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 $ [Reference Vopěnka, Pultr and Hedrlín11], 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 [Reference Adámek and Rosický2, 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 $ .

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.

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.

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.

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 [Reference Wilson12] 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 [Reference Wilson12] 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 $T^{\mathcal {M}}$ .⊣

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 $(6)\Rightarrow (1)$ .

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 $ .⊣

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

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

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 $ .

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 $[\kappa ]^{|a|}$ .

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

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 [Reference Kanamori9, 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) $[a,[f]] =_{\mathcal E} [b,[g]] \mbox { iff } k(f)(a) = k(g)(b)$ .

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 $M^\ast _{\mathcal E}$ .

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, $[\{\delta \},[f]] =_{\mathcal E} [a,[g]]$ .

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., $j_N=\pi \circ j_{\mathcal E}$ .

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.