THE WEAK VOPĚNKA PRINCIPLE FOR DEFINABLE CLASSES OF STRUCTURES

Abstract

We give a level-by-level analysis of the Weak Vopěnka Principle for definable classes of relational structures ( $\mathrm {WVP}$ ), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular, we show that $\mathrm {WVP}$ for $\Sigma _2$ -definable classes is equivalent to the existence of a strong cardinal. The main theorem (Theorem 5.11) shows, more generally, that $\mathrm {WVP}$ for $\Sigma _n$ -definable classes is equivalent to the existence of a $\Sigma _n$ -strong cardinal (Definition 5.1). Hence, $\mathrm {WVP}$ is equivalent to the existence of a $\Sigma _n$ -strong cardinal for all $n<\omega $ .

1 Introduction

The Vopěnka Principle ( $\mathrm {VP}$ ), which asserts that there is no rigid proper class of graphs, is a well-known strong large-cardinal principle ¹ (see [9]). Properly formulated as a first-order assertion, $\mathrm {VP}$ is a schema, that is, an infinite collection of statements, one for each first-order formula defining a proper class of graphs and asserting that the class defined by the formula is not rigid, i.e., there is some non-identity morphism. An equivalent formulation of $\mathrm {VP}$ as a first-order schema is given by restricting $\mathrm {VP}$ to proper classes of graphs that are definable with a certain degree of complexity, according to the Levy hierarchy of formulas $\Sigma _n$ , $n<\omega $ (see [8]). Thus, writing $\mathbf {\Sigma _n}$ - $\mathrm {VP}$ for the first-order assertion that every $\Sigma _n$ -definable (with parameters) proper class of graphs is rigid, we have that $\mathrm {VP}$ is equivalent to the schema consisting of $\mathbf {\Sigma _n}$ - $\mathrm {VP}$ for every n. As shown in [4, 5], $\mathbf {\Sigma _1}$ - $\mathrm {VP}$ is provable in ZFC, while $\mathbf {\Sigma _2}$ - $\mathrm {VP}$ is equivalent to the existence of a proper class of supercompact cardinals, $\mathbf {\Sigma _3}$ - $\mathrm {VP}$ is equivalent to the existence of a proper class of extendible cardinals, and $\mathbf {\Sigma _n}$ - $\mathrm {VP}$ is equivalent to the existence of a proper class of $C^{(n-2)}$ -extendible cardinals, for $n\geq 3$ . The level-by-level analysis of $\mathbf {\Sigma _n}$ - $\mathrm {VP}$ , $n<\omega $ , and the determination of their large-cardinal strength yielded new results in category theory, homology theory, homotopy theory, and universal algebra (see [5]). For example, the existence of cohomological localizations in the homotopy category of simplicial sets (Bousfield Conjecture) follows from $\mathbf {\Sigma _2}$ - $\mathrm {VP}$ .

The role of $\mathrm {VP}$ in category theory has a rich history. The first equivalences of $\mathrm {VP}$ with various category-theoretic statements were announced by Fisher in [6]. Further equivalences were proved over the next two decades by Adámek, Rosický, Trnková, and others. Their work showed that under $\mathrm {VP}$ “the structure of locally presentable categories becomes much more transparent” [2, p. 241]. For example, the statement that a category is locally presentable if and only if it is complete and bounded is equivalent to $\mathrm {VP}$ . And so is the statement that every orthogonality class in a locally presentable category is a small-orthogonality class [2, 6.9 and 6.14], [10]. Of the many category-theoretic statements now known to be equivalent to $\mathrm {VP}$ , the following one (see [2, 6.D]) turned out to be of particular interest:

1. (1) Every full subcategory of a locally presentable category $\mathcal {K}$ closed under colimits is coreflective in $\mathcal {K}$ .

What made (1) particularly interesting is that its dual statement

1. (2) Every full subcategory of a locally presentable category $\mathcal {K}$ closed under limits is reflective in $\mathcal {K}$ ,

while being a consequence of (1), could not be proved equivalent to it. Since $\mathrm {VP}$ —hence also (1)—was known to be equivalent to

$$ \begin{align*}\mathbf{Ord}\mbox{ cannot be fully embedded into }\mathbf{Gra}\end{align*} $$

(see [2, 6.3]), while statement (2) was proved equivalent to

$$ \begin{align*}\mathbf{Ord}^{\textit{op}}\mbox{ cannot be fully embedded into }\mathbf{Gra}\end{align*} $$

(see [2, 6.22 and 6.23]), the latter assertion was then called the Weak Vopěnka Principle (WVP). The term Weak was aptly given, for it is readily shown that $\mathrm {VP}$ implies $\mathrm {WVP}$ [3] (Proposition 2.1). The question then remained if $\mathrm {WVP}$ implied $\mathrm {VP}$ . Using a result of Isbell [7], which showed that $\mathbf {Ord}^{\textit{op}}$ is bounded iff there is no proper class of measurable cardinals, Adámek and Rosický [2] proved that $\mathrm {WVP}$ implies the existence of a proper class of measurable cardinals. This was seen as a first step in showing that $\mathrm {WVP}$ was indeed a strong large-cardinal principle, perhaps even equivalent to $\mathrm {VP}$ . Much work was devoted to trying to obtain stronger large cardinals from it, e.g., strongly compact or supercompact cardinals, but to no avail. A further natural principle, between $\mathrm {VP}$ and $\mathrm {WVP}$ , called the Semi-Weak Vopěnka Principle ( $\mathrm {SWVP}$ ), was introduced in [1] and the further question of the equivalence between the three principles, $\mathrm {WVP}$ , $\mathrm {SWVP}$ , and $\mathrm {VP}$ , remained open. The problem was finally solved in 2019 by the second author of the present paper. In [13] he showed that $\mathrm {WVP}$ and $\mathrm {SWVP}$ are equivalent, and they are also equivalent to the large-cardinal principle “ $\mathrm {OR}$ is Woodin,” whose consistency strength is known to be well below the existence of a supercompact cardinal, thereby showing that $\mathrm {WVP}$ cannot imply $\mathrm {VP}$ (if consistent with ZFC).

In the present paper we carry out a level-by-level analysis of $\mathrm {WVP}$ and $\mathrm {SWVP}$ similar to the analysis of $\mathrm {VP}$ done in [4, 5]. Thus, for every $n\geq 2$ we prove the equivalence of both $\mathbf {\Sigma _n}$ - $\mathrm {WVP}$ and $\mathbf {\Sigma _n}$ - $\mathrm {SWVP}$ (see Definition 2.2) with the existence of certain large cardinals. In particular, we show that $\mathbf {\Sigma _2}$ - $\mathrm {WVP}$ and $\mathbf {\Sigma _2}$ - $\mathrm {SWVP}$ are equivalent to the existence of a proper class of strong cardinals. The main theorems (Theorems 5.11 and 5.13) show, more generally, that $\mathbf {\Sigma _n}$ - $\mathrm {WVP}$ and $\mathbf {\Sigma _n}$ - $\mathrm {SWVP}$ are equivalent to the existence of a proper class of $\Sigma _n$ -strong cardinals (Definition 5.1). It follows that $\mathrm {WVP}$ and $\mathrm {SWVP}$ are equivalent to the schema asserting the existence of a $\Sigma _n$ -strong cardinal for every $n<\omega $ . Our arguments yield also a new proof of the second author’s result from [13] that $\mathrm {WVP}$ implies “ $\mathrm {OR}$ is Woodin” (Corollary 5.15). The main difference between the two proofs is that while in the present paper we derive the extenders witnessing “ $\mathrm {OR}$ is Woodin” from homomorphisms on products of relational structures with universe of the form $V_\alpha $ , the proof in [13] uses homomorphisms of so-called $\mathcal {P}$ -structures. We think, however, that it should be possible to do a similar level-by-level analysis as done here by using $\mathcal {P}$ -structures instead. A number of consequences in category theory should follow from our results. For instance, the statement that every $\mathbf {\Sigma _2}$ -definable full subcategory of a locally presentable category $\mathcal {K}$ closed in $\mathcal {K}$ under limits is reflective in $\mathcal {K}$ , should be equivalent to the existence of a proper class of strong cardinals. See [2, Chapter 6] for more examples.

2 Preliminaries

Recall that a graph is a structure $G=\langle G, E_G\rangle $ , where G is a non-empty set and $E_G$ is a binary relation on G. If $G=\langle G, E_G\rangle $ and $H=\langle H, E_H\rangle $ are graphs, a map $h:G\to H$ is a homomorphism if it preserves the binary relation, meaning that for all $x,y\in G$ , if $xE_G y$ , then $h(x)E_H h(y)$ .

A class $\mathcal {G}$ of graphs is called rigid if there are no non-trivial homomorphisms between graphs in $\mathcal {G}$ , i.e., the only homomorphisms are the identity morphisms $G\to G$ , for $G\in \mathcal {G}$ .

The original formulation of the Vopěnka Principle ( $\mathrm {VP}$ ) (P. Vopěnka, ca. 1960) asserts that there is no rigid proper class of graphs. As shown in [2, 6.A], $\mathrm {VP}$ is equivalent to the statement that the category $\mathbf {Ord}$ of ordinals cannot be fully embedded into the category $\mathbf {Gra}$ of graphs. That is, there is no sequence $\langle G_\alpha :\alpha \in \mathrm {OR}\rangle $ of graphs such that for every $\alpha \leq \beta $ there exists exactly one homomorphism $G_\alpha \to G_\beta $ , and no homomorphism $G_\beta \to G_\alpha $ whenever $\alpha < \beta $ .

The Weak Vopěnka Principle ( $\mathrm {WVP}$ ) (first introduced in [3]) is the statement dual to $\mathrm {VP}$ , namely that the opposite category of ordinals, $\mathbf {Ord}^{\textit{op}}$ , cannot be fully embedded into $\mathbf {Gra}$ . That is, there is no sequence $\langle G_\alpha :\alpha \in \mathrm {OR}\rangle $ of graphs such that for every $\alpha \leq \beta $ there exists exactly one homomorphism $G_\beta \to G_\alpha $ , and no homomorphism $G_\alpha \to G_\beta $ whenever $\alpha < \beta $ .

The Semi-Weak Vopěnka Principle ( $\mathrm {SWVP}$ ) [1] asserts that there is no sequence $\langle G_\alpha :\alpha \in \mathrm {OR}\rangle $ of graphs such that for every $\alpha \leq \beta $ there exists some (not necessarily unique) homomorphism $G_\beta \to G_\alpha $ , and no homomorphism $G_\alpha \to G_\beta $ whenever $\alpha < \beta $ .

Clearly, $\mathrm {SWVP}$ implies $\mathrm {WVP}$ . The second author showed in [12] that $\mathrm {SWVP}$ is in fact equivalent to $\mathrm {WVP}$ . As shown in [3], $\mathrm {VP}$ implies $\mathrm {WVP}$ , and the same argument also shows that $\mathrm {VP}$ implies $\mathrm {SWVP}$ . In fact, the argument shows the following:

Proposition 2.1. $\mathrm {VP}$ implies that for every sequence $\langle G_\alpha : \alpha \in \mathrm {OR} \rangle $ of graphs there exist $\alpha <\beta $ with a homomorphism $G_\alpha \to G_\beta $ .

Proof Suppose $\langle G_\alpha : \alpha \in \mathrm {OR} \rangle $ is a sequence of graphs. Without loss of generality, if $\alpha < \beta $ , then $G_\alpha $ and $G_\beta $ are not isomorphic. Since there are only set-many (as opposed to proper-class-many) non-isomorphic graphs of any given cardinality, there exists a proper class $C\subseteq \mathrm {OR}$ such that $|G_\alpha | <|G_\beta |$ whenever $\alpha <\beta $ are in C. For each $\alpha \in C$ , add to $G_\alpha =\langle G_\alpha , E_\alpha \rangle $ a rigid binary relation $S_\alpha $ on $G_\alpha $ [11], as well as the non-identity relation $\not =$ , and consider the structure $A_\alpha =\langle G_\alpha , E_\alpha , S_\alpha , \not =\rangle $ . Since the cardinalities are strictly increasing, by the $\not =$ relation there cannot be any homomorphism $A_\beta \to A_\alpha $ with $\alpha <\beta $ . Also, because of the rigid relation $S_\alpha $ , the identity is the only homomorphism $A_\alpha \to A_\alpha $ . Since the category whose objects are the $A_\alpha $ , $\alpha \in C$ and the morphisms are the homomorphisms can be fully embedded into $\mathbf {Gra}$ (see [2, 2.65]), by $\mathrm {VP}$ the class $\{ A_\alpha :\alpha \in C\}$ is not rigid, and so there must exist $\alpha <\beta $ with a homomorphism $A_\alpha \to A_\beta $ , hence also a homomorphism $G_\alpha \to G_\beta $ .

The definitions of $\mathrm {VP}$ , $\mathrm {WVP}$ , and $\mathrm {SWVP}$ given above quantify over arbitrary classes, so they are not first-order. Thus, a proper study of these principles must be carried out in some adequate class theory, such as NBG. In particular, the proof of the last proposition can only be formally given in such class theory. We shall however be interested in the forthcoming in the first-order versions of $\mathrm {VP}$ , $\mathrm {WVP}$ , and $\mathrm {SWVP}$ , which require us to restrict our attention to definable classes.

2.1 The VP, WVP, and SWVP for definable classes

Each of $\mathrm {VP}$ , $\mathrm {WVP}$ , and $\mathrm {SWVP}$ can be formulated in the first-order language of set theory as a definition schema, namely as an infinite list of definitions, one for every natural number n, as follows:

Definition 2.2. Let n be a natural number, and let P be a set or a proper class.

The $\Sigma _n(P)$ -Vopěnka Principle ( $\Sigma _n(P)$ - $\mathrm {VP}$ for short) asserts that there is no $\Sigma _n$ -definable, with parameters in P, sequence $\langle G_\alpha :\alpha \in \mathrm {OR}\rangle $ of graphs such that for every $\alpha \leq \beta $ there exists exactly one homomorphism $G_\alpha \to G_\beta $ , and no homomorphism $G_\beta \to G_\alpha $ whenever $\alpha <\beta $ .

The $\Sigma _n(P)$ -Weak Vopěnka Principle ( $\Sigma _n(P)$ - $\mathrm {WVP}$ for short) asserts that there is no $\Sigma _n$ -definable, with parameters in P, sequence $\langle G_\alpha :\alpha \in \mathrm {OR}\rangle $ of graphs such that for every $\alpha \leq \beta $ there exists exactly one homomorphism $G_\beta \to G_\alpha $ , and no homomorphism $G_\alpha \to G_\beta $ whenever $\alpha <\beta $ .

The boldface versions ${\mathbf \Sigma _n}$ - $\mathrm {VP}$ and ${\mathbf \Sigma _n}$ - $\mathrm {WVP}$ are defined as $\Sigma _n(V)$ - $\mathrm {VP}$ and $\Sigma _n(V)$ - $\mathrm {WVP}$ respectively, i.e., any set is allowed as a parameter in the definitions.

$\Pi _n(P)$ - $\mathrm {VP}$ and $\Pi _n(P)$ - $\mathrm {WVP}$ , as well as $\mathbf {\Pi _n}$ - $\mathrm {VP}$ and $\mathbf {\Pi _n}$ - $\mathrm {WVP}$ , and the lightface (i.e., without parameters) versions $\Sigma _n$ - $\mathrm {VP}$ , $\Sigma _n$ - $\mathrm {WVP}$ and $\Pi _n$ - $\mathrm {VP}$ , $\Pi _n$ - $\mathrm {WVP}$ , are defined similarly.

The Vopěnka Principle ( $\mathrm {VP}$ ) is the schema asserting that the $\mathbf {\Sigma _n}$ - $\mathrm {VP}$ holds for every $n < \omega $ . And the Weak Vopěnka Principle ( $\mathrm {WVP}$ ) is the schema asserting that the $\mathbf {\Sigma _n}$ - $\mathrm {WVP}$ holds for every $n < \omega $ .

If instead of requiring that for $\alpha \leq \beta $ there is exactly one homomorphism ${G_\beta \to G_\alpha }$ we only require that there is at least one, then we obtain the Semi-Weak Vopěnka Principle ( $\mathrm {SWVP}$ ), formulated as the first-order schema consisting of $\mathbf {\Sigma _n}$ - $\mathrm {SWVP}$ for all $n<\omega $ .

It is well-known that the category of structures in any fixed (many-sorted, infinitary) relational language can be fully embedded into $\mathbf {Gra}$ (see [2, 2.65]). Thus, if in the original definitions of $\mathrm {VP}$ , $\mathrm {WVP}$ , and $\mathrm {SWVP}$ one replaces “graphs” by “structures in a fixed (many-sorted, infinitary) relational language,” one obtains equivalent notions. The same is true for the first-order formulations of these principles, but some extra care is needed to ensure there is no increase in the complexity of the definitions. In particular, in the case of infinite language signatures, an extra parameter for the language signature $\tau $ , as well as a parameter for a rigid binary relation on a binary signature associated with $\tau $ , may be needed in the definition. Namely, suppose $\Gamma $ is one of the definability classes $\Sigma _n$ , $\Pi _n$ , with $n\geq 1$ , P is a set or a proper class, and ${\mathcal C}$ is a $\Gamma $ -definable, with parameters in P, class of (possibly many-sorted) relational structures in a language type $\tau $ , i.e., $\tau =\langle R_\alpha :\alpha <\lambda \rangle $ , where each $R_\alpha $ is an $n_\alpha $ -ary relation symbol, $n_\alpha $ being some ordinal, possibly infinite. As in [2, 2.65], there is a $\Delta _1$ -definable (i.e., both $\Sigma _1$ -definable and $\Pi _1$ -definable), using $\tau $ as a parameter, one-sorted binary type $\tau '$ (meaning that all the relations are binary), and also a $\Gamma $ -definable, with parameters in P plus $\tau $ as an additional parameter, full embedding of ${\mathcal C}$ into the category $\mathbf {Rel} \, \tau '$ of $\tau '$ -structures and homomorphisms. Furthermore, there is a $\Delta _1$ -definable, using $\tau $ and a rigid binary relation r on $\tau '$ as parameters, full embedding of $\mathbf {Rel} \, \tau '$ into $\mathbf {Gra}$ . Hence, there is a $\Gamma $ -definable (with parameters in P, plus $\tau $ and r as additional parameters) full embedding of ${\mathcal C}$ into $\mathbf {Gra}$ . Therefore, in the definitions of $\Gamma $ - $\mathrm {VP}$ , $\Gamma $ - $\mathrm {WVP}$ , and $\Gamma $ - $\mathrm {SWVP}$ we may replace “graphs” by “structures in a fixed (many-sorted, infinitary) relational language” and obtain equivalent principles, provided we allow for the additional parameters ( $\tau $ and r) involved. Let us, however, stress the fact that in the case of finite $\tau $ , or even if $\tau $ is countable infinite and definable without parameters (e.g., recursive), then no additional parameters are involved, and therefore the versions of $\Gamma $ - $\mathrm {VP}$ , $\Gamma $ - $\mathrm {WVP}$ , and $\Gamma $ - $\mathrm {SWVP}$ for graphs and for relational structures are equivalent.

2.2 Strong cardinals

Recall that a cardinal $\kappa $ is $\lambda $ -strong, where $\lambda $ is a cardinal greater than $\kappa $ , if there exists an elementary embedding $j: V\to M$ , with M transitive, with critical point $\kappa $ , and with $V_\lambda $ contained in M. A cardinal $\kappa $ is strong if it is $\lambda $ -strong for every cardinal $\lambda>\kappa $ .

If $\kappa $ is a strong cardinal, then for every cardinal $\lambda>\kappa $ there exists an elementary embedding $j: V\to M$ , with M transitive, critical point $\kappa $ , $V_\lambda $ contained in M, and $j(\kappa )>\lambda $ . Moreover, if $\kappa $ is strong, then $V_\kappa \preceq _{\Sigma _2}V$ . (See [9].)

It is well-known that the notion of strong cardinal can be formulated in terms of extenders (see [9, Section 26]). Namely,

Definition 2.3. Given a cardinal $\kappa $ , and $\beta>\kappa $ , a $(\kappa ,\beta )$ -extender is a collection ${\mathcal E}:=\{ E_a:a\in [\beta ]^{<\omega }\}$ such that:

1. (1) Each $E_a$ is a $\kappa $ -complete ultrafilter over $[\kappa ]^{|a|}$ , and $E_{a}$ is not $\kappa ^+$ -complete for some a.

2. (2) For each $\xi <\kappa $ , there is some a with $\{ s\in [\kappa ]^{|a|}:\xi \in s\}\in E_a$ .

3. (3) Coherence: If $a\subseteq b$ are in $[\beta ]^{<\omega }$ , with $b=\{ \alpha _1,\ldots , \alpha _n\}$ and $a=\{ \alpha _{i_1}, \ldots , \alpha _{i_n}\}$ , and $\pi _{ba}:[\kappa ]^{|b|}\to [\kappa ]^{|a|}$ is the map given by $\pi _{ba}(\{ \xi _1,\ldots ,\xi _n\})=\{ \xi _{i_1},\ldots ,\xi _{i_n}\}$ , then

   $$ \begin{align*}X\in E_a \quad \mbox{ if and only if } \quad \{ s\in [\kappa]^{|b|}: \pi_{ba}(s)\in X\}\in E_b\, .\end{align*} $$

4. (4) Normality: Whenever $a\in [\beta ]^{<\omega }$ and $f:[\kappa ]^{|a|} \to V$ are such that $\{ s\in [\kappa ]^{|a|}: f(s)\in {\mathrm {max}}(s)\}\in E_a$ , there is $b\in [\beta ]^{<\omega }$ with $a\subseteq b$ such that

   $$ \begin{align*}\{ s\in [\kappa]^{|b|}: f(\pi_{ba}(s))\in s\}\in E_{b}\, .\end{align*} $$

5. (5) Well-foundedness: Whenever $a_m\in [\beta ]^{<\omega }$ and $X_m\in E_{a_m}$ for $m\in \omega $ , there is a function $d:\bigcup _m a_m \to \kappa $ such that $d“ a_m \in X_m$ for every m.

Proposition 2.4. A cardinal $\kappa $ is $\lambda $ -strong if and only if there exists a $(\kappa , |V_{\lambda }|^+)$ -extender $\mathcal {E}$ such that $V_{\lambda }\subseteq \overline {M}_{\mathcal {E}}$ and $\lambda <j_{\mathcal {E}}(\kappa )$ (where $\overline {M}_{\mathcal {E}}$ is the transitive collapse of the direct limit ultrapower $M_{\mathcal {E}}$ of V by $\mathcal {E}$ , and $j_{\mathcal {E}}:V\to \overline {M}_{\mathcal {E}}$ is the corresponding elementary embedding).

Proof See [9, Exercise 26.7].

3 The Product Reflection Principle

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 pointwise.

Definition 3.1 (The Product Reflection Principle ( $\mathrm {PRP}$ ))

For $\Gamma $ a definability class (i.e., one of $\Sigma _n$ , $\Pi _n$ , some $n>0$ ), and P a set or proper class, $\Gamma (P)$ - $\mathrm {PRP}$ asserts that for every class ${\mathcal C}$ of graphs that is $\Gamma $ -definable with parameters in P, the following holds:

• PRP: There is a non-empty subset S of ${\mathcal C}$ such that for every G in ${\mathcal C}$ there is a homomorphism $\prod S \to G$ .

If $P=\varnothing $ , then we simply write $\Gamma $ - $\mathrm {PRP}$ . If $P=V$ , then we write $\Gamma $ in boldface, e.g., $\mathbf {\Sigma _n}$ - $\mathrm {PRP}$ .

Note that in the case where ${\mathcal C}$ is a set, PRP is trivial because we may take $S = {\mathcal C}$ and use the coordinate projection homomorphisms from $\prod S$ , so the nontrivial case is the one where ${\mathcal C}$ is a proper class (and S is still required to be a set.)

In the definition of $\Gamma (P)$ - $\mathrm {PRP}$ we may replace “graphs” by “structures in a fixed (many-sorted, infinitary) relational language” and obtain equivalent principles, provided we allow for some additional parameters (see our remarks after Definition 2.2). Thus, the boldface principle $\mathbf {\Sigma _n}$ - $\mathrm {PRP}$ for classes of graphs is equivalent to its version for classes of relational structures.

We shall denote by $C^{(n)}$ the $\Pi _n$ -definable closed and unbounded class of ordinals $\kappa $ that are $\Sigma _n$ -correct in V, i.e., $V_\kappa \preceq _{\Sigma _n}V$ . (See [4].)

Proposition 3.2. $\mathbf {\Sigma _1}$ - $\mathrm {PRP}$ holds. In fact, for every $\kappa \in C^{(1)}$ and every $\Sigma _1$ -definable with parameters in $V_\kappa $ proper class ${\mathcal C}$ of structures in a fixed relational language $\tau \in V_\kappa $ , the set $S:={\mathcal C}\cap V_\kappa $ witnesses $\Sigma _1(V_\kappa )$ - $\mathrm {PRP}$ .

Proof Let $\kappa \in C^{(1)}$ , and let ${\mathcal C}$ be a $\Sigma _1$ -definable, with a set of parameters $P\in V_\kappa $ , proper class of structures in a relational language $\tau \in V_\kappa $ . Note that since $\kappa \in C^{(1)}$ , $V_\kappa =H_\kappa $ , and hence $|\mathrm {TC}(\{ \tau \}\cup P)|<\kappa $ . Let $\varphi (x)$ be a $\Sigma _1$ formula, with parameters in P, defining ${\mathcal C}$ . We claim that $S:={\mathcal C}\cap V_\kappa $ satisfies $\mathrm {PRP}$ . Given $\mathcal {A}\in {\mathcal C}$ , let $\lambda \in C^{(1)}$ be greater than $\kappa $ and such that $\mathcal {A}\in V_\lambda $ . Let $N\preceq V_\lambda $ be of cardinality less than $\kappa $ and such that $\mathcal {A}\in N$ and $\mathrm {TC}(\{ \tau \}\cup P)\subseteq N$ . Let $\pi :M\to N$ be the inverse transitive collapse isomorphism, and let $\mathcal {B}\in M$ be such that $\pi (\mathcal {B})=\mathcal {A}$ . Notice that $\pi $ fixes $\tau $ and the parameters of $\varphi (x)$ . Since M is transitive and of cardinality less than $\kappa $ , $\mathcal {B}\in H_\kappa =V_\kappa $ . Also, since $V_\lambda \models \varphi (\mathcal {A})$ , we have $N\models \varphi (\mathcal {A})$ , and therefore $M\models \varphi (\mathcal {B})$ . Hence, since M is transitive and $\varphi $ is upwards absolute for transitive sets, $\mathcal {B}\in {\mathcal C}$ . Thus, $\mathcal {B}\in S$ . Then the composition of $\pi $ with the projection $\prod S\to \mathcal {B}$ yields the desired homomorphism.

Proposition 3.3. If $\kappa $ is a strong cardinal, then $\Sigma _2(V_\kappa )$ - $\mathrm {PRP}$ holds.

Proof Let $\kappa $ be a strong cardinal and let ${\mathcal C}$ be a $\Sigma _2$ -definable, with parameters in $V_\kappa $ , proper class of structures in a fixed relational language $\tau \in V_\kappa $ . Let $\varphi (x)$ be a $\Sigma _2$ formula defining it. We will show that $S:={\mathcal C} \cap V_\kappa $ witnesses $\mathrm {PRP}$ .

Given any $\mathcal {A}\in {\mathcal C}$ , let $\lambda \in C^{(2)}$ be greater than or equal to $\kappa $ and with $\mathcal {A}\in V_\lambda $ .

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

By elementarity, the restriction of j to $\prod S$ yields a homomorphism

$$ \begin{align*}h:\prod S\to \prod (\{ X: M\models \varphi(X)\} \cap V^M_{j(\kappa)}).\end{align*} $$

Since $\mathcal {A}\in V_\lambda $ , and $\lambda \in C^{(2)}$ , we have that $V_\lambda \models \varphi (\mathcal {A})$ . Since $\lambda \in C^{(1)}$ 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, letting

$$ \begin{align*}g: \prod (\{ X: M\models \varphi(X)\} \cap V^M_{j(\kappa)})\to \mathcal{A}\end{align*} $$

be the projection map, we have that

$$ \begin{align*}g\circ h:\prod S\to \mathcal{A}\end{align*} $$

is a homomorphism, as wanted.

Corollary 3.4. If there exists a proper class of strong cardinals, then ${\mathbf \Sigma _2}$ - $\mathrm {PRP}$ holds.

We shall next show that $\mathrm {SWVP}$ is equivalent to the assertion that $\mathrm {PRP}$ holds for all definable proper classes of structures. Let $\Gamma _n$ be either $\Sigma _n$ or $\Pi _n$ .

Proposition 3.5. $\Gamma _n (P)$ - $\mathrm {PRP}$ implies $\Gamma _{n}(P)$ - $\mathrm {SWVP}$ , for every $n>0$ and every class P.

Proof Assume $\mathcal {G} = \langle G_\alpha : \alpha \in \text {OR}\rangle $ is a sequence of graphs that is definable by a $\Gamma _n$ formula $\varphi $ from a parameter $p \in P$ , such that whenever $\alpha \le \beta $ there is a homomorphism $G_\beta \to G_\alpha $ . We shall use $\Gamma _n(P)$ -PRP to produce a homomorphism $G_\beta \to G_{\gamma }$ for some $\gamma> \beta $ , thereby witnessing the desired instance of SWVP.

In the case $\Gamma _n = \Sigma _n$ we define the class of graphs $\mathcal {C} = \{G_\alpha : \alpha \in \text {OR}\}$ , which is $\Sigma _n$ -definable from the parameter $p \in P$ because a graph G is in $\mathcal {C}$ if and only if $\exists \alpha \in \text {OR}\;\varphi (\alpha , G, p)$ and the formula $\varphi $ is $\Sigma _n$ . Therefore by $\Sigma _n(P)$ -PRP, there is a subset $\mathcal {S}$ of the class $\mathcal {C}$ such that for every $G \in \mathcal {C}$ there is a homomorphism $\prod \mathcal {S} \to G$ . This gives a subset I of the class $\text {OR}$ such that for every $\gamma \in \text {OR}$ there is a homomorphism $\prod _{\alpha \in I} G_\alpha \to G_\gamma $ . In particular, letting $\beta = \sup I$ and $\gamma> \beta $ (for example, $\gamma = \beta + 1$ ), there is a homomorphism $\prod _{\alpha \in I} G_\alpha \to G_{\gamma }$ . Composing this with a homomorphism $G_\beta \to \prod _{\alpha \in I} G_\alpha $ (which exists because for each $\alpha \in I$ there is a homomorphism $G_\beta \to G_\alpha $ ) we obtain a homomorphism $G_\beta \to G_{\gamma }$ , as desired.

In the case $\Gamma _n = \Pi _n$ , we will need the following preliminary observation. Namely, we may assume without loss of generality that for every $\kappa $ the class $\{\alpha \in \text {OR} : |G_\alpha | \le \kappa \}$ is bounded by some ordinal $f(\kappa )$ , as otherwise it would contain two distinct $\alpha $ and $\alpha '$ such that $G_\alpha $ and $G_{\alpha '}$ are isomorphic, trivially witnessing the desired instance of SWVP. It follows that the class $\{\alpha \in \text {OR} : |G_\alpha | \ge \alpha \}$ contains a club class (namely the class of all closure points of this bounding function f) and in particular it is a proper class. We therefore define the class of graphs

$$\begin{align*}\mathcal{C} = \{G_\alpha : \alpha \in \text{OR} \text{ and } |G_\alpha| \ge \alpha\},\end{align*}$$

which is $\Pi _n$ -definable from the parameter $p \in P$ because a graph G is in $\mathcal {C}$ if and only if

$$\begin{align*}\forall \beta \in \text{OR}\;\big(|\beta| = |G| \implies \exists \alpha \le \beta\; \varphi(\alpha, G, p)\big)\end{align*}$$

and the formula $\varphi $ is $\Pi _n$ . (The main point is that the existential quantification over $\alpha $ is bounded.) Therefore, by $\Pi _n(P)$ -PRP, there is a subset $\mathcal {S}$ of the class $\mathcal {C}$ such that for every graph $G \in \mathcal {C}$ there is a homomorphism $\prod \mathcal {S} \to G$ . This gives a subset I of the class $\{\alpha \in \text {OR} : |G_\alpha | \ge \alpha \}$ such that for every ordinal $\gamma $ in that class, there is a homomorphism $\prod _{\alpha \in I} G_\alpha \to G_\gamma $ . Letting $\beta = \sup I$ , because the class $\{\alpha \in \text {OR} : |G_\alpha | \ge \alpha \}$ is a proper class, it contains some ordinal $\gamma> \beta $ . We may then proceed as in the $\Sigma _n$ case to obtain a homomorphism $G_\beta \to G_{\gamma }$ , as desired.

The converse also holds, and in fact more is true. Namely,

Proposition 3.6. $\Pi _{n}(P)$ - $\mathrm {WVP}$ implies $\Sigma _{n+1} (P)$ - $\mathrm {PRP}$ , for every $n\in \omega $ , and every P.

Proof Let $\mathcal {C}$ be a $\Sigma _{n+1}$ -definable, with parameter $p\in P$ , class of graphs that is a counterexample to PRP. We may represent this class as an increasing union: $\mathcal {C} = \bigcup _{\beta \in C^{(n)}} \mathcal {C}^{V_{\beta }}$ where $\mathcal {C}^{V_{\beta }}$ is the relativization of $\mathcal {C}$ to $V_\beta $ . Note that $\mathcal {C}^{V_{\beta }}$ is a subset of $\mathcal {C} \cap V_\beta $ for any $\beta \in C^{(n)}$ , and (unless $\mathcal {C}$ happens to be $\Pi _n$ ) it may be a proper subset, since the least witness to a $\Sigma _{n+1}$ property of a graph may have much larger rank than the graph itself.

We recursively define a function $f : \text {Ord} \to \text {Ord}$ by letting $f(0)$ be the least $\beta \in C^{(n)}$ such that $\mathcal {C}^{V_\beta } \not = \varnothing $ , letting $f(\alpha +1)$ be the least $\beta \in C^{(n)}$ such that there is no homomorphism $\prod \mathcal {C}^{V_{f(\alpha )}} \to \prod \mathcal {C}^{V_{\beta }}$ , and letting $f(\lambda ) = \sup _{\alpha < \lambda } f(\alpha )$ if $\lambda $ is a limit ordinal. To see that f is a total function, note that the ordinal $\beta $ in the definition of $f(\alpha +1)$ exists: otherwise for any graph $G \in \mathcal {C}$ we could take $\beta \in C^{(n)}$ sufficiently large that $G \in \mathcal {C}^{V_{\beta }}$ and then compose a homomorphism $\prod \mathcal {C}^{V_{f(\alpha )}} \to \prod \mathcal {C}^{V_{\beta }}$ with a projection homomorphism to obtain a homomorphism $\prod \mathcal {C}^{V_{f(\alpha )}} \to G$ , thereby witnessing PRP for $\mathcal {C}$ .

For every ordinal $\alpha $ we define the product graph

$$\begin{align*}H_\alpha = \prod \mathcal{C}^{V_{f(\alpha)}}.\end{align*}$$

Note that the sequence $\langle H_\alpha : \alpha \in \mathrm {Ord}\rangle $ is a counterexample to SWVP (which we do not claim to be $\Pi _n$ or even $\Sigma _{n+1}$ ): for every pair of ordinals $\alpha \le \alpha '$ there is a homomorphism from $H_{\alpha '}$ to $H_{\alpha }$ given by restriction, and for every ordinal $\alpha $ the definition of f implies there is no homomorphism from $H_{\alpha }$ to $H_{\alpha +1}$ (nor to $H_{\alpha '}$ for any larger $\alpha '$ , or else we could compose with a restriction homomorphism to get a homomorphism to $H_{\alpha +1}$ .) We will use this sequence to build a counterexample to WVP, which moreover will be $\Pi _n$ .

Let $\Lambda $ be the class of all limit ordinals that are fixed points of f. Equivalently because f is continuous and increasing, $\Lambda $ is the class of all limit ordinals that are closed under f. Note that $\Lambda $ is a closed unbounded class and $\Lambda \subseteq C^{(n)}$ (so in particular the elements of $\lambda $ are limit cardinals, although we won’t use this fact directly.) For every pair of ordinals $\lambda \le \lambda '$ in $\Lambda $ we define the function $h_{\lambda '\lambda } : V_{\lambda '+1} \to V_{\lambda +1}$ by

$$\begin{align*}h_{\lambda'\lambda} (x) = x \cap V_{\lambda}.\end{align*}$$

We will define some structure (constants and relations) on the sets $V_{\lambda +1}$ ( ${\lambda \in \Lambda }$ ) that is preserved by these functions $h_{\lambda '\lambda }$ and not by any other functions. To ensure that the structure is not preserved by any other functions, it will encode our counterexample $\langle H_\alpha : \alpha \in \mathrm {Ord}\rangle $ to SWVP (among other things.) The coding will use a “stratified” version $H_\alpha ^*$ of the product graph $H_\alpha $ , which we define as a double product of length $1+\alpha $ , having $\prod \mathcal {C}^{V_{f(0)}}$ as initial factor, namely:

$$ \begin{align*}H^*_\alpha = \prod \mathcal{C}^{V_{f(0)}} \times \prod (\mathcal{C}^{V_{f(1)}} \setminus \mathcal{C}^{V_{f(0)}}) \times \prod (\mathcal{C}^{V_{f(2)}} \setminus \mathcal{C}^{V_{f(1)}}) \times \cdots.\end{align*} $$

Note that the graph $H_\alpha ^*$ is isomorphic to $H_\alpha $ (the initial factor was added to ensure this in the case $\alpha =0$ ), and it has the advantage that for every pair of ordinals $\lambda \le \lambda '$ in $\Lambda $ , the function $h_{\lambda '\lambda }$ is a homomorphism from $H^*_{\lambda '}$ to $H^*_{\lambda }$ because it just restricts the outermost product from $\prod _{\alpha _0 < \lambda '}$ to $\prod _{\alpha _0 < \lambda }$ , whereas it might not be a homomorphism from $H_{\lambda '}$ to $H_{\lambda }$ because $\mathcal {C}^{V_{\lambda '}} \cap V_{\lambda }$ might not be equal to $\mathcal {C}^{V_\lambda }$ .

Our structures will be defined as in [12] except using the graph $H_\alpha ^*$ in place of $H_\alpha $ . Namely, we let $\Sigma $ be the signature with a constant symbol c and ternary relation symbols R, S, and T, and for every ordinal $\lambda \in \Lambda $ we define a corresponding $\Sigma $ -structure

$$\begin{align*}\mathcal{M}_\lambda = \big\langle V_{\lambda+1}, c^{\mathcal{M}_\lambda}, R^{\mathcal{M}_\lambda}, S^{\mathcal{M}_\lambda}, T^{\mathcal{M}_\lambda}\big\rangle,\end{align*}$$

where $c^{\mathcal {M}_\lambda } = \lambda $ and the interpretations of R, S, and T are defined as follows:

$$ \begin{align*} R^{\mathcal{M}_\lambda}(\alpha, x,y) &\iff \big(\alpha = \mathrm{rank}(x) \text{ and } x \in y\big) \text{ or } \alpha = \lambda,\\ S^{\mathcal{M}_\lambda}(\alpha, x,y) &\iff \big(\alpha = \mathrm{rank}(x) \text{ and } x \notin y\big) \text{ or } \alpha = \lambda,\\ T^{\mathcal{M}_\lambda}(\alpha, x,y) &\iff x\ \text{is adjacent to}\ y\ \text{in}\ H_\alpha^*. \end{align*} $$

(In the definition of $T^{\mathcal {M}_\lambda }$ we take x and y to be vertices of $H_\alpha ^*$ .)

Essentially the same argument as in [12] shows that the only homomorphisms among the structures $\mathcal {M}_\lambda $ for $\lambda \in \Lambda $ are the homomorphisms $h_{\lambda '\lambda }$ for $\lambda \le \lambda '$ . Here we will just remind the reader of the main idea of that argument, which is that for any ordinals $\lambda $ and $\lambda '$ in $\Lambda $ :

1. (1) If $\lambda < \lambda '$ , then preservation of the T relation and the constant c ensures that any “forward” homomorphism from $\mathcal {M}_\lambda $ to $\mathcal {M}_{\lambda '}$ would produce a homomorphism from $H^*_{\lambda }$ to $H^*_{\lambda '}$ , or equivalently from $H_{\lambda }$ to $H_{\lambda '}$ , contradicting the fact that $\langle H_\alpha : \alpha \in \mathrm {Ord}\rangle $ is a counterexample to SWVP.

2. (2) If $\lambda \le \lambda '$ , then preservation of the R and S relations (which encode the membership relation $\in $ ) ensures that any “reverse” homomorphism from $\mathcal {M}_{\lambda '}$ to $\mathcal {M}_{\lambda }$ that is not equal to $h_{\lambda '\lambda }$ would have a critical point that is mapped forward, which would then yield a contradiction using preservation of the T relation by an argument similar to (1).

Enumerating $\Lambda $ in increasing order as $\langle \lambda _\xi : \xi \in \mathrm {Ord}\rangle $ , or in other words letting $\lambda _\xi $ be the $\xi $ th fixed point of f, the sequence of structures

$$\begin{align*}\langle \mathcal{M}_{\lambda_\xi} : \xi \in \mathrm{Ord}\rangle\end{align*}$$

is therefore a counterexample to WVP. It remains to check that this sequence of structures is $\Pi _n$ . For any ordinal $\xi $ and any structure $\mathcal {M}$ , the condition $\mathcal {M} = \mathcal {M}_{\lambda _\xi }$ is equivalent to the conjunction of the following conditions, each of which can be expressed by a $\Pi _n$ formula (or simpler):

1. (1) $\mathcal {M}$ is a $\Sigma $ -structure whose underlying set M contains a largest ordinal $\lambda $ . This condition can be expressed by a bounded formula.

2. (2) M is a rank initial segment of V (which must therefore be equal to $V_{\lambda + 1}$ .) This condition is $\Pi _1$ .

3. (3) $\lambda $ is in the class $C^{(n)}$ . This condition is $\Pi _n$ .

4. (4) $\lambda $ is equal to $\lambda _\xi $ , the $\xi $ th closure point of f. Given that $M = V_{\lambda +1}$ where $\lambda \in C^{(n)}$ , this condition can be expressed by a bounded formula over M. To see this, first note that the definition of the class $C^{(n)}$ is $\Pi _n$ and therefore absolute to $V_\lambda $ . Second, note that the definition of f (as a relation and in particular as a partial function) using $C^{(n)}$ is also absolute to $V_\lambda $ . For the absoluteness of f, note that the complexity of $\mathcal {C}$ doesn’t make f complex, because f is defined using relativizations of $\mathcal {C}$ that can be computed locally; also note that the construction of product graphs is absolute to $V_\lambda $ , as is the existence of a homomorphism between any two given graphs.

   Third, note that because f is absolute to $V_\lambda $ , it follows that $V_\lambda $ can see whether or not $\lambda $ is a closure point of f. (This holds if and only if the relativization of f to $V_\lambda $ is a total function on $\lambda $ .) Finally, note that M can see more specifically whether or not $\lambda $ is the $\xi $ th closure point of f: in the case that $\xi \in M$ , this again follows from the absoluteness of f to $V_\lambda $ , but actually the case that $\xi \in M$ is the only case we need to consider, because $M = V_{\lambda +1}$ and for $\lambda $ to be the $\xi $ th closure point of f would require $\xi \le \lambda $ and therefore $\xi \in M$ .

5. (5) The constant and the three relations of the $\Sigma $ -structure $\mathcal {M}$ are defined correctly, meaning $c^{\mathcal {M}} = c^{\mathcal {M}_\lambda }$ , $R^{\mathcal {M}}= R^{\mathcal {M}_\lambda }$ , $S^{\mathcal {M}} =S^{\mathcal {M}_\lambda }$ , and $T^{\mathcal {M}} =T^{\mathcal {M}_\lambda }$ , defined as above. Given that the underlying set M of this structure is equal to $V_{\lambda +1}$ where $\lambda $ is a closure point of f (and is therefore in $C^{(n)}$ ) these four conditions can be expressed by bounded formulas over the structure $\mathcal {M}$ . We show this only for the condition $T^{\mathcal {M}} =T^{\mathcal {M}_\lambda }$ , since the other three are relatively straightforward.

   First, note that the definition of the sequence of graphs $\langle H^*_\alpha : \alpha < \lambda \rangle $ is absolute to $V_\lambda $ : again the complexity of the class $\mathcal {C}$ doesn’t matter because f and $H^*_\alpha $ are defined using local relativizations of it. Second, note that the vertex set and edge relation of the graph $H^*_\lambda $ are definable by bounded formulas over $V_{\lambda +1}$ for a similar reason. These two observations show that the correct relation $T^{\mathcal {M}_\lambda }$ is definable by a bounded formula over the set $M = V_{\lambda +1}$ , so the structure $\mathcal {M}$ can see whether or not $T^{\mathcal {M}_\lambda }$ agrees with its own relation $T^{\mathcal {M}}$ .⊣

The following is now an immediate consequence of Propositions 3.5 and 3.6.

Theorem 3.7. For every $n>0$ and every set or proper class P, the following are equivalent:

1. (1) $\Sigma _{n+1}(P)$ - $\mathrm {PRP}$ .

2. (2) $\Pi _n(P)$ - $\mathrm {PRP}$ .

3. (3) $\Sigma _{n+1}(P)$ - $\mathrm {SWVP}$ .

4. (4) $\Pi _n(P)$ - $\mathrm {SWVP}$ .

5. (5) $\Sigma _{n+1}(P)$ - $\mathrm {WVP}$ .

6. (6) $\Pi _n(P)$ - $\mathrm {WVP}$ .

Proof The implications $(1)\Rightarrow (2)$ , $(3)\Rightarrow (4)$ , and $(5)\Rightarrow (6)$ are trivial. The implications $(3)\Rightarrow (5)$ and $(4)\Rightarrow (6)$ are clear. Proposition 3.5 yields $(1)\Rightarrow (3)$ and $(2)\Rightarrow (4)$ . Finally, Proposition 3.6 yields $(6)\Rightarrow (1)$ .

In the next two sections we shall prove, for each $n>0$ , exact equivalences of (1)–(6) above with large cardinals.

4 The main theorem for strong cardinals

Theorem 4.1. The following are equivalent:

1. (1) There exists a strong cardinal.

2. (2) $\Sigma _2$ - $\mathrm {PRP}$ .

3. (3) $\Pi _1$ - $\mathrm {PRP}$ .

4. (4) $\Sigma _2$ - $\mathrm {SWVP}$ .

5. (5) $\Pi _1$ - $\mathrm {SWVP}$ .

6. (6) $\Sigma _2$ - $\mathrm {WVP}$ .

7. (7) $\Pi _1$ - $\mathrm {WVP}$ .

Proof (1) $\Rightarrow $ (2) is given by Proposition 3.3. The equivalence of (2)–(7) is given by Theorem 3.7. So, it will be sufficient to prove (3) $\Rightarrow $ (1).

(3) $\Rightarrow $ (1): Let $\mathcal {A}$ be the class of all structures

$$ \begin{align*}{\mathcal A}_\alpha:=\langle V_{ \alpha+1}, \in , \alpha ,\{ R^{\alpha}_\varphi \}_{\varphi \in \Pi_1} \rangle,\end{align*} $$

where the constant $ \alpha $ is the $\alpha $ -th element of $C^{(1)}$ and $\{ R^{\alpha }_\varphi \}_{\varphi \in \Pi _1}$ is the $\Pi _1$ relational diagram for $V_{\alpha +1}$ , i.e., if $\varphi (x_1,\ldots ,x_n)$ is a $\Pi _1$ formula in the language of $\langle V_{ \alpha +1},\in ,\alpha \rangle $ , then

$$ \begin{align*}R^\alpha_\varphi =\{ \langle x_1,\ldots ,x_n\rangle: \langle V_{ \alpha +1},\in,\alpha \rangle\models “\varphi(x_1,\ldots ,x_n)”\}\, .\end{align*} $$

We claim that $\mathcal {A}$ is $\Pi _1$ -definable without parameters. For $X\in \mathcal {A}$ if and only if $X=\langle X_0,X_1,X_2,X_3 \rangle $ , where:

1. (1) $X_2$ belongs to $C^{(1)}$ ,

2. (2) $X_0=V_{X_2 +1}$ ,

3. (3) $X_1 = \mathord {\in } \restriction X_0$ ,

4. (4) $X_3$ is the $\Pi _1$ relational diagram of $\langle X_0,X_1,X_2 \rangle $ , and

5. (5) $\langle X_0, X_1, X_2 \rangle \models “X_2$ is the $X_2$ -th element of $C^{(1)}$ .”

Note that $\mathcal {A}$ is a proper class. In fact, the class C of ordinals $\alpha $ such that ${\mathcal A}_\alpha \in {\mathcal A}$ is a closed and unbounded proper class. By $\Pi _1$ - $\mathrm {PRP}$ there exists a subset S of C such that for every $\beta \in C$ there is a homomorphism $j_\beta : \prod _{\alpha \in S} {\mathcal A}_\alpha \to {\mathcal A}_\beta $ . By enlarging S, if necessary, we may assume that $\mathrm {sup(S)\in S}$ . Let us denote $\prod _{\alpha \in S}{\mathcal A}_\alpha $ by M. Notice that

$$ \begin{align*}M =\langle \prod_{\alpha \in S} V_{\alpha +1}, \overline{\in}, \langle \alpha \rangle_{\alpha \in S}, \{ \overline{R}^\alpha_\varphi\}_{\varphi \in \Pi_1} \rangle,\end{align*} $$

where $\overline {\in }$ is the pointwise membership relation, and $\overline {R}^\alpha _\varphi $ is the pointwise $R^\alpha _\varphi $ relation. Let $\kappa := \mathrm {sup(S)}$ .

Now assume, aiming for a contradiction, that no cardinal $\leq\!\kappa $ is strong, and fix some $\beta \in C$ greater than $\kappa $ , of uncountable cofinality, such that no cardinal $\leq\!\kappa $ is $\beta $ -strong. Let $j=j_\beta $ .⊣

Claim 4.2 j preserves the Boolean operations $\cap $ , $\cup $ , $-$ , and also the $\subseteq $ relation.

Proof of claim For every $X,Y,Z\in M$ ,

$$ \begin{align*}M\models “X=Y\cap Z”\quad \mbox{ iff }\quad V_{\alpha +1}\models “X(\alpha)=Y(\alpha)\cap Z(\alpha),\!” \mbox{ all } \alpha \in S\, .\end{align*} $$

So, letting $\varphi (x,y,z)$ be the bounded formula expressing $x=y\cap z$ , we have that $\langle X(\alpha ),Y(\alpha ),Z(\alpha )\rangle \in R^\alpha _\varphi $ , for all $\alpha \in S$ . Hence $\langle X,Y,Z\rangle \in \overline {R}_\varphi $ , and since j is a homomorphism $\langle j(X),j(Y),j(Z)\rangle \in R^\beta _\varphi $ , which yields ${\mathcal A}_\beta \models “j(X)=j(Y)\cap j(Z).”$

Similarly for the operations $\cup , -$ , and for the relation $\subseteq $ .

Now define $k:V_{\kappa +1}\to V_{\beta +1}$ by

$$ \begin{align*}k(X)=j(\langle X\cap V_{\alpha}\rangle_{\alpha \in S})\, .\end{align*} $$

Claim 4.3 k also preserves the Boolean operations, as well as the $\subseteq $ relation.

Proof of claim Suppose $V_{\kappa +1}\models “X=Y\cap Z.”$ Then $X\cap V_{\alpha } =(Y\cap V_{\alpha })\cap (Z\cap V_{\alpha })$ , for every $\alpha \in S$ . Hence,

$$ \begin{align*}M\models “\langle X\cap V_{\alpha} \rangle_{\alpha \in S} =\langle Y\cap V_{\alpha}\rangle_{\alpha \in S}\cap \langle Z\cap V_{\alpha}\rangle_{\alpha \in S}.”\, \end{align*} $$

Since j preserves the $\cap $ operation,

$$ \begin{align*}{\mathcal A}_{\beta}\models “k(X)=k(Y)\cap k(Z).”\, \end{align*} $$

Hence,

$$ \begin{align*}V_{\beta +1}\models “k(X)=k(Y)\cap k(Z).”\, \end{align*} $$

Similarly for the operations $\cup , -$ , and the relation $\subseteq $ .

Claim 4.4 k maps ordinals to ordinals, and is the identity on $\omega +1$ .

Proof of claim Let $\varphi (x)$ be the bounded formula expressing that x is an ordinal. Let $\gamma \leq \kappa $ . Then $\gamma \cap V_{\alpha }$ is an ordinal, for all $\alpha <\kappa $ , and so

$$ \begin{align*}M\models “\langle \gamma \cap V_{\alpha}\rangle_{\alpha \in S} \, \overline{\in} \, \, \overline{R}_\varphi.”\, \end{align*} $$

Since j is a homomorphism,

$$ \begin{align*}{\mathcal A}_\beta \models “j(\langle \gamma \cap V_{\alpha}\rangle_{\alpha \in S}) \in R^\beta_\varphi,\!”\end{align*} $$

which yields that $k(\gamma )=j(\langle \gamma \cap V_{\alpha }\rangle _{\alpha \in S})$ is an ordinal in ${\mathcal A}_\beta $ , hence also in $V_{\beta +1}$ .

For every ordinal $\gamma \leq \omega $ , we have that $\gamma \cap V_{\alpha } =\gamma $ , for all $\alpha \in S$ . Moreover, $\gamma $ is definable by some bounded formula $\varphi _\gamma $ . Hence,

$$ \begin{align*}M\models “\langle \gamma \cap V_{\alpha}\rangle_{\alpha \in S} \, \overline{\in}\, \, \overline{R}_{\varphi_\gamma}”\end{align*} $$

and therefore

$$ \begin{align*}{\mathcal A}_\beta \models “j(\langle \gamma \cap V_{\alpha}\rangle_{\alpha \in S}) \in R^\beta_{\varphi_\gamma},\!”\end{align*} $$

which yields $k(\gamma )=\gamma $ .

Note that $k(\kappa )=j(\langle \alpha \rangle _{\alpha \in S})= \beta $ .

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

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

Since $k(\kappa )=\beta $ and $k(|a|)=|a|$ , we also have $k([\kappa ]^{|a|})=[\beta ]^{|a|}$ , and hence ${[\kappa ]^{|a|} \in E_a}$ . Moreover, since k preserves Boolean operations and the $\subseteq $ relation, $E_a$ is an ultrafilter over $[\kappa ]^{|a|}$ .

Claim 4.5 $E_a$ is $\omega _1$ -complete.

Proof of claim Given $\{ X_n:n<\omega \}\subseteq E_a$ , let $Y=\{ \langle n,x\rangle : x\in X_n\}$ . So, $Y\subseteq V_\kappa $ . We can express that $X=\bigcap _{n<\omega }X_n$ by a bounded sentence $\varphi $ in the parameters X, Y, and $\omega $ . Moreover, since $\alpha $ is a limit ordinal, for every $\alpha \in S$ , the sentence $\varphi (X\cap V_{\alpha }, Y\cap V_{\alpha }, \omega )$ expresses that $X\cap V_{\alpha } =\bigcap _{n<\omega }X_n\cap V_{\alpha }$ . So,

$$ \begin{align*}M\models “\langle X \cap V_{\alpha}, Y\cap V_{\alpha}, \omega \rangle_{\alpha \in S} \, \overline{\in}\, \, \overline{R}_\varphi.”\, \end{align*} $$

Since j is a homomorphism,

$$ \begin{align*}{\mathcal A}_\beta \models “\langle j( X \cap V_{\alpha}), j(Y\cap V_{\alpha}), j(\omega) \rangle_{\alpha \in S} \in R^\beta_\varphi”\end{align*} $$

and so $\langle k(X),k(Y),k(\omega )\rangle $ satisfies $\varphi $ . Since $k(\omega )=\omega $ , we thus have $k(X)=\bigcap _{n<\omega }k(X_n)$ . Hence, $a\in k(X)$ , and so $X\in E_a$ .

Let $\mathcal {E}:=\{ E_a: a\in [\beta ]^{<\omega }\}$ .

Claim 4.6 $\mathcal {E}$ is normal. That is, whenever $a\in [\beta ]^{<\omega }$ and f is a function with domain $[\kappa ]^{|a|}$ such that $\{ s\in [\kappa ]^{|a|}: f(s)\in {\mathrm {max}}(s)\}\in E_a$ , there is $b\supseteq a$ such that $\{ s\in [\kappa ]^{|b|}: f(\pi ^\kappa _{ba}(s))\in s\}\in E_{b}$ , where $\pi ^\kappa _{ba}:[\kappa ]^{|b|}\to [\kappa ]^{|a|}$ is the standard projection function.

Proof Fix a and f, and suppose the set

$$ \begin{align*}X:= \{ s\in [\kappa]^{|a|}: f(s)\in \mathrm{max}(s)\}\end{align*} $$

belongs to $E_a$ . For every $\alpha \in S$ ,

$$ \begin{align*}V_{\alpha +1}\models “X\cap V_\alpha =\{ s\in [\alpha ]^{|a|}: (f\cap V_\alpha )(s)\in \mathrm{max}(s)\}.”\end{align*} $$

Thus, letting $\varphi [X\cap V_\alpha , \alpha , |a|, f\cap V_\alpha ]$ be the sentence

$$ \begin{align*}\forall x (x\in (X\cap V_\alpha) \leftrightarrow x\in [\alpha]^{|a|} \wedge (f\cap V_\alpha) (x)\in \mathrm{max}(x)),\end{align*} $$

we have that

$$ \begin{align*}V_{\alpha +1}\models \varphi [X\cap V_\alpha, \alpha, |a|, f\cap V_\alpha].\end{align*} $$

Since the formula $\varphi (x,y,z)$ is equivalent to a bounded formula, we have $\langle X\cap V_\alpha , \alpha , |a|, f\cap V_\alpha \rangle \in R^\alpha _\varphi $ , for every $\alpha \in S$ . Hence,

$$ \begin{align*}M\models “\langle \langle X\cap V_\alpha\rangle_{\alpha \in S}, \langle \alpha\rangle_{\alpha \in S}, \langle |a| \rangle_{\alpha \in S} , \langle f \cap V_\alpha\rangle_{\alpha \in S} \rangle \, \, \overline{\in} \, \,\overline{R}^\alpha_\varphi”\end{align*} $$

and therefore

$$ \begin{align*}V_{\beta +1}\models “\langle j(\langle X\cap V_\alpha\rangle_{\alpha \in S}), j(\langle \alpha\rangle_{\alpha \in S}), j(\langle |a|\rangle_{\alpha \in S}), j(\langle f \cap V_\alpha\rangle_{\alpha \in S})\rangle \in R^\beta_\varphi.”\end{align*} $$

Since $j(\langle \alpha \rangle _{\alpha \in S})=k(\kappa )=\beta $ and $k(|a|)=|a|$ ,

$$ \begin{align*}V_{\beta +1}\models “\langle k(X), \beta, |a|, k(f)\rangle \in R^\beta_\varphi,\!”\end{align*} $$

which yields

$$ \begin{align*}k(X)=\{ s\in[\beta]^{|a|}: k(f)(s)\in \mathrm{max}(s)\}\, .\end{align*} $$

Also, since $X\in E_a$ , we have that $k(f)(a)\in \mathrm {max}(a)$ .

Let $\delta =k(f)(a)$ , and let $b=a\cup \{ \delta \}$ . Thus,

$$ \begin{align*}b\in \{ s\in [\beta]^{|b|}: k(f)(\pi^\beta_{ba}(s))\in s\},\end{align*} $$

where $\pi ^\beta _{ba}:[\beta ]^{|b|}\to [\beta ]^{|a|}$ is the standard projection function. So, since $\{ s\in [\beta ]^{|b|}: k(f)(\pi ^\beta _{ba}(s))\in s\}=k(\{ s\in [\kappa ]^{|b|}: f(\pi ^\kappa _{ba}(s))\in s\})$ , we have

$$ \begin{align*}\{ s\in [\kappa]^{|b|}: f(\pi^\kappa_{ba}(s))\in s\}\in E_b,\end{align*} $$

which shows that ${\mathcal E}$ is normal.

For each $a\in [\beta ]^{<\omega }$ , the ultrapower ${\mathrm {Ult}}(V,E_a)$ of V by the $\omega _1$ -complete ultrafilter $E_a$ is well-founded. So, let

$$ \begin{align*}j_a:V\to M_a \cong {\mathrm{Ult}}(V, E_a),\end{align*} $$

with $M_a$ transitive, be the corresponding ultrapower embedding. As usual, we denote the elements of $M_a$ by their corresponding elements in $\mathrm {Ult}(V, E_a)$ .

Claim 4.7 $\mathcal {E}$ is coherent. That is, for every $a\subseteq b$ in $[\beta ]^{<\omega }$ ,

$$ \begin{align*}X\in E_a \mbox{ if and only if } \{ s\in [\kappa]^{|b|}: \pi_{ba}(s) \in X\}\in E_b\, .\end{align*} $$

Proof Let $a\subseteq b$ in $[\beta ]^{<\omega }$ , and suppose $X\in E_a$ . Thus, $X\subseteq [\kappa ]^{|a|}$ and $a\in k(X)$ . We need to see that $b\in k(\{ s\in [\kappa ]^{|b|}:\pi _{ba}(s) \in X\})$ . Now notice that, since k is the identity on natural numbers, and $k(\kappa )=\beta $ ,

$$ \begin{align*}k(\{ s\in [\kappa]^{|b|}:\pi_{ba}(s) \in X\})=\{ s\in [\beta]^{|b|}: \pi_{ba}(s)\in k(X)\}.\end{align*} $$

Hence, since $\pi _{ba}(b)=a$ , and $a\in k(X)$ , we have that $b\in \{ s\in [\beta ]^{|b|}: \pi _{ba}(s)\in k(X)\}$ , as wanted.

Conversely, if $\{ s\in [\kappa ]^{|b|}: \pi _{ba}(s) \in X\}\in E_b$ , we have that $b\in k(\{ s\in [\kappa ]^{|b|}:\pi _{ba}(s) \in X\})=\{ s\in [\beta ]^{|b|}: \pi _{ba}(s)\in k(X)\}.$ Hence, $\pi _{ba}(b)=a\in k(X)$ , and therefore $X\in E_a$ .

For each $a\subseteq b$ in $[\beta ]^{<\omega }$ , let $i_{ab}:M_a\to M_b$ be given by

$$ \begin{align*}i_{ab}([f]_{E_a})=[f\circ \pi_{ba}]_{E_b}\end{align*} $$

for all $f:[\kappa ]^{|a|}\to V$ . By coherence, the maps $i_{ab}$ are well-defined and commute with the ultrapower embeddings $j_a$ (see [9, Section 26]).

Let $M_{\mathcal E}$ be the direct limit of

$$ \begin{align*}\langle \langle M_a:a\in [\beta]^{<\omega}\rangle, \langle i_{ab}:a\subseteq b\rangle\rangle.\end{align*} $$

For notational simplicity, whenever we write $[a, [f]]$ what we mean is that $[f]=[f]_{E_a}$ , which belongs to $M_a$ . Thus, when we say, e.g., that $[a, [f]]\in [b,[g]]$ in $M_{\mathcal E}$ , what we mean is that $[f]=[f]_{E_a}\in M_a$ , $[g]=[g]_{E_b}\in M_b$ , and $[\langle a, [f]_{E_a}\rangle ]_{\mathcal E} \in _{\mathcal E} [\langle b, [g]_{E_b}\rangle ]_{\mathcal E}$ .

Let $j_{\mathcal E} :V\to M_{\mathcal E}$ be the corresponding limit elementary embedding, i.e.,

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

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

Let $k_a:M_a\to M_{\mathcal E}$ be given by

$$ \begin{align*}k_a([f]_{E_a})=[a, [f]_{E_a}] \, .\end{align*} $$

It is easily checked that $j_{\mathcal E} =k_a \circ j_a$ and $k_b \circ i_{ab}=k_a$ , for all $a\subseteq b$ , $a,b \in [\beta ]^{<\omega }$ . Thus, letting $\mathrm {Id}_{|a|}: [\kappa ]^{|a|}\to [\kappa ]^{|a|}$ be the identity function, we have

$$ \begin{align*}M_{\mathcal E} =\{ j_{\mathcal E} (f)(k_a([\mathrm{id}_{|a|}]_{E_a})) : a\in [\beta]^{<\omega} \mbox{ and } f : [\kappa]^{|a|}\to V\}\, .\end{align*} $$

Let $M^\ast _{\mathcal E} :=\{ [a,[f]]\in M_{\mathcal E} : f \cap V_{\alpha }: [\alpha ]^{|a|}\to V_{\alpha } , \mbox { all }\alpha \in S\}$ . Suppose $[a,[f]], [b,[g]]\in M^\ast _{\mathcal E}$ . Then the following can be easily verified:

1. (1) $[a,[f]]\in _{\mathcal E} [b,[g]] \mbox { iff } k(f)(a)\in k(g)(b)$ .

2. (2) $[a,[f]] =_{\mathcal E} [b,[g]] \mbox { iff } k(f)(a) = k(g)(b)$ .

Claim 4.8 $M^\ast _{\mathcal E}$ is well-founded and downward closed under $\in _{\mathcal E}$ .

Proof Well-foundedness follows from items (1) and (2) above, as any infinite $\in _{\mathcal E}$ -descending sequence in $M^\ast _{\mathcal E}$ would yield an infinite $\in $ -descending sequence in $V_{\beta +1}$ .

Now suppose $[a,[f]]\in _{\mathcal E} [b,[g]]$ , with $[b,[g]]\in M^\ast _{\mathcal E}$ . Then, for some $c\supseteq a,b$ , and some $X\in E_c$ ,

$$ \begin{align*}(f\circ \pi_{ca})(s)\in (g\circ \pi_{cb})(s)\end{align*} $$

for every $s\in X$ . Let $Y=\{ \pi _{ca}(s):s\in X\} \in E_a$ . Define $h:[\kappa ]^{|a|}\to V$ by: $h(s)=f(s)$ for all $s\in Y$ , and $h(s)=0$ , otherwise. Then $[h]_{E_a}=[f]_{E_a}$ , and $[a,[f]]=[a,[h]]\in M^\ast _{\mathcal E}$ .

By the last claim, $M^\ast _{\mathcal E}$ is well-founded and extensional. So, let $M^\ast $ be the transitive collapse of $M^\ast _{\mathcal E}$ .

Claim 4.9 $V_\beta \subseteq M^\ast $ .

Proof of claim Since $\kappa $ and $\alpha $ , for $\alpha \in S$ , belong to $C^{(1)}$ , we have that $|V_\kappa |=\kappa $ and $|V_{\alpha } |=\alpha $ , all $\alpha \in S$ . Let $f\in V$ be a bijection between $[\kappa ]^1$ and $V_\kappa $ such that $f\restriction [\alpha ]^1$ is a bijection between $[\alpha ]^1$ and $V_{\alpha }$ , all $\alpha \in S$ . Let $\varphi (x,y,z)$ be a $\Pi _1$ formula expressing that $x=[u]^1$ , with u an ordinal, $y=V_u$ , and $z:x\to V_u$ is a bijection. Thus,

$$ \begin{align*}M\models “\langle [\alpha]^1 , V_{\alpha}, f\cap V_{\alpha}\rangle_{\alpha \in S} \, \overline{\in}\, \, \overline{R}_\varphi.”\, \end{align*} $$

Hence,

$$ \begin{align*}{\mathcal A}_\beta \models “\langle k([\kappa]^1), k(V_\kappa), k(f)\rangle \in R^\beta_\varphi”\end{align*} $$

and so $k(f): [\beta ]^1 \to V_\beta $ is a bijection. Therefore, for every $x\in V_\beta $ there exists $\gamma <\beta $ such that $k(f)(\{ \gamma \})=x$ .

Thus, letting $D:=\{ [\{\gamma \}, [f]]:\gamma <\beta \}$ , we have just shown that the map $i:\langle D, \in _{\mathcal E} \restriction D\rangle \to \langle V_\beta , \in \rangle $ given by

$$ \begin{align*}i([\{ \gamma\},[f]])=k(f)(\{\gamma\})\end{align*} $$

is onto. Moreover, if $[\{ \gamma \},[f]]\in _{\mathcal E} [\{ \delta \},[f]]$ , then for some $X\in E_{\{\gamma ,\delta \} }$ , we have

$$ \begin{align*}(f\circ \pi_{\{ \gamma,\delta\}\{\gamma\}})(s)\in (f\circ \pi_{\{\gamma ,\delta\}\{\delta\}})(s)\end{align*} $$

for every $s\in X$ . Letting $\varphi $ be the bounded formula expressing this, since $V_{\alpha }$ is closed under f for every $\alpha $ , we have

$$ \begin{align*}M\models “\langle X\cap V_{\alpha}, (f\circ \pi_{\{ \gamma,\delta\}\{\gamma\}})\cap V_{\alpha}, (f\circ \pi_{\{\gamma ,\delta\}\{\delta\}})\cap V_{\alpha}\rangle_{\alpha \in S} \, \overline{\in}\,\, \overline{R}_\varphi .”\, \end{align*} $$

Hence, in ${\mathcal A}_{\beta }$ , for every $s\in k(X)$ ,

$$ \begin{align*}(k(f)\circ \pi_{\{ \gamma,\delta\}\{\gamma\}})(s)\in (k(f)\circ \pi_{\{\gamma ,\delta\}\{\delta\}})(s)\, .\end{align*} $$

In particular, since $\{\gamma ,\delta \}\in k(X)$ ,

$$ \begin{align*}k(f)(\{\gamma\})\in k(f)(\{\delta\})\, .\end{align*} $$

A similar argument shows that i is one-to-one. Hence, i is an isomorphism, and so i is just the transitive collapsing map. Since $D\subseteq M^\ast _{\mathcal E}$ , to conclude that $V_\beta \subseteq M^\ast $ it will be sufficient to show that the transitive collapse of D is the same as the restriction to D of the transitive collapse of $M^*_{\mathcal E}$ . For this, it suffices to see that every $\in _{\mathcal E}$ -element of an element of D is $=_{\mathcal E}$ -equal to an element of D. So, suppose $[\{ \gamma \}, [f]]\in D$ and $[a,[g]] \in _{\mathcal E} [\{\gamma \}, [f]]$ , with $[a,[g]] \in M^*_{\mathcal E}$ . Then $k(g)(a)\in k(f)({\gamma })$ , by (1) above (just before Claim 4.8). Now $k(f):[\beta ]^1\to V_\beta $ is surjective and $V_\beta $ is transitive, so there is some $\delta < \beta $ such that $k(f)(\{\delta \})=k(g)(a)$ . Hence, by (2) above, $[\{\delta \},[f]] =_{\mathcal E} [a,[g]]$ .

Claim 4.10 $M_{\mathcal E}$ is closed under $\omega $ -sequences, and hence it is well-founded.

Proof of claim Let $\langle j_{\mathcal E} (f_n)(k_{a_n}([\mathrm {id}_{|a_n|}]_{E_{a_n}}))\rangle _{n<\omega }$ be a sequence of elements of $M_{\mathcal E}$ . On the one hand, the sequence $\langle j_{\mathcal E} (f_n)\rangle _{n<\omega } =j_{\mathcal E}(\langle f_n\rangle _{n<\omega })$ belongs to $M_{\mathcal E}$ . On the other hand, $k_{a_n}([\mathrm {Id}_{|a_n|}]_{E_{a_n}})=[a_n, [\mathrm {Id}_{|a_n|}]_{E_{a_n}}]$ belongs to $M^\ast _{\mathcal E}$ for all $n<\omega $ . Since ${\mathcal E}$ is normal (Claim 4.6), as in [9, Lemma 26.2(a)] we can show that the transitive collapse of $[a_n, [\mathrm {Id}_{|a_n|}]_{E_{a_n}}]$ is precisely $a_n$ . The sequence $\langle a_n\rangle _{n<\omega }$ belongs to $V_\beta $ , because $\beta $ has uncountable cofinality. Hence, since $V_\beta \subseteq M^\ast $ , the preimage of $\langle a_n\rangle _{n<\omega }\in V_\beta $ under the transitive collapsing map of $M^\ast _{\mathcal E}$ to $M^\ast $ is precisely the sequence $\langle k_{a_n}([\mathrm {Id}_{|a_n|}]_{E_{a_n}})\rangle _{n<\omega }$ and belongs to $M_{\mathcal E}$ . It now follows that the sequence $\langle j_{\mathcal E} (f_n)(k_{a_n}([\mathrm {id}_{|a_n|}]_{E_{a_n}}))\rangle _{n<\omega }$ is also in $M_{\mathcal E}$ .

Let $\pi :M_{\mathcal E} \to N$ be the transitive collapsing isomorphism, and let $j_N:V\to N$ be the corresponding elementary embedding, i.e., $j_N=\pi \circ j_{\mathcal E}$ .

Claim 4.11 $j_N(\kappa )\geq \beta $ .

Proof of claim Let $\alpha <\beta $ . Let $\mathrm {Id}_1$ be the identity function on $[\kappa ]^1$ , and let $c_{[\kappa ]^1}:[\kappa ]^1 \to \{ [\kappa ]^1\}$ and $c_\kappa :[\kappa ]^1\to \{ \kappa \}$ . In $M_{\{ \alpha \}}$ , we have

$$ \begin{align*}[\mathrm{Id}_1]_{E_{\{ \alpha \}}}\in [c_{[\kappa]^1}]_{E_{\{ \alpha \}}}=[[c_\kappa]_{E_{\{ \alpha \}}}]^1=[j_{\{ \alpha \}}(\kappa)]^1,\end{align*} $$

and hence in $M_{\mathcal E}$ ,

$$ \begin{align*}k_{\{ \alpha \}}([\mathrm{Id}_1]_{E_{\{ \alpha \}}})\in k_{\{ \alpha \}}([j_{\{ \alpha \}}(\kappa)]^1) =[j_{\mathcal E} (\kappa)]^1,\end{align*} $$

and therefore, since $\pi (k_{\{ \alpha \}}([\mathrm {Id}_1]_{E_{\{ \alpha \}}}))=\{ \alpha \}$ , in N we have

$$ \begin{align*}\{ \alpha \} \in [j_N(\kappa)]^1,\end{align*} $$

that is, $\alpha <j_N(\kappa )$ .

Since $\beta>\kappa $ , the last claim implies that the critical point of $j_N$ is less than or equal to $\kappa $ . Since $V_\beta \subseteq N$ by Claim 4.9, $j_N$ witnesses that its critical point is a $\beta $ -strong cardinal, in contradiction to our choice of $\beta $ . This completes the proof of Theorem 4.1.

The boldface version of Theorem 4.1, i.e., with parameters, also holds by essentially the same arguments. Namely,

Theorem 4.12. The following are equivalent:

1. (1) There exists a proper class of strong cardinals.

2. (2) $\mathbf {\Sigma _2}$ - $\mathrm {PRP}$ .

3. (3) $\mathbf {\Pi _1}$ - $\mathrm {PRP}$ .

4. (4) $\mathbf {\Sigma _2}$ - $\mathrm {SWVP}$ .

5. (5) $\mathbf {\Pi _1}$ - $\mathrm {SWVP}$ .

6. (6) $\mathbf {\Sigma _2}$ - $\mathrm {WVP}$ .

7. (7) $\mathbf {\Pi _1}$ - $\mathrm {WVP}$ .

For the proof of (3) implies (1), in order to show that there exists a strong cardinal greater than or equal to a fixed ordinal $\gamma $ we need to consider the class of structures

$$ \begin{align*}{\mathcal A}_\alpha:=\langle V_{ \alpha+1}, \in , \alpha ,\{ R^{\alpha}_\varphi \}_{\varphi \in \Pi_1}, \langle \delta \rangle_{\delta <\gamma} \rangle,\end{align*} $$

where the structure

$$ \begin{align*}\langle V_{ \alpha+1}, \in , \alpha ,\{ R^{\alpha}_\varphi \}_{\varphi \in \Pi_1} \rangle\end{align*} $$

is as in the proof of Theorem 4.1, and we have a constant $\delta $ for every $\delta <\gamma $ .

5 The general case

We shall now consider the general case of definable proper classes of structures with any degree of definable complexity. For this we shall need the following new kind of large cardinals.

If $j:V\to M$ is an elementary embedding, with M transitive and critical point $\kappa $ , and A is a class definable by a formula $\varphi $ (possibly with parameters in $V_\kappa $ ), we define

$$ \begin{align*}j(A):=\{ X\in M: M\models \varphi(X)\}\, .\end{align*} $$

Note that

$$ \begin{align*}j(A)= \bigcup \{ j(A\cap V_\alpha): \alpha \in \mathrm{OR}\}\end{align*} $$

as $j(A\cap V_\alpha )=\{ X \in M: M\models \varphi (X)\}\cap V^M_{j(\alpha )}$ . Also note that if A is a class of structures of the same type $\tau \in V_\kappa $ , then by elementarity $j(A)$ is a subclass of M of structures of type $\tau $ .

5.1 $\Gamma _n$ -strong cardinals

In the sequel, let $\Gamma _n$ stand for one of the definability classes $\Sigma _n$ , $\Pi _n$ .

Definition 5.1. For $n\geq 1$ , a cardinal $\kappa $ is $\lambda $ - $\Gamma _n$ -strong if for every $\Gamma _n$ -definable (without parameters) class A there is an elementary embedding $j:V\to M$ with M transitive, $\mathrm {crit}(j)=\kappa $ , $j(\kappa )>\lambda $ , $V_\lambda \subseteq M$ , and $A\cap V_\lambda \subseteq j(A)$ .

$\kappa $ is $\Gamma _n$ -strong if it is $\lambda $ - $\Gamma _n$ -strong for every ordinal $\lambda $ .

Note that in the definition above, $A \cap V_\lambda $ is only required to be contained in $j(A) \cap V_\lambda $ and not equal to it. The reason is that in the $\Sigma _2$ case, if A is the class of non-strong cardinals (which is $\Sigma _2$ ) and $\kappa $ is the least strong cardinal, then $\kappa \notin A$ , but $\kappa \in j(A)$ . See however the equivalence given in Proposition 5.9.

As with the case of strong cardinals, standard arguments show (cf. [9, Exercise 26.7(b)]) that $\kappa $ is $\Gamma _n$ -strong if and only if for every $\Gamma _n$ -definable (without parameters) class A and every ordinal $\lambda $ there is an elementary embedding $j:V\to M$ with M transitive, $\mathrm {crit}(j)=\kappa $ , $V_\lambda \subseteq M$ , and $A\cap V_\lambda \subseteq j(A)$ .

Proposition 5.2. Every strong cardinal is $\Sigma _2$ -strong.

Proof Let $\kappa $ be a strong cardinal and let A be a class that is $\Sigma _2$ -definable (even allowing for parameters in $V_\kappa $ ). Let $\lambda \in C^{(2)}$ be greater than $\kappa $ . Let $j:V\to M$ be elementary, with M transitive, $\mathrm {crit}(j)=\kappa $ , and $V_\lambda \subseteq M$ . Let $\varphi $ be a $\Sigma _2$ formula defining A. If $a\in A\cap V_\lambda $ , then $V_\lambda \models \varphi (a)$ . Hence, since $V_\lambda \preceq _{\Sigma _1}M$ , $M\models \varphi (a)$ , and so $a\in j(A)=\{ x: M\models \varphi (x)\}$ .

Proposition 5.3. If $\lambda \in C^{(n+1)}$ , then a cardinal $\kappa $ is $\lambda $ - $\Pi _n$ -strong if and only if it is $\lambda $ - $\Sigma _{n+1}$ -strong.

Proof Assume $\kappa $ is $\lambda $ - $\Pi _n$ -strong, with $\lambda \in C^{(n+1)}$ , and let A be a $\Sigma _{n+1}$ -definable class. Let $\varphi (x)\equiv \exists y\, \psi (x,y)$ be a $\Sigma _{n+1}$ formula, with $\psi (x,y)$ being $\Pi _n$ , that defines A. Now define B as the class of all structures of the form $\langle V_\alpha , \in , a\rangle $ , where $\alpha \in C^{(n)}$ , $a\in V_\alpha $ , and $V_\alpha \models \varphi (a)$ . Then B is $\Pi _n$ -definable. By our assumption, let $j:V\to M$ be an elementary embedding with M transitive, $\mathrm {crit} (j)=\kappa $ , $V_\lambda \subseteq M$ , and $B\cap V_\lambda \subseteq j(B)$ . We just need to show that $A\cap V_\lambda \subseteq j(A)$ . So, suppose $a\in A\cap V_\lambda $ . Since $\lambda \in C^{(n+1)}$ , we have that $V_\lambda \models \varphi (a)$ . Let $b\in V_\lambda $ be a witness, so that $V_\lambda \models \psi (a,b)$ . For some $\alpha <\lambda $ in $C^{(n)}$ we have that $a, b\in V_\alpha $ . Hence, $V_\alpha \models \varphi (a)$ . So $\langle V_\alpha , \in ,a\rangle \in B\cap V_\lambda $ , and therefore $\langle V_\alpha ,\in , a\rangle \in j(B)$ . Thus, $M\models “\alpha \in C^{(n)}, \, a\in V_\alpha , \mbox { and }V_\alpha \models \varphi (a).”$ Hence, $M\models \varphi (a)$ , i.e., $a\in j(A)$ .

Corollary 5.4. A cardinal $\kappa $ is $\Pi _n$ -strong if and only if it is $\Sigma _{n+1}$ -strong.

Proposition 5.5. Suppose that $n\geq 2$ and $\lambda \in C^{(n)}$ . Then the following are equivalent for a cardinal $\kappa <\lambda $ :

1. (1) $\kappa $ is $\lambda $ - $\Sigma _{n}$ -strong.

2. (2) There is an elementary embedding $j:V\to M$ with M transitive, $\mathrm {crit}(j)=\kappa $ , $V_\lambda \subseteq M$ , $j(\kappa )>\lambda $ , and $M\models “\lambda \in C^{(n-1)}.”$

Proof (1) $\Rightarrow $ (2): Suppose $\kappa $ is $\lambda $ - $\Sigma _n$ -strong. Let $A=C^{(n-1)}$ . Since A is $\Pi _{n-1}$ -definable, hence also $\Sigma _n$ -definable, by (1) there is an elementary embedding $j:V\to M$ with M transitive, $\mathrm {crit} (j)=\kappa $ , $j(\kappa )>\lambda $ , $V_\lambda \subseteq M$ , and $A\cap V_\lambda \subseteq j(A)$ . Since $\lambda \in C^{(n)}$ , $C^{(n-1)}\cap \lambda $ is a club subset of $\lambda $ . For every $\alpha <\lambda $ in $C^{(n-1)}$ , $\alpha \in j(A)$ , and hence $M\models “\alpha \in C^{(n-1)}”$ and so $M\models “\lambda \mbox { is a limit point of }C^{(n-1)},\!”$ which yields $M\models “\lambda \in C^{(n-1)}.”$

(2) $\Rightarrow $ (1): Let A be a class definable by a $\Sigma _n$ formula $\varphi $ , and let $j:V\to M$ be an elementary embedding with M transitive, $\mathrm {crit}(j)=\kappa $ , $j(\kappa )>\lambda $ , $V_{\lambda } \subseteq M$ , and $M\models “\lambda \in C^{(n-1)}.$ ” Let $a\in A\cap V_{\lambda }$ . Since $\lambda \in C^{(n)}$ , $V_{\lambda } \models \varphi (a)$ . And since $V_{\lambda } \subseteq M$ and $M\models “\lambda \in C^{(n-1)},$ ” $M\models \varphi (a)$ , i.e., $a\in j(A)$ .

Proposition 5.6. If $\kappa $ is $\Pi _n$ -strong, then $\kappa \in C^{(n+1)}$ .

Proof By induction on n. So let $\kappa $ be $\Pi _n$ -strong and assume, inductively, that $\kappa \in C^{(n)}$ . Let $\exists x \varphi (x)$ be a formula, with $\varphi (x)$ being a $\Pi _n$ formula which may contain parameters in $V_\kappa $ , and suppose that $\exists x\varphi (x)$ holds in V. Pick a witness b and let $\lambda \in C^{(n)}$ be such that $b\in V_\lambda $ . Thus, $V_\lambda \models \exists x \varphi (x)$ . By Corollary 5.4 and Proposition 5.5, let $j:V\to M$ be an elementary embedding, with M transitive, $\mathrm {crit}(j)=\kappa $ , $V_\lambda \subseteq M$ , $j(\kappa )>\lambda $ , and $M\models “\lambda \in C^{(n)}.”$ Then by elementarity of j, in V there exists some $\lambda '\in C^{(n)}$ less than $\kappa $ such that $V_{\lambda '}\models “\exists x\varphi (x),\!”$ and since $\kappa \in C^{(n)}$ , $V_\kappa \models “\exists x \varphi (x),\!”$ as wanted.

The last proposition suggests the following definition and the ensuing characterization of $\Sigma _n$ -strong cardinals in terms of extenders.

Definition 5.7. Given $n\geq 1$ and given cardinals $\kappa <\lambda $ , a $\Sigma _n$ -strong $(\kappa ,\lambda )$ -extender is a $(\kappa ,|V_{\lambda }|^+)$ -extender ${\mathcal E}$ (see Definition 2.3) such that $V_{\lambda }\subseteq \overline {M}_{\mathcal {E}}$ , $\lambda <j(\kappa )$ , and $\overline {M}_{\mathcal E}\models “\lambda \in C^{(n-1)},\!”$ where $\overline {M}_{\mathcal E}$ is the transitive collapse of the direct limit ultrapower $M_{\mathcal E}$ of V by ${\mathcal E}$ , and $j:V\to \overline {M}_{\mathcal E}$ is the corresponding elementary embedding.

Proposition 5.8. If $n\geq 2$ and $\lambda \in C^{(n)}$ , then a cardinal $\kappa <\lambda $ is $\lambda $ - $\Sigma _n$ -strong if and only if there exists a $\Sigma _n$ -strong $(\kappa , \lambda )$ -extender.

Proof If ${\mathcal E}$ is a $\Sigma _n$ -strong $(\kappa ,\lambda )$ -extender, then the extender embedding $j:V\to \overline {M}_{\mathcal E}$ witnesses that $\kappa $ is $\lambda $ - $\Sigma _n$ -strong by Proposition 5.5.

Conversely, suppose $j:V\to M$ is an elementary embedding, with M transitive, $\mathrm {crit}(j)=\kappa $ , $j(\kappa )>\lambda $ , $V_\lambda \subseteq M$ , and $M\models “\lambda \in C^{(n-1)}.”$ Note that since $\lambda \in C^{(1)}$ , $|V_\lambda |=\lambda $ . Let ${\mathcal E}$ be the $(\kappa , \lambda ^+)$ -extender derived from j. Namely, for every $a\in [\lambda ^+]^{<\omega }$ let $E_a$ be defined by

$$ \begin{align*}X\in E_a \mbox{ if and only if } X\subseteq [\kappa]^{|a|} \mbox{ and }a \in j(X).\end{align*} $$

One can easily check that ${\mathcal E}$ satisfies conditions $(1)$ – $(5)$ of Definition 2.3 (see [9, Exercise 26.7]). So we only need to check that $\overline {M}_{\mathcal E}\models “\lambda \in C^{(n-1)}.”$

Let $j_{\mathcal E}:V\to M_{\mathcal E}$ and $k_{\mathcal E}:\overline {M}_{\mathcal E}\to M$ be the standard maps given by: $j_{\mathcal E}(x)=[a, [c^a_x]]$ (any a), where $c^a_x :[\kappa ]^{|a|} \to \{ x\}$ ; and $k_{\mathcal E}(\pi ([a,[f]]))=j(f)(a)$ , for $f:[\kappa ]^{|a|}\to V$ , where $\pi :M_{\mathcal E}\to \overline {M}_{\mathcal E}$ is the transitive collapse isomorphism. The maps $j_{\mathcal E}$ and $k_{\mathcal E}$ are elementary and $j=k_{\mathcal E} \circ \pi \circ j_{\mathcal E}$ . Moreover, $k_{\mathcal E}\restriction V_\lambda $ is the identity.

Since $M\models “\lambda \in C^{(n-1)},\!”$ for each $\mu <\lambda $ in $C^{(n-1)}$ , we have that $M\models “\mu \in C^{(n-1)}.”$ So, since $k_{\mathcal E}$ is elementary and is the identity on $V_\lambda $ , we have that $M_{\mathcal E}\models “\mu \in C^{(n-1)}.”$ Hence, $M_{\mathcal E}\models “\lambda \mbox { is a limit point of }C^{(n-1)},\!”$ which yields $M_{\mathcal E}\models “\lambda \in C^{(n-1)}.”$

Similar characterizations may also be given for $\Pi _n$ -strong cardinals. Namely, if $n\geq 2$ and $\lambda \in C^{(n)}$ , then a cardinal $\kappa <\lambda $ is $\lambda $ - $\Pi _n$ -strong if and only if there exists a $\Pi _n$ -strong $(\kappa , \lambda )$ -extender. Notice that (3) of the following proposition characterizes $\Pi _n$ -strong cardinals as witnessing “ $\mathrm {OR}$ is Woodin” restricted to $\Pi _n$ -definable classes (see Definition 5.14). In particular, it says $A\cap V_\lambda = j(A)\cap V_\lambda $ , not just $A\cap V_\lambda \subseteq j(A)\cap V_\lambda $ .

Proposition 5.9. Suppose that $n\geq 1$ and $\lambda $ is a limit point of $C^{(n)}$ . 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, $\mathrm {crit}(j)=\kappa $ , $j(\kappa )=>\lambda $ , $V_\lambda \subseteq M$ , and $M\models “\lambda \in C^{(n)}.”$

3. (3) For every $\Pi _n$ -definable class A there is an elementary embedding $j:V\to M$ with M transitive, $\mathrm {crit} (j)=\kappa $ , $j(\kappa )>\lambda $ , $V_\lambda \subseteq M$ , and $A\cap V_\lambda = j(A)\cap V_\lambda $ .

Proof (1) $\Rightarrow $ (2): Suppose $\kappa $ is $\lambda $ - $\Pi _n$ -strong. Let $A=C^{(n)}$ . Since A is $\Pi _n$ -definable, by (1) there is an elementary embedding $j:V\to M$ with M transitive, $\mathrm {crit} (j)=\kappa $ , $j(\kappa )>\lambda $ , $V_\lambda \subseteq M$ , and $A\cap V_\lambda \subseteq j(A)$ . Thus, for every $\alpha <\lambda $ in A, $\alpha \in j(A)$ , and hence $M\models “\alpha \in C^{(n)}”$ and so $M\models “\lambda \mbox { is a limit point of }C^{(n)},\!”$ which yields $M\models “\lambda \in C^{(n)}.”$

(2) $\Rightarrow $ (3): Let A be a class definable by a $\Pi _n$ formula $\varphi (x)$ , and let $j:V\to M$ be an elementary embedding with M transitive, $\mathrm {crit}(j)=\kappa $ , $j(\kappa )>\lambda $ , $V_\lambda \subseteq M$ , and $M\models “\lambda \in C^{(n)}.”$ Let $a\in A\cap V_\lambda $ . Since $\lambda \in C^{(n)}$ , $V_\lambda \models \varphi (a)$ . And since $V_\lambda \subseteq M$ and $M\models “\lambda \in C^{(n)},\!” M\models \varphi (a)$ , i.e., $a\in j(A)$ . Conversely, suppose $a\in j(A)\cap V_\lambda $ , i.e., $M\models “\varphi (a).”$ Since $M\models “\lambda \in C^{(n)},\!” V_\lambda \models \varphi (a)$ . And since $\lambda \in C^{(n)}$ , $a\in A$ .

(3) $\Rightarrow $ (1) is immediate.

Let us remark that the implication (2) $\Rightarrow $ (3) above also holds for classes A that are $\Pi _n$ -definable with parameters in $V_\kappa $ . Thus a corollary of Proposition 5.9 is that a cardinal $\kappa $ is $\Pi _n$ -strong if and only if it is $\lambda $ - $\Pi _n$ -strong for every $\Pi _n$ -definable, with parameters in $V_\kappa $ , class A.

It easily follows from the last proposition that being a $\Pi _n$ -strong cardinal is a $\Pi _{n+1}$ property. Moreover, if $\kappa $ is $\Pi _n$ -strong, then $\kappa \in C^{(n+1)}$ . Hence, if $\kappa $ is $\Pi _{n+1}$ -strong, then there are many $\Pi _n$ -strong cardinals below $\kappa $ , which shows that the $\Pi _n$ -strong cardinals, $n>0$ , form a hierarchy of strictly increasing strength.

Similarly as in Proposition 3.3 we can prove the following.

Proposition 5.10. If $\kappa $ is a $\Sigma _n$ -strong cardinal, where $n\geq 2$ , then $\Sigma _n(V_\kappa )$ - $\mathrm {PRP}$ holds.

Proof Let $n\geq 2$ . Let $\kappa $ be $\Sigma _n$ -strong and let ${\mathcal C}$ be a definable, by a $\Sigma _n$ formula with parameters in $V_\kappa $ , proper class of structures in a fixed countable relational language. We will show that the set $S:={\mathcal C}\cap V_\kappa $ witnesses PRP.

Given any ${\mathcal A}\in {\mathcal C}$ , let $\lambda \geq \kappa $ be an ordinal with ${\mathcal A}\in V_\lambda $ .

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

By elementarity, the restriction of j to $\prod S$ yields a homomorphism

$$ \begin{align*}h:\prod S\to \prod (j({\mathcal C}) \cap V^M_{j(\kappa)}).\end{align*} $$

Since ${\mathcal A}\in {\mathcal C} \cap V_\lambda $ , we have that ${\mathcal A} \in j({\mathcal C})$ . Moreover ${\mathcal A} \in V_\lambda \subseteq V^M_{j(\kappa )}$ . Thus, letting

$$ \begin{align*}g: \prod (j({\mathcal C}) \cap V^M_{j(\kappa)})\to {\mathcal A}\end{align*} $$

be the projection map, we have that

$$ \begin{align*}g\circ h:\prod S\to {\mathcal A}\end{align*} $$

is a homomorphism, as wanted.

5.2 The main theorem for $\Gamma _n$ -strong cardinals

Using similar arguments as in Theorem 4.1, we can now prove the main theorem of this section.

Theorem 5.11. The following are equivalent for $n\geq 2$ :

1. (1) There exists a $\Sigma _n$ -strong cardinal.

2. (2) There exists a $\Pi _{n-1}$ -strong cardinal.

3. (3) $\Sigma _n$ - $\mathrm {PRP}$ .

4. (4) $\Pi _{n-1}$ - $\mathrm {PRP}$ .

5. (5) $\Sigma _n$ - $\mathrm {SWVP}$ .

6. (6) $\Pi _{n-1}$ - $\mathrm {SWVP}$ .

7. (7) $\Sigma _n$ - $\mathrm {WVP}$ .

8. (8) $\Pi _{n-1}$ - $\mathrm {WVP}$ .

Proof (1) $\Rightarrow $ (3) is given by Proposition 5.10. The equivalence of (3)–(8) is given by Theorem 3.7. So, we only need to prove (4) $\Rightarrow $ (2).

The proof is analogous to the proof of Theorem 4.1, so we shall only indicate the relevant differences. Theorem 4.1 proves the case $n=2$ (see Proposition 5.2). Thus, we shall assume in the sequel that $n> 2$ .

Let $\mathcal {A}$ be the class of all structures

$$ \begin{align*}{\mathcal A}_\alpha:=\langle V_{\alpha +1}, \in , \alpha , C^{(n-1)}\cap\alpha , \{ R^{\alpha}_\varphi \}_{\varphi \in \Pi_1} \rangle,\end{align*} $$

where the constant $\alpha $ is the $\alpha $ -th element of $C^{(n-1)}$ , and $\{ R^{\alpha }_\varphi \}_{\varphi \in \Pi _1}$ is the $\Pi _1$ relational diagram for $V_{\alpha +1}$ , i.e., if $\varphi (x_1,\ldots ,x_n)$ is a $\Pi _1$ formula in the language of ${\langle V_{\alpha +1},\in ,\alpha , C^{(n-1)}\cap \alpha \rangle }$ , then

$$ \begin{align*}R^\alpha_\varphi =\{ \langle x_1,\ldots ,x_n\rangle: \langle V_{\alpha +1},\in,\alpha, C^{(n-1)}\cap \alpha \rangle\models “\varphi(x_1,\ldots ,x_n)”\}\, .\end{align*} $$

Then $\mathcal {A}$ is $\Pi _{n-1}$ -definable without parameters. For $X\in \mathcal {A}$ if and only if ${X=\langle X_0,X_1,X_2,X_3, X_4 \rangle }$ , where:

1. (1) $X_2$ belongs to $C^{(n-1)}$ .

2. (2) $X_0=V_{X_2 +1}$ .

3. (3) $X_1 = \mathord {\in } \restriction X_0$ .

4. (4) $X_0$ satisfies that $X_3=C^{(n-1)}\cap X_2$ .

5. (5) $X_4$ is the $\Pi _1$ relational diagram of $\langle X_0,X_1,X_2,X_3 \rangle $ .

6. (6) $\langle X_0, X_1, X_2, X_3 \rangle \models “X_2$ is the $X_2$ -th element of $C^{(n-1)}$ .”

Note that the class C of ordinals $\alpha $ such that ${\mathcal A}_\alpha \in {\mathcal A}$ is a closed and unbounded proper class. By $\Pi _{n-1}$ - $\mathrm {PRP}$ there exists a subset S of C such that for every $\beta \in C$ there is a homomorphism $j_\beta : \prod _{\alpha \in S} {\mathcal A}_\alpha \to {\mathcal A}_\beta $ . By enlarging S, if necessary, we may assume that $\kappa := \mathrm {sup(S)\in S}$ .

Now fix some $\beta \in C$ greater than $\kappa $ , of uncountable cofinality, and assume, towards a contradiction, that no cardinal $\leq\!\kappa $ is $\beta $ - $\Pi _{n-1}$ -strong. Let $j=j_\beta $ .

From this point, the proof proceeds as in Theorem 4.1. Namely, we define ${k:V_{\kappa +1}\to V_{\beta +1}}$ by

$$ \begin{align*}k(X)=j(\langle X\cap V_{\alpha}\rangle_{\alpha \in S})\end{align*} $$

and note that $k(\kappa ) =\beta $ .

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

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

As in Theorem 4.1, $E_a$ is an $\omega _1$ -complete ultrafilter over $[\kappa ]^{|a|}$ . Moreover, ${\mathcal E} :=\{ E_a: a\in [\beta ]^{<\omega }\}$ is normal and coherent.

For each $a\in [\beta ]^{<\omega }$ , the ultrapower ${\mathrm {Ult}}(V,E_a)$ is well-founded by $\omega _1$ -completeness. So, let

$$ \begin{align*}j_a:V\to M_a \cong {\mathrm{Ult}}(V, E_a),\end{align*} $$

with $M_a$ transitive, be the corresponding ultrapower embedding, and let $M_{\mathcal E}$ be the direct limit of

$$ \begin{align*}\langle \langle M_a:a\in [\beta]^{<\omega}\rangle, \langle i_{ab}:a\subseteq b\rangle\rangle,\end{align*} $$

where the $i_{ab}:M_a\to M_b$ are the usual commuting maps. The corresponding limit embedding $j_{\mathcal E}:V\to M_{\mathcal E}$ is elementary. As in Theorem 4.1, $M_{\mathcal E}$ is closed under $\omega $ -sequences, and hence it is well-founded. Moreover, letting $\pi :M_{\mathcal E} \to N$ be the transitive collapsing isomorphism, and $j_N:V\to N$ the corresponding elementary embedding, i.e., $j_N=\pi \circ j_{\mathcal E}$ , we have that $V_\beta \subseteq N$ and $j_N(\kappa )\geq \beta $ . Since $\beta>\kappa $ , this implies that the critical point of $j_N$ is less than or equal to $\kappa $ . The only additional argument needed, with respect to the proof of Theorem 4.1, is the following:

Claim 5.12 $N\models “\beta \in C^{(n-1)}.”$

Proof Since $\beta $ is a limit point of $C^{(n-1)}$ , it suffices to show that if $\gamma <\beta $ belongs to $C^{(n-1)}$ , then $N\models “\gamma \in C^{(n-1)}.”$ So, fix some $\gamma <\beta $ in $C^{(n-1)}$ .

Let $f :[\kappa ]^1\to \kappa $ be such that $f (\{ x\})=x$ . It is well known that $k_{\{ \gamma \}}( [f]_{E_{\{\gamma \}}})=\gamma $ , where $k_{\{ \gamma \}}:M_{\{\gamma \}}\to N$ is the standard map given by $k_{\{ \gamma \}}([f]_{E_{\{\gamma \}}})=\pi ([\{ \gamma \}, [f]_{E_{\{\gamma \}}}])$ (see [9, Lemma 26.2(a)]).

Let $X:=\{ \{ x\}\in [\kappa ]^1: x\in C^{(n-1)}\}$ . Note that, since being a singleton is a property expressible by a bounded formula, and $C^{(n-1)}\cap \alpha $ is a predicate in the language of every structure ${\mathcal A}_{\alpha }$ , the homomorphism k maps X to the set $\{\{ x\}\in [\beta ]^1: x\in C^{(n-1)}\}$ . Thus, $\{ \gamma \} \in k(X)$ , and therefore $X\in E_{\{\gamma \}}$ . Hence, $M_{\{ \gamma \}}\models “[f]\in C^{(n-1)},\!”$ and therefore $M_{\mathcal E} \models “[\{ \gamma \}, [f]]\in C^{(n-1)},\!”$ which yields $N\models “\gamma \in C^{(n-1)},\!”$ as wanted.

Thus, by Proposition 5.9, $j_N$ witnesses that the critical point of $j_N$ is less than or equal to $\kappa $ and is $\beta $ - $\Pi _{n-1}$ -strong, in contradiction to our choice of $\beta $ .

In a similar way we may obtain the following parameterized version of Theorem 5.11. For the proof of (4) implies (2), we need to consider the class of structures

$$ \begin{align*}{\mathcal A}_\alpha:=\langle V_{\alpha +1}, \in , \alpha , C^{(n-1)}\cap\alpha , \{ R^{\alpha}_\varphi \}_{\varphi \in \Pi_1} , \langle \delta \rangle_{\delta <\gamma} \rangle,\end{align*} $$

where the structure

$$ \begin{align*}\langle V_{\alpha +1}, \in , \alpha , C^{(n-1)}\cap\alpha , \{ R^{\alpha}_\varphi \}_{\varphi \in \Pi_1}\rangle\end{align*} $$

is as in the proof of Theorem 5.11, and we have a constant $\delta $ for every $\delta <\gamma $ .

Theorem 5.13. The following are equivalent for $n\geq 2$ :

1. (1) There exists a proper class of $\ \Sigma _n$ -strong cardinals.

2. (2) There exists a proper class of $\ \Pi _{n-1}$ -strong cardinals.

3. (3) $\mathbf {\Sigma _n}$ - $\mathrm {PRP}$ .

4. (4) $\mathbf {\Pi _{n-1}}$ - $\mathrm {PRP}$ .

5. (5) $\mathbf {\Sigma _n}$ - $\mathrm {SWVP}$ .

6. (6) $\mathbf {\Pi _{n-1}}$ - $\mathrm {SWVP}$ .

7. (7) $\mathbf {\Sigma _n}$ - $\mathrm {WVP}$ .

8. (8) $\mathbf {\Pi _{n-1}}$ - $\mathrm {WVP}$ .

Recall that a cardinal $\kappa $ is Woodin if for every $A\subseteq V_\kappa $ there is $\alpha <\kappa $ such that $\alpha $ is $<\!\kappa $ -A-strong, i.e., for every $\gamma <\kappa $ there is an elementary embedding $j:V\to M$ with $\mathrm {crit} (j)=\alpha $ , $\gamma <j(\alpha )$ , $V_\gamma \subseteq M$ , and $A\cap V_\gamma =j(A)\cap V_\gamma $ . (See [9, Theorem 26.14].)

Definition 5.14. $\mathrm {OR}$ is Woodin if for every definable (with set parameters) class A there exists some $\alpha $ which is A-strong, i.e., for every $\gamma $ there is an elementary embedding $j:V\to M$ with $\mathrm {crit} (j)=\alpha $ , $\gamma <j(\alpha )$ , $V_\gamma \subseteq M$ , and $A\cap V_\gamma =j(A)\cap V_\gamma $ .

The statement “ $\mathrm {OR}$ is Woodin” is first-order expressible as a schema, namely as “There exists $\alpha $ which is A-strong,” for each definable, with parameters, class A. Or equivalently, by Proposition 5.9 and the remark that follows it, as the schema “There exists $\alpha $ which is $\Pi _n$ -strong,” for every n. Let us note that, by Theorem 5.13, “ $\mathrm {OR}$ is Woodin” is also equivalent to the schema asserting “There exist a proper class of $\alpha $ which are $\Pi _n$ -strong,” for every n. Thus, Theorem 5.13 yields the following corollary, first proved by the second author in [13] for arbitrary classes (not necessarily definable), which gives the exact large-cardinal strength of $\mathrm {WVP}$ and $\mathrm {SWVP}$ .

Corollary 5.15. The following are equivalent:

1. (1) $\mathrm {OR}$ is Woodin.

2. (2) $\mathrm {SWVP}$ .

3. (3) $\mathrm {WVP}$ .