kappa $ is either a $C^{(n)}$ -extendible cardinal or a limit of $C^{(n)}$ -extendible cardinals.

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

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 $\mathrm {VP}$ can be characterized in terms of $\mathrm {SR}$ . 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 -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 $\mathrm {SR}$ restricted to particular (collections of) -definable classes of structures, or for classes of structures whose definitional complexity is between $\Sigma _1$ and (e.g., $\Sigma _1$ -definability with additional predicates), or for weaker forms of structural reflection. Let us consider first the $\mathrm {SR}$ 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 -definable in V, without parameters, and such that $\mathrm {SR} ({\mathcal C})$ 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) $\mathrm {SR}({\mathcal C}).$

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 '<\beta '<\alpha $ and an elementary embedding

$$ \begin{align*} j:\langle L_{\beta'},\in ,\gamma'\rangle \to \langle L_{\beta} ,\in ,\gamma\rangle. \end{align*} $$

Since $j(\gamma ')=\gamma $ , j is not the identity. Let $\kappa $ be the critical point of j. Thus, $\kappa \leq \gamma '<\beta '$ . Hence by Kunen’s Theorem [Reference Kunen18] $0^{\sharp }$ 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 '<\beta '<\alpha $ be uncountable cardinals in V such that $\gamma ' \leq \gamma $ . Let I denote the class of Silver indiscernibles. Let $j:I\cap [\gamma ',\beta ']\to I\cap [\gamma ,\beta ]$ be order-preserving and such that $j(\gamma ')=\gamma $ and $j(\beta ')=\beta $ . Then j generates an elementary embedding

$$ \begin{align*} j:\langle L_{\beta'},\in ,\gamma'\rangle \to \langle L_{\beta} ,\in ,\gamma\rangle, \end{align*} $$

as required.

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

Theorem 4.2. If $0^{\sharp }$ exists, then $\mathrm {SR}({\mathcal C})$ 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 $\alpha _0 <\cdots <\alpha _m$ 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

, without parameters, such that

Choose indiscernibles $j_0 <\cdots < j_n <j_{n+1} <\kappa $ , with $\alpha _m < j_0$ , and let

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

By indiscernibility,

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 $\mathrm {SR}$ 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 [Reference Bagaria, Gitman and Schindler4].

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 $V^{\mathrm {{Coll}(\omega , <\kappa )}}$ there exists an elementary embedding $j:V_{\bar {\lambda }}\to V_{\lambda }$ with $j(\mathrm {{crit}}(j))=\kappa $ and $a\in \mathrm {{range}}(j)$ .

A cardinal $\kappa $ is $1$ -remarkable if and only if it is remarkable, in the sense of Schindler (see [Reference Bagaria, Gitman and Schindler4] and Definition 7.2).

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 [Reference Bagaria, Gitman and Schindler4]).

A weaker notion than n-remarkability is obtained by eliminating from its definition the requirement that $j(\mathrm {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 $V^{\mathrm {{Coll}(\omega , <\kappa )}}$ there exists an elementary embedding $j:V_{\bar {\lambda }}\to V_{\lambda }$ with $a\in \mathrm {{range}}(j)$ .

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^{\mathrm {{Coll}(\omega ,<\kappa )}} \ \kappa $ witnesses $\mathrm {SR}({\mathcal C})$ for every class ${\mathcal C}$ that is -definable in V with parameters in $V_{\kappa }$ .

Proof Assume $\kappa $ is almost n-remarkable. Fix $\mathcal {C}$ and a 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 $V^{\mathrm {{Coll}(\omega ,<\kappa )}}$ there exists an elementary embedding $j:V_{\bar {\lambda }}\to V_{\lambda }$ with $B \in \mathrm {{range}}(j)$ . 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 $V^{\mathrm {{Coll}(\omega ,<\kappa )}}$ , $\kappa $ witnesses $\mathrm {SR}({\mathcal C})$ for every class ${\mathcal C}$ that is -definable in V with parameters in $V_{\kappa }$ . Let ${\mathcal C}$ be the -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 $V^{\mathrm {{Coll}(\omega ,<\kappa )}}$ there exists some $\langle V_{\bar {\lambda }}, \in ,b\rangle \in {\mathcal C}$ together with an elementary embedding

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

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

Corollary 4.6. If $\kappa $ is an almost completely remarkable cardinal in L, then in $L^{\mathrm {{Coll}(\omega ,<\kappa )}} \ \kappa $ witnesses $\mathrm {SR}({\mathcal C})$ 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 definable with X as a parameter.

Theorem 4.7.

1. (1) $\mathrm {SR}({\mathcal C}_X)$ holds if and only if $X^{\sharp }$ exists.

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

These results suggest the following forms of $\mathrm {SR}$ 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:

1. Γ-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 exist 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 exists 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 $\mathrm {SR}(L[X])$ , i.e., $\mathbf {\Sigma _n}$ - $\mathrm {SR}(L[X])$ , 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 -definable in V, with U as a parameter. By using well-known facts about $0^{\dagger }$ due to Solovay (see [Reference Kanamori15] 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) $\mathrm {SR}({\mathcal C}^U)$ holds if and only if $0^{\dagger }$ exists.

2. (2) If $0^{\dagger }$ exists, then $\mathrm {SR}({\mathcal C})$ 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 -definable with U and X as parameters.

Theorem 4.9.

1. (1) $\mathrm {SR}({\mathcal C}^U_X)$ holds if and only if $X^{\dagger }$ exists.

2. (2) If $X^{\dagger }$ exists, then $\mathrm {SR}(\mathcal {C})$ 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. For example, for the canonical inner model $L[\mathcal {E}]$ for a strong cardinal, as in [Reference Koepke17] or [Reference Mitchell27], and its sharp, zero pistol . Also for a canonical inner model for a proper class of strong cardinals, as in [Reference Schindler32], 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 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 $(\mathrm {PRP})$ introduced in [Reference Bagaria and Wilson9]:

Similarly as in the case of $\mathrm {SR}$ (see Section 2), we may formally define $\mathrm {PSR}$ as a schema. Thus, we say that an ordinal $\alpha $ witnesses $\Gamma (P)$ - $\mathrm {PSR}$ (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 - $\mathrm {PSR}$ if and only if it witnesses $\mathbf {\Sigma _{n+1}}$ - $\mathrm {PSR}$ . Moreover, we obtain equivalent principles by restricting to classes of natural structures. Thus, for $\Gamma $ a lightface definability class, we define:

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

As in [Reference Bagaria and Wilson9, Proposition 3.2] one can show that every cardinal $\kappa $ in $C^{(1)}$ witnesses $\mathbf {\Sigma _1}$ - $\mathrm {PSR}$ . The converse also holds, and in fact we have the following:

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

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

-definable, with a as a parameter, class of structures of the form

with $\alpha \in C^{(1)}$ , and where

is the

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

formula in the language of $\langle V_{\alpha },\in ,a \rangle $ , then

$$ \begin{align*} R^{\alpha}_{\varphi} =\{ \langle a_1,\ldots ,a_n\rangle: \langle V_{\alpha},\in,a\rangle\models \text{"}\varphi[a_1,\ldots ,a_n, a]\text{"}\}\,. \end{align*} $$

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

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

and therefore, by elementarity of j,

which implies that ${\mathcal A}_{\alpha } \models R^{\alpha }_{\varphi } \ne \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 , 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 , 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, $\mathrm {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 [Reference Bagaria and Wilson9, Proposition 3.3].

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

Let $S:= \{ X: M\models \varphi (X)\} \cap V^M_{j(\kappa )}$ . Since $\mathcal {A}\in V_{\lambda }$ and $\varphi (x)$ is

, by downward absoluteness $V_{\lambda } \models \varphi (\mathcal {A})$ . Hence, since the fact that $\lambda \in C^{(1)}$ is

-expressible and therefore downwards absolute for transitive classes, and since $V_{\lambda } \subseteq M$ , it follows that

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 ( $\mathrm {PRP}$ ) introduced in [Reference Bagaria and Wilson9] when restricted to -definable classes is an easy consequence of - $\mathrm {PSR}$ , from [Reference Bagaria and Wilson9] we obtain the following:

as well as its boldface version:

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

As with the case of strong cardinals, standard arguments show (cf. [Reference Kanamori15] 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, $\mathrm {crit}(j)=\kappa $ , $V_{\lambda } \subseteq M$ , $j(\kappa )>\lambda $ , and $A\cap V_{\lambda } \subseteq j(A)$ . As shown in [Reference Bagaria and Wilson9], every strong cardinal is $\Sigma _2$ -strong. Also, a cardinal is -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 $ - -strong.

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

Similarly as in Proposition 5.2, one can prove the following (see [Reference Bagaria and Wilson9] for details):

The following theorem then follows from the main result in [Reference Bagaria and Wilson9]:

The corresponding boldface version also holds. Namely,

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

Note that if $\delta $ is a Woodin cardinal (see [Reference Kanamori15] for the definition of Woodin cardinal and its equivalent formulation in terms of A-strength), then $V_{\delta }$ satisfies $\mathrm {Ord}$ is Woodin. The following equivalences then follow (cf. Theorem 3.9):

A close inspection of the proofs of Propositions 5.2 and 5.6 reveals that, for $n>0$ , if $\kappa $ is a -strong cardinal, then for every -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 with the following properties:

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

2. (2) $\subseteq $ -chain-preserving: If is so that $f({\mathcal A})\subseteq f({\mathcal A}')$ whenever $A \subseteq A'$ , then so is $h(f)$ .

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

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

The next theorem implies that strong cardinals can be characterized in terms of - $\mathrm {PSR}$ . The proof follows closely [Reference Bagaria and Wilson9, Theorem 5.1], with the properties of faithfulness and $\subseteq $ -chain preservation ((1) and (2) above) playing now a key role.

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 -definable class of all structures

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

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 $\mathrm {PSR}$ there is a faithful $\subseteq $ -chain-preserving elementary embedding , 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

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

$$ \begin{align*} k_{\beta}:V_{\kappa +1}\to V_{\beta+1} \end{align*} $$

by

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

Since j is elementary, for all formulas $\varphi (x_1,\ldots ,x_n)$ and all

, if

, then

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

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

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 $\mathrm {{Ult}}(V,E^{\beta }_a)$ is well-founded. Furthermore, since j is faithful, if $\beta <\kappa $ , then $E^{\beta }_a$ is the principal ultrafilter generated by $\{ a\}$ . Let

$$ \begin{align*}j^{\beta}_a:V\to M^{\beta}_a \cong \mathrm{{Ult}}(V, E^{\beta}_a),\end{align*} $$

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 [Reference Bagaria and Wilson9], 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

for all $f:[\kappa ]^{|a|}\to V$ , are well-defined and commute with the ultrapower embeddings $j^{\beta }_a$ (see [Reference Kanamori15, p. 26]).

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

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

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

$$ \begin{align*}j_{{\mathcal E}_{\beta}}(x)=[a,[c^a_x]_{E^{\beta}_a}]_{{\mathcal E}_{\beta}}\end{align*} $$

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

As in [Reference Bagaria and Wilson9] one can also show that $M_{{\mathcal E}_{\beta }}$ is well-founded. So, let be the transitive collapse, and let $j_{N_{\beta }}:V\to N_{\beta }$ be the corresponding elementary embedding, i.e., . Then, as in [Reference Bagaria and Wilson9] we can show that $V_{\beta } \subseteq N_{\beta }$ and $j_{N_{\beta }}(\kappa )\geq \beta $ . If $\beta>\kappa $ , this implies that $\mathrm {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 $\beta \leq \beta '$ are in $I_S$ , then $E^{\beta }_a = E^{\beta '}_a$ , for every $a\in [\beta ]^{<\omega }$ .

Proof of claim

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

By the claim above, for every $\beta <\beta '$ in $I_S$ the map

$$ \begin{align*}k_{\beta ,\beta'}:M_{{\mathcal E}_{\beta}}\to M_{{\mathcal E}_{\beta'}}\end{align*} $$

given by

$$ \begin{align*}k_{\beta ,\beta'}([a, [f]_{E^{\beta}_a}]_{{\mathcal E}_{\beta}})=[a,[f]_{E^{\beta'}_a}]_{{\mathcal E}_{\beta'}}\end{align*} $$

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

$$ \begin{align*}\langle \langle M_{{\mathcal E}_{\beta}}:\beta \in I' \rangle, \langle k_{\beta, \beta '}: \beta <\beta' \mbox{ in }I_S\rangle\rangle,\end{align*} $$

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

$$ \begin{align*}j_M(x)=[\beta , [a,[c^a_x]_{E^{\beta}_a}]_{{\mathcal E}_{\beta}}]\end{align*} $$

for some (any) $a\in [\beta ]^{<\omega }$ . Let be the transitive collapse, and let .

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 [Reference Bagaria and Wilson9], 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

$$ \begin{align*}j_N(\kappa)=\ell_{\beta,N}(j_{N_{\beta}}(\kappa))\geq \ell_{\beta , N}(\beta)\geq \beta.\end{align*} $$

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 $\mathrm {crit}(j_N)\leq \kappa $ . But since for $\beta <\kappa $ the map $j_{N_{\beta }}$ is the identity, we must have $\mathrm {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.

Similar results can be proven for $\Gamma $ -strong cardinals (Definition 5.5). On the one hand, Proposition 5.6 shows that if $\kappa $ is -strong, then $\kappa $ witnesses - $\mathrm {PSR}$ . On the other hand, similarly as in Theorem 5.11, we can prove that if $\kappa $ witnesses - $\mathrm {PSR}$ , then $\kappa $ is -strong. This yields the following characterization of -strong cardinals:

5.1 Strong Product Structural Reflection

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

1. 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 exist an ordinal $\beta $ with ${\mathcal A} \in V_{\beta }$ and an elementary embedding .

Similarly as in the case of $\mathrm {PSR}$ , let us say that a cardinal $\kappa $ witnesses $\Gamma (P)$ - $\mathrm {SPSR}$ if $\kappa $ strongly product-reflects all classes ${\mathcal C}$ that are $\Gamma $ -definable (with parameters in P). Also, a cardinal $\kappa $ witnesses - $\mathrm {SPSR}$ if and only if it witnesses $\mathbf {\Sigma _{n+1}}$ - $\mathrm {SPSR}$ , 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 $ - $\mathrm {SPSR}$ , for $\Gamma $ a lightface definability class, as follows:

1. Γ-SPSR: ( $\Gamma $ -Strong Product Structural Reflection) There exists a cardinal $\kappa $ that strongly product-reflects all $\Gamma $ -definable classes ${\mathcal C}$ of natural structures.

The boldface version is as follows:

• There exists 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 $ - $\mathrm {SPSR}$ implies $\Gamma $ - $\mathrm {PSR}$ , for any definability class $\Gamma $ .

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

Definition 5.16 [Reference Cai and Tsaprounis10].

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, $\mathrm {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, $\mathrm {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 $C^{(n+2)}$ [Reference Cai and Tsaprounis10]. As shown in [Reference Cai and Tsaprounis10], on the one hand, if $\kappa $ is $C^{(n)}$ -globally 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 - $\mathrm {SPSR}$ .

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 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

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 $\kappa \in C^{(n+2)}$ ) [Reference Cai and Tsaprounis10], by elementarity

, 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

, 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

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

Observe that since the function h in the proof of the last proposition is the restriction of j to , and j is elementary, it preserves all first-order properties. In particular, it is $\subseteq $ -chain-preserving, and since $\kappa =\mathrm {crit} (j)$ , 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 $\mathrm {PSR}$ schema, we may reformulate $\Gamma $ - $\mathrm {SPSR}$ for $\Gamma $ a lightface definability class as follows:

1. Γ-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 exist an ordinal $\beta $ with ${\mathcal A} \in V_{\beta }$ and a faithful and $\subseteq $ -chain-preserving elementary embedding .

The corresponding version for boldface $\Gamma $ being:

• There exists 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 [Reference Bagaria and Wilson9]), we can now prove the following:

Theorem 5.18. For every $n>0$ , there is a -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 -definable class of all structures

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

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 $\mathrm {SPSR}$ for ${\mathcal C}$ . Let $I:=\{ \alpha :{\mathcal A}_{\alpha } \in V_{\kappa } \}$ . Since $\kappa \in C^{(n+1)}$ (Proposition 5.1), $\mathrm {{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 $\kappa '$ be such that ${\mathcal A}_{\beta } \in V_{\kappa '}$ and there is a faithful $\subseteq $ -chain-preserving elementary embedding

Let

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

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

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

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

Then $E^{\beta }_a$ is an $\omega _1$ -complete proper ultrafilter over $[\kappa ]^{|a|}$ , and so the ultrapower $\mathrm {{Ult}}(V,E^{\beta }_a)$ is well-founded. Furthermore, since j is faithful, if $\beta \in I$ , then $E^{\beta }_a$ is the principal ultrafilter generated by $\{ a\}$ . Let

$$ \begin{align*}j^{\beta}_a:V\to M^{\beta}_a \cong \mathrm{{Ult}}(V, E^{\beta}_a),\end{align*} $$

and let $\mathcal {E}_{\beta }:=\{ E^{\beta }_a: a\in [\beta ]^{<\omega }\}$ . As in [Reference Bagaria and Wilson9], $\mathcal {E}_{\beta }$ is normal and coherent. Let $M_{{\mathcal E}_{\beta }}$ be the direct limit of

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

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 [Reference Bagaria and Wilson9], $M_{{\mathcal E}_{\beta }}$ is well-founded. So, let

be the transitive collapse, and let

. We have that $V_{\beta } \subseteq N_{\beta }$ and $j_{N_{\beta }}(\kappa )\geq \beta $ (see [Reference Bagaria and Wilson9]).

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

$$ \begin{align*} k_{\beta ,\beta'}:M_{{\mathcal E}_{\beta}}\to M_{{\mathcal E}_{\beta'}} \end{align*} $$

given by

$$ \begin{align*} k_{\beta ,\beta'}([a, [f]_{E_a}]_{{\mathcal E}_{\beta}})=[a,[f]_{E_a}]_{{\mathcal E}_{\beta'}} \end{align*} $$

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

$$ \begin{align*} \langle \langle M_{{\mathcal E}_{\beta}}:\beta \in I' \rangle, \langle k_{\beta, \beta '}: \beta <\beta' \mbox{ in }I'\rangle\rangle, \end{align*} $$

and let $j_M :V\to M$ be the corresponding limit elementary embedding. Let be the transitive collapse, and let .

Let $\xi =\mbox {sup}(I')$ . Note that $\xi \in C^{(n)}$ and $\xi>\kappa $ . As in Claim 5.13, $j_N(\kappa )=\xi $ , hence $\mathrm {crit}(j_N)\leq \kappa $ . But since for $\beta \in I$ the map $j_{N_{\beta }}$ is the identity, $\mathrm {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 - $\mathrm {SPSR}.$

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

Corollary 5.20. The following are equivalent:

1. (1) $\mathrm {SPSR}$ , i.e., - $\mathrm {SPSR}$ for every n.

2. (2) - $\mathrm {SPSR}$ for every n.

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

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

5.2 Bounded Product Structural Reflection

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

1. Γ-PSR_β: 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 exist a set S with ${\mathcal A}$ in S and an elementary embedding .

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

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

Theorem 5.21. The following are equivalent:

1. (1) - $\mathrm {PSR}_1.$

2. (2) There exists a measurable cardinal.

Proof (1) implies (2): Let ${\mathcal C}$ be the

-definable class of $\langle V_{\gamma }, \in \rangle $ , $\gamma \geq \omega $ . Let $\kappa $ witness $\mathrm {PSR}_1$ for ${\mathcal C}$ , and let S be a set that contains $V_{\kappa +1}$ such that there exists an elementary embedding

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

$$ \begin{align*}k(X)=h_{\kappa +1}(h(\{ X\cap V_{\gamma}\}_{\gamma <\kappa})),\end{align*} $$

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

$$ \begin{align*}X\in{\mathcal U}_a \quad \mbox{ iff }\quad X\subseteq \kappa \mbox{ and } a \in k(X)\, .\end{align*} $$

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 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, $\mathrm {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

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

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 $\mathrm {PSR}_1$ for all classes ${\mathcal C}$ that are -definable with parameters in $V_{\kappa }$ .