The Weak Vop\V {E} Nka Principle for Definable Classes of Structures

$The Weak Vop\V {E} Nka Principle for Definable Classes of Structures$

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Thus, writing Σn-VP for the first- order assertion that every Σn-definable (with parameters), proper class of graphs is rigid, we have that VP is equivalent to Σn-VP, for every n. As shown in [Bag12, BCMR15], Σ1-VP is provable in ZFC, while Σ2-VP is equivalent to the existence of a proper class of supercompact cardinals, Σ3-VP is equivalent to the existence of a proper class of extendible cardinals, and Σn-VP is equivalent to the existence of a proper class of C(n−2)-extendible cardinals, for n ≥ 3. The level-by-level analysis of Σn-VP, n<ω, and the determination of their large-cardinal strength yielded new results in category theory, homology theory, homotopy theory, and universal algebra (see [BCMR15]). For example, the existence of cohomological localizations in the homotopy category of simplicial sets (Bousfield Conjecture) follows from Σ2-VP. The role of VP in category theory has a rich history. The first equivalences of VP with various category-theoretic statements were an- nounced by E. R. Fisher in [Fis77]. Further equivalences were proved over the next two decades by Adámek, Rosický, Trnková, and others. Their work showed that under VP “the structure of locally presentable categories becomes much more transparent” ([AR94] p. 241). For example, the statement that a category is locally presentable if and only if it is complete and bounded is equivalent to VP. And so is the statement that every orthogonality class in a locally presentable category is a small-orthogonality class ([RTA90], [AR94] 6.9, 6.14). Of the many category-theoretic statements now known to be equivalent to VP, the following one (see [AR94] 6.D) turned out to be of particular interest: (1) Every full subcategory of a locally presentable category K closed under colimits is coreflective in K. What made (1) particularly interesting is that its dual statement (2) Every full subcategory of a locally presentable category K closed under limits is reflective in K. while being a consequence of (1), could not be proved equivalent to it. Since VP – hence also (1) – was known to be equivalent to Ord cannot be fully embedded into Gra (see [AR94] 6.3), while statement (2) was proved equivalent to Ordop cannot be fully embedded into Gra (see [AR94] 6.22, 6.23), the latter assertion was then called The Weak Vopˇenka Principle (WVP). The term Weak was aptly given, for it is readily shown that VP implies WVP ([ART88]; Proposition 2.1 below). The question then remained if WVP implied VP. Using a result of J. R. Isbell [Isb60], which showed that Ordop is bounded iff there is no proper class of measurable cardinals, Adámek-Rosický[AR94] proved that WVP implies the existence of a proper class of measurable cardinals. This was seen as a first step in showing that WVP was indeed a THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 3 strong large-cardinal principle, perhaps even equivalent to 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 VP and WVP, called the Semi-Weak Vopˇenka Principle (SWVP), was introduced in [AR93] and the further question of the equivalence between the three principles, WVP, SWVP and VP, remained open. The problem was finally solved in 2019 by the second author of the present paper. In [Wil19] he showed that WVP and SWVP are equivalent, and they are also equivalent to the large-cardinal principle “OR is Woodin”, whose consistency strength is known to be well below the existence of a supercompact cardinal, thereby showing that WVP cannot imply VP (if consistent with ZFC). In the present paper we carry out a level-by-level analysis of WVP and SWVP similar to the analysis of VP done in [Bag12, BCMR15]. Thus, we prove the equivalence of both Σn-WVP and Σn-SWVP (see definition 2.2 below) with large-cardinals, for every n ≥ 2. In particular, we show that Σ2-WVP and Σ2-SWVP are equivalent to the existence of a proper class of strong cardinals. The main theorems (6.13, 6.15) show, more generally, that Σn-WVP and Σn-SWVP are equivalent to the existence of a proper class of Σn-strong cardinal (definition 6.1). Moreover, WVP and SWVP are equivalent to the existence of a Σn-strong cardinal, for every n<ω. Our arguments yield also a new proof of the second author’s result from [Wil19] that WVP implies “OR is Woodin” (corollary 6.17 below). The main difference between the two proofs is that while in the present paper we derive the extenders witnessing “OR is Woodin” from homomorphisms on products of relational structures with universe of the form Vα, the proof in [Wil19] uses homomorphisms of so-called P-structures. We think, however, that it should be possible to do a similar level-by-level analysis as done here by using P-structures instead. A number of consequences in category theory should follow from our results. For instance, the statement that every Σ2-definable full subcategory of a locally-presentable category K closed in K under limits is reflective in K, should be equivalent to the existence of a proper class of strong cardinals. See [AR94] Chapter 6 for more examples.

2. Preliminaries

Recall that a graph is a structure G = hG, EGi, where G is a non- empty set and EG is a binary relation on G. If G = hG, EGi and H = hH, EH i are graphs, a map h : G → H is a homomorphism if it preserves the binary relation, meaning that for every x, y ∈ G, if xEGy, then h(x)EH h(y). A class G of graphs is called rigid if there are no non-trivial homomorphisms between graphs in G, i.e., the only homomorphisms are the identity morphisms G → G, for G ∈ G. THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 4

The original formulation of Vopˇenka Principle (VP) (P. Vopˇenka, ca. 1960) asserts that there is no rigid proper class of graphs. As shown in [AR94] 6.A, VP is equivalent to the statement that the category Ord of ordinals cannot be fully embedded into the category Gra of graphs. That is, there is no sequence hGα : α ∈ ORi of graphs such that for every α ≤ β there exists exactly one homomorphism Gα → Gβ, and no homomorphism Gβ → Gα whenever α < β. The Weak Vopˇenka Principle (WVP) (ﬁrst introduced in [ART88]) is the statement dual to VP, namely that the opposite category of ordinals, Ordop, cannot be fully embedded into Gra. That is, there is no sequence hGα : α ∈ ORi of graphs such that for every α ≤ β there exists exactly one homomorphism Gβ → Gα, and no homomorphism Gα → Gβ whenever α < β. The Semi-Weak Vopˇenka Principle (SWVP) ([AR93]) asserts that there is no sequence hGα : α ∈ ORi of graphs such that for every α ≤ β there exists some (not necessarily unique) homomorphism Gβ → Gα, and no homomorphism Gα → Gβ whenever α < β. Clearly, SWVP implies WVP. The second author showed in [Wil20] that SWVP is in fact equivalent to WVP. As shown in [ART88], WVP is a consequence of VP, and the same argument also shows that VP implies SWVP. In fact, the argument shows the following:

Proposition 2.1. VP implies that for every sequence hGα : α ∈ ORi of graphs there exist α < β with a homomorphism Gα → Gβ.

Proof. Suppose hGα : α ∈ ORi is a sequence of graphs. Without loss of generality, if α < β, then Gα and Gβ 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 ⊆ OR such that |Gα| < |Gβ| whenever α < β are in C. For each α ∈ C, add to Gα = hGα, Eαi a rigid binary relation Sα ([VPH65]), as well as the non- identity relation =,6 and consider the structure Aα = hGα, Eα,Sα, =6 i, which can be easily seen as a graph. Since the cardinalities are strictly increasing, by the =6 relation there cannot be any homomorphism Aβ → Aα with α < β. Also, because of the rigid relation Sα, the identity is the only homomorphism Aα → Aα. Since by VP the class {Aα : α ∈ C} is not rigid, there must exist α < β with a homomorphism Aα → Aβ, hence a homomorphism Gα → Gβ.

The definitions of VP, WVP and 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 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 VP, WVP and SWVP, which require to restrict to definable classes. THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 5

2.1. The VP, WVP and SWVP for definable classes. Each of VP, WVP and 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 Σn(P )-Vopˇenka Principle (Σn(P )-VP for short) asserts that there is no Σn-definable, with parameters in P , sequence hGα : α ∈ ORi of graphs such that for every α ≤ β there exists exactly one homomorphism Gα → Gβ, and no homomorphism Gβ → Gα whenever α < β. The Σn(P )-Weak Vopˇenka Principle (Σn(P )-WVP for short) asserts that there is no Σn-definable, with parameters in P , sequence hGα : α ∈ ORi of graphs such that for every α ≤ β there exists exactly one homomorphism Gβ → Gα, and no homomorphism Gα → Gβ whenever α < β. The boldface versions Σn-VP and Σn-WVP are defined as Σn(V )-VP and Σn(V )-WVP respectively, i.e., any set is allowed as a parameter in the definitions. Πn(P )-VP and Πn(P )-WVP, as well as Πn-VP and Πn-WVP, and the lightface (i.e., without parameters) versions Σn−VP, Σn−WVP and Πn−VP, Πn−WVP, are defined similarly. The Vopˇenka Principle (VP) is the schema asserting that the Σn−VP holds for every n. And the Weak Vopˇenka Principle (WVP) is the schema asserting that the Σn−WVP holds for every n. If instead of requiring that for α ≤ β there is exactly one homomorphism Gβ → Gα we only require that there is at least one, then we obtain the Semi-Weak Vopˇenka Principle (SWVP), formulated as the first-order schema Σn−SWVP, n<ω. It is well-known that the category of structures in any fixed (many- sorted, infinitary) relational language can be fully embedded into Gra (see [AR94] 2.65). Thus, if in the original definitions of VP, WVP and 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 τ, as well as a parameter for a rigid binary relation on a binary signature associated to τ, may be needed in the definition. Namely, suppose Γ is one of the definability classes Σn, Πn, with n ≥ 1, P is a set or a proper class, and C is a Γ-definable, with parameters in P , class of (possibly many-sorted) relational structures in a language type τ, i.e., τ = hRα : α<λi, where each Rα is an nα-ary relation symbol, nα being some ordinal, possibly infinite. As in [AR94] 2.65, there is a ∆1-definable (i.e., both THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 6

Σ1-definable and Π1-definable), using τ as a parameter, one-sorted binary type τ ′ (meaning that all the relations are binary), and also a Γ-definable, with parameters in P plus τ as an additional parameter, full embedding of C into the category Rel τ ′ of τ ′-structures and homomorphisms. Furthermore, there is a ∆1-definable, using τ and a rigid binary relation r on τ ′ as parameters, full embedding of Rel τ ′ into Gra. Hence, there is a Γ-definable (with parameters in P , plus τ and r as additional parameters) full embedding of C into Gra. There- fore, in the definitions of Γ-VP, Γ-WVP, and Γ-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 (τ and r) involved. Let us, however, stress the fact that in the case of finite τ, or even if τ is countable infinite and definable without parameters (e.g., recursive), then no additional parameters are involved, and therefore the versions of Γ-VP, Γ-WVP, and Γ-SWVP for graphs and for relational structures are equivalent.

2.2. Strong cardinals. Recall that a cardinal κ is λ-strong, where λ is a cardinal greater than κ, if there exists an elementary embedding j : V → M, with M transitive, with critical point κ, and with Vλ contained in M. A cardinal κ is strong if it is λ-strong for every cardinal λ > κ. If κ is a strong cardinal, then for every cardinal λ > κ there exists an elementary embedding j : V → M, with M transitive, critical point κ, Vλ contained in M, and j(κ) > λ. Moreover, if κ is strong, then

Vκ Σ2 V . (See [Kan03].) It is well-known that the notion of strong cardinal can be formulated in terms of extenders (see [Kan03], section 26). Namely,

Deﬁnition 2.3. Given a cardinal κ, and β > κ, a (κ, β)-extender is a <ω collection E := {Ea : a ∈ [β] } such that |a| (1) Each Ea is a κ-complete ultraﬁlter over [κ] , and Ea is not κ+-complete for some a. |a| (2) For each ξ < κ, there is some a with {s ∈ [κ] : ξ ∈ s} ∈ Ea. <ω (3) Coherence: If a ⊆ b are in [β] , with b = {α1,...,αn} and |b| |a| a = {αi1 ,...,αin }, and πba :[κ] → [κ] is the map given by

πba({ξ1,...,ξn})= {ξi1 ,...,ξin }, then

|b| X ∈ Ea if and only if {s ∈ [κ] : πba(s) ∈ X} ∈ Eb .

(4) Normality: Whenever a ∈ [β]<ω and f :[κ]|a| → V are such |a| <ω that {s ∈ [κ] : f(s) ∈ max(s)} ∈ Ea, there is b ∈ [β] with a ⊆ b such that

|b| {s ∈ [κ] : f(πba(s)) ∈ s} ∈ Eb . THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 7

<ω (5) Well-foundedness: Whenever am ∈ [β] and Xm ∈ Eam for m ∈ ω, there is a function d : m am → κ such that d“am ∈ Xm for every m. S Proposition 2.4. A cardinal κ is λ-strong if and only if there exists a + (κ, |Vλ| )-extender E such that Vλ ⊆ M E and λ < jE (κ) (where M E is the transitive collapse of the direct limit ultrapower ME of V by E, and jE : V → M E is the corresponding elementary embedding). Proof. See [Kan03] 26.7.

3. The Product Reflection Principle For any set S of relational structures A = hA, . . .i of the same type, the set-theoretic product S is the structure whose universe is the set of all functions f with domain S such that f(A) ∈ A, for every A ∈ S, and whose relations are definedQ pointwise. Definition 3.1 (The Product Reflection Principle (PRP)). For Γ a definability class (i.e., one of Σn, Πn, some n > 0), and a set or a class P , Γ(P )-PRP asserts that for every Γ-definable, with parameters in P , proper class C of graphs the following holds: PRP: There is a subset S of C such that for every G in C there is a homomorphism S → G. If P = ∅, then we simply write Γ-PRP. If P = V , then we write Γ in Q boldface, e.g., Σn-PRP. In the definition of Γ(P )-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 Σn-PRP for classes of graphs is equivalent to its version for classes of relational structures. (n) We shall denote by C the Πn-definable closed and unbounded class of ordinals κ that are Σn-correct in V , i.e., Vκ Σn V . (See [Bag12].)

(1) Proposition 3.2. Σ1-PRP holds. In fact, for every κ ∈ C and every Σ1-definable with parameters in Vκ proper class C of structures in a fixed relational language τ ∈ Vκ, the set S := C ∩ Vκ witnesses Σ1(Vκ)-PRP. (1) Proof. Let κ ∈ C , and let C beaΣ1-definable, with a set of parameters P ∈ Vκ, proper class of structures in a relational language τ ∈ Vκ. (1) Note that since κ ∈ C , Vκ = Hκ, hence |TC({τ} ∪ P )| < κ. Let ϕ(x)beaΣ1 formula, with parameters in P , defining C. We claim that (1) S := C ∩ Vκ satisfies PRP. Given A ∈ C, let λ ∈ C be greater than κ and such that A ∈ Vλ. Let N Vλ be of cardinality less than κ and such that A ∈ N and TC({τ} ∪ P ) ⊆ N. Let π : M → N be the THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 8 inverse transitive collapse isomorphism, and let B ∈ M be such that π(B)= A. Notice that π fixes τ and the parameters of ϕ(x). Since M is transitive and of cardinality less than κ, B ∈ Hκ = Vκ. Also, since Vλ |= ϕ(A), we have N |= ϕ(A), and therefore M |= ϕ(B). Hence, since M is transitive and ϕ is upwards absolute for transitive sets, B ∈ C. Thus, B ∈ S. Then the composition of π with the projection S → B yields the desired homomorphism. Q Proposition 3.3. If κ is a strong cardinal, then Σ2(Vκ)-PRP holds.

Proof. Let κ be a strong cardinal and let C be a Σ2-definable, with parameters in Vκ, proper class of structures in a fixed relational language τ ∈ Vκ. Let ϕ(x)beaΣ2 formula defining it. We will show that S := C ∩ Vκ witnesses PRP. Given any A ∈ C, let λ ∈ C(2) be greater than or equal to κ and with A ∈ Vλ. Let j : V → M be an elementary embedding, with crit(j) = κ, Vλ ⊆ M, and j(κ) >λ. By elementarity, the restriction of j to S yields a homomorphism M h : S → ({X : M |= ϕQ(X)} ∩ Vj(κ)). (2) Since A ∈ Vλ, andYλ ∈ C Y, we have that Vλ |= ϕ(A). Hence, since the (1) fact that λ ∈ C is Π1-expressible and therefore downwards absolute for transitive classes, and since Vλ ⊆ M, it follows that Vλ Σ1 M and M therefore M |= ϕ(A). Moreover A ∈ Vλ ⊆ Vj(κ). Thus, letting M g : ({X : M |= ϕ(X)} ∩ Vj(κ)) →A be the projection map,Y we have that g ◦ h : S →A is a homomorphism, as wanted. Y Corollary 3.4. If there exist a proper class of strong cardinals, then Σ2-PRP holds.

We shall next show that SWVP is equivalent to the assertion that PRP holds for all deﬁnable proper classes of structures.

Proposition 3.5. Γn(P )-PRP implies Γn(P )-SWVP, for all n > 0 and every P .

Proof. Assume G = hGα : α ∈ ORi is a Γn-deﬁnable, with parameter p ∈ P , sequence of graphs such that for every α ≤ β there exists some homomorphism Gβ → Gα. Without loss of generality, the sequence is injective, i.e., Gα =6 Gβ whenever α =6 β. We shall produce a homomorphism Gα → Gβ, for some α < β. Notice that the collection of all G such that G = Gα, some α ∈ OR, is a proper class. Let ϕ(x) be a Γn formula, with p as a parameter, that deﬁnes G. THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 9

Let H be the class of all G such that G = Gα, for some ordinal α such that the rank of Gα is greater than or equal to α. Notice that H is also a proper class. We claim that H is Γn-definable with p as a parameter. This is clear if Γ = Σ, because G ∈ H iff there exists α such that ϕ(hα,Gi) and rank(G) ≥ α. If Γ = Π, and n = 1, then this is also clear since G ∈ H iff for all transitive M with G,p ∈ M, if M |= “{x : ϕ(x)} is a sequence of graphs indexed by the ordinals”, then M |= “There exists α such that ϕ(hα,Gi) and rank(G) ≥ α”. If (n−1) Γ = Π, and n > 1, then G ∈ H iff for all β ∈ C with G,p ∈ Vβ, if Vβ |= “ {x : ϕ(x)} is a sequence of graphs indexed by the ordinals”, then Vβ |= “There exists α such that ϕ(hα,Gi) and rank(G) ≥ α”. Hence by PRP there is a subset S of H and, for every G ∈ H, a homomorphism S → G. Thus, for every β, there is a homomorphism

Q hβ : S → Gβ.

Let κ be the supremum of allYα such that Gα ∈ S. Pick any α > κ. By our assumption, for every γ ≤ κ there is a homomorphism kγ : Gα → Gγ. Then the map ℓα : Gα → S, given by ℓα(x) = {hGγ,kγ(x)i : Gγ ∈ S} is a homomorphism. Q Now, given any α < β with α greater than κ, the composition hβ ◦ℓα : Gα → Gβ is a homomorphism, as wanted. The converse is also true. Namely,

Proposition 3.6. Γn(P )-SWVP implies Γn(P )-PRP, for every n> 1, and every P .

Proof. Let n> 0andﬁx aΓn-deﬁnable, with parameter p in P , proper class C of graphs and, aiming for a contradiction, suppose that PRP fails for C. We build by induction on γ a sequence C = hCγ : γ ∈ ORi, where Cγ = (C ∩Vαγ ), some αγ, such that γ ≤ η implies αγ ≤ αη, and such that there is no homomorphism h : Cγ → Cη, whenever γ < η. Q ∅ Namely, let α0 be the least ordinal such that C ∩ Vα0 =6 and let

C0 = (C ∩ Vα0 ). Given Cδ and αδ for δ<γ, let β be the least ordinal greater than sup{αδ : δ<γ} such that for some A∈C∩ Vβ there is no homomorphismQ

(C ∩ Vsup{αδ :δ<γ}) →A.

Then let αγ = β and deﬁneY

Cγ = (C ∩ Vαγ ) .

Since n> 1, it is easily seenY that C is Γn-definable (with parameter p). For if C is Πn-definable, then X = Cγ if and only if for every ordinal (n−1) ξ ∈ C with p ∈ Vξ, , if X ∈ Vξ, then Vξ |= “X = Cγ”. And if C is (n−1) Σn-definable, then X = Cγ if and only if for some ordinal ξ ∈ C with p and X ∈ Vξ, Vξ |=“X = Cγ”. THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 10

By the construction of C, there is no homomorphism Cγ → Cδ whenever γ < δ. So, by Γn(P )-SWVP there are γ < δ such that there is no homomorphism Cδ → Cγ. But the projection is such a homomorphism.

Suppose now C is Π1-deﬁnable. Let D be the class of all structures

Dγ := h Vα+1, ∈, Eγi 0 α∈[Yα ,αγ ) where (1) ∈ is the pointwise membership relation, (2)

Eγ ⊆ ( (C ∩ Vα)) 0 α∈[Yα ,αγ )Y consists of all f such that f(α) ⊆ f(α′), for all α<α′ in their domain, and 0 (1) (3) α is the least ordinal in C such that C ∩ Vα0 =6 ∅. (1) (4) αγ is the γ-th element of the class of ordinals η in C greater than α0 such that

∀δ < η∃A ∈ C ∩ Vη(¬∃h(h : (C ∩ Vδ) → A is a homomorphism)). Y Since PRP fails for C, Dγ exists for every ordinal γ. Also D is Π1, for X ∈ D if and only if every transitive model of a sufficiently rich finite fragment of ZFC that contains X satisfies “X ∈D”. ′ If γ<γ , then there is no homomorphism h : Dγ → Dγ′ , for the restriction of such an h to E would yield a homomorphism Eγ → Eγ′ , which in turn would yield (by (2) above) a homomorphism

′ h : (C ∩ Vαγ ) → (C ∩ Vαγ′ ) by letting Y Y

′ 0 h (f)= (h(hf ↾ Vαiα∈[α ,αγ ))). [ But by (3)-(b) above, there is some A ∈ C ∩ Vαγ′ for which there is no homomorphism

(C ∩ Vαγ ) → A.

′ So, by Π1(P )-SWVP thereY must exist γ<γ for which there is no homomorphism Dγ′ → Dγ. But the projection is such a homomorphism.

The following is an immediate corollary of Propositions 3.6 and 3.5.

Corollary 3.7. Γn(P )-PRP is equivalent to Γn(P )-SWVP, for every n> 0. Hence, SWVP is equivalent to Σn-PRP holds for every n. THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 11

4. The equivalence of WVP and SWVP Following the proof due to the second author [Wil20] of the equivalence of WVP and SWVP, we prove next the equivalence of their corresponding versions for definable classes of graphs. We only give a sketch of the proof, emphasizing only the definability aspects of it, and we refer to [Wil20] for additional details. Theorem 4.1. For Γ a definability class, and every set or proper class P , the principles Γ(P )-WVP and Γ(P )-SWVP are equivalent. Proof. That Γ(P )-SWVP implies Γ(P )-WVP is immediate. For the converse, fix Γ and P , and assume that hGα : α ∈ ORi is a sequence of graphs that is a counterexample to Γ(P )-SWVP. Thus, the sequence ′ is Γ(P )-definable, for every α ≤ α there is a homomorphism from Gα′ ′ to Gα, and for every α<α there is no homorphism from Gα to Gα′ . Let

α≤β Gα if β =0 or β is a successor ordinal, Hβ = (Qα<β Gα if β is a limit ordinal.

We claim that theQ sequence hHβ : β ∈ ORi yields also a counterexample to Γ(P )-SWVP. First, since Σ1(P )-SWVP is provable in ZFC (see propositions 3.2, 3.5), we may assume Γ is either Σn with n> 1, or Πn with n ≥ 1. In either case, since hGα : α ∈ ORi is Γ(P )-deﬁnable, so is ′ hHβ : β ∈ ORi (see the proof of 3.6). Second, for every β ≤ β there is ′ a restriction homomorphism from Hβ′ to Hβ, and for every β < β there is no homomorphism from Hβ to Hβ′ , for otherwise, arguing similarly as in proposition 3.5, we could compose homomorphisms

Gβ → Hβ → Hβ′ → Gβ+1 to obtain a homomorphism from Gβ to Gβ+1, thus yielding a contradiction. The class Λ of limit ordinals λ such that Gα ∈ Vλ for all α<λ is closed and unbounded. If the sequence hGα : α ∈ ORi is Π1(P )- definable, then so is Λ, and if it is Σn(P )-definable, with n> 1, then so ′ is Λ. Now for each λ ≤ λ in Λ, define the function hλ′,λ : Vλ′+1 → Vλ+1 by

hλ,λ′ (x)= x ∩ Vλ. For each λ ∈ Λ let the structure

Mλ = hVλ+1, ∈,λ,Rλ,Sλ, Tλi be such that

Rλ = {hβ,x,yi :(β = rank(x) and x ∈ y) or β = λ} Sλ = {hβ,x,yi :(β = rank(x) and x 6∈ y) or β = λ} Rλ = {hβ,x,yi : x is adjacent to y in Hβ} THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 12

If the sequence hGα : α ∈ ORi is Π1(P )-deﬁnable, then as argued above so are the sequence hHβ : β ∈ ORi and Λ, hence also hMλ : λ ∈ Λi; and if it is Σn(P )-deﬁnable, with n> 1, then so is hMλ : λ ∈ Λi. Now as in [Wil20] one can show that for all λ ≤ λ′ in Λ, the map hλ′,λ : Mλ′ → Mλ is a homomorphism, and is the unique one. More- ′ over, for every λ<λ in Λ there is no homomorphism from Mλ to Mλ′ . By re-enumerating the class Λ in increasing order as hλα : α ∈ ORi we obtain a sequence of relational structures hMλα : α ∈ ORi which yields a counterexample to Γ(P )-WVP.

5. The main theorem for strong cardinals Theorem 5.1. The following are equivalent: (1) There exists a strong cardinal. (2) Σ2-PRP (3) Π1-PRP (4) Σ2-SWVP (5) Π1-SWVP (6) Σ2-WVP (7) Π1-WVP. Proof. (1)⇒(2) is given by proposition 3.3; (2)⇒(3), (4)⇒(5), and (6)⇒(7) are immediate; the equivalence of (2) and (4), and also of (3) and (5), are given by corollary 3.7. The equivalence of (4) and (6), and also of (5) and (7), is given by theorem 4.1. So, it will be suﬃcient to prove (3)⇒(1). (3)⇒(1): Let A be the class of all structures α Aα := hVα+1, ∈, α, {Rϕ}ϕ∈Π1 i (1) α where the constant α is the α-th element of C and {Rϕ}ϕ∈Π1 is the Π1 relational diagram for Vα+1, i.e., if ϕ(x1,...,xn) is a Π1 formula in the language of hVα+1, ∈,αi, then α Rϕ = {hx1,...,xni : hVα+1, ∈,αi|=“ϕ(x1,...,xn)”} .

We claim that A is Π1-deﬁnable without parameters. For X ∈A if and only if X = hX0,X1,X2,X3i, where (1) (1) X2 belongs to C

(2) X0 = VX2+1 (3) X1 =∈↾ X0 (4) X3 is the Π1 relational diagram of hX0,X1,X2i, and (1) (5) hX0,X1,X2i|=“X2 is the X2-th element of C ”. Note that A is a proper class. In fact, the class C of ordinals α such that Aα ∈ A is a closed and unbounded proper class.. By Π1- PRP there exists a subset S of C such that for every β ∈ C there is a homomorphism jβ : α∈S Aα → Aβ. By enlarging S, if necessary, we Q THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 13 may assume that sup(S) ∈ S. Let us denote α∈S Aα by M. Notice that αQ M = h Vα+1, ∈, hαiα∈S, {Rϕ}ϕ∈Π1 i α∈S Y α where ∈ is the pointwise membership relation, and Rϕ is the pointwise α Rϕ relation. Let κ := sup(S). Now ﬁx some β ∈ C greater than κ, of uncountable coﬁnality, and assume, aiming for a contradiction, that no cardinal ≤ κ is β-strong. Let j = jβ. Claim 5.2. j preserves the Boolean operations ∩, ∪, −, and also the ⊆ relation. Proof of claim. For every X,Y,Z ∈ M,

M |=“X = Y ∩ Z” iﬀ Vα+1 |=“X(α)= Y (α) ∩ Z(α)”, all α ∈ S. So, letting ϕ(x, y, z) be the bounded formula expressing x = y ∩ z, we α have that hX(α),Y (α),Z(α)i ∈ Rϕ, for all α ∈ S. Hence hX,Y,Zi ∈ β Rϕ, and since j is a homomorphism hj(X), j(Y ), j(Z)i ∈ Rϕ, which yields Aβ |=“j(X)= j(Y ) ∩ j(Z)”. Similarly for the operations ∪, −, and for the relation ⊆.

Now deﬁne k : Vκ+1 → Vβ+1 by

k(X)= j(hX ∩ Vαiα∈S) . Claim 5.3. k also preserves the Boolean operations, as well as the ⊆ relation.

Proof of claim. Suppose Vκ+1 |= “X = Y ∩ Z”. Then X ∩ Vα = (Y ∩ Vα) ∩ (Z ∩ Vα), for every α ∈ S. Hence,

M |=“hX ∩ Vαiα∈S = hY ∩ Vαiα∈S ∩hZ ∩ Vαiα∈S” . Since j preserves the ∩ operation,

Aβ |=“k(X)= k(Y ) ∩ k(Z)” .

Let ψ(x)betheΠ1 formula (in the language with an additional constant for α) asserting that x = Vα. Then

hVαiα∈S ∈ Rψ β and so j(hVαiα∈S) ∈ Rψ, which yields j(hVαiα∈S) = Vβ. Hence, since j preserves the subset relation, and sends hαiα∈S to β, we have that k(X),k(Y ),k(Z) ⊆ Vβ. So,

Vβ+1 |=“k(X)= k(Y ) ∩ k(Z)” . Similarly for the operations ∪, −, and the relation ⊆. Claim 5.4. k maps ordinals to ordinals, and is the identity on ω +1. THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 14

Proof of claim. Let ϕ(x) be the bounded formula expressing that x is an ordinal. Let γ ≤ κ. Then γ ∩ Vα is an ordinal, for all α < κ, and so

M |=“hγ ∩ Vαiα∈S ∈ Rϕ” . Since j is a homomorphism, β Aβ |=“j(hγ ∩ Vαiα∈S) ∈ Rϕ” which yields that k(γ)= j(hγ ∩ Vαiα∈S) is an ordinal in Aβ, hence also in Vβ+1. For every ordinal γ ≤ ω, we have that γ ∩ Vα = γ, for all α ∈ S. Moreover, γ is deﬁnable by some bounded formula ϕγ. Hence,

M |=“hγ ∩ Vαiα∈S ∈ Rϕγ ” and therefore β Aβ |=“j(hγ ∩ Vαiα∈S) ∈ Rϕγ ” which yields k(γ)= γ.

Note that k(κ)= j(hαiα∈S)= β. <ω For each a ∈ [β] , define Ea by |a| X ∈ Ea iff X ⊆ [κ] and a ∈ k(X) . Since k(κ) = β and k(|a|) = |a|, we also have k([κ]|a|) = [β]|a|, hence |a| [κ] ∈ Ea. Moreover, since k preserves Boolean operations and the ⊆ |a| relation, Ea is an ultrafilter over [κ] .

Claim 5.5. Ea is ω1-complete.

Proof of claim. Given {Xn : n<ω} ⊆ Ea, let Y = {hn, xi : x ∈ Xn}. So, Y ⊆ Vκ. We can express that X = n<ω Xn by a bounded sentence ϕ in the parameters X, Y and ω. Moreover, since α is a limit ordinal, T for every α ∈ S, the sentence ϕ(X ∩ Vα,Y ∩ Vα,ω) expresses that X ∩ Vα = n<ω Xn ∩ Vα. So,

T M |=“hX ∩ Vα,Y ∩ Vα,ωiα∈S ∈ Rϕ” . Since j is a homomorphism, β Aβ |=“hj(X ∩ Vα), j(Y ∩ Vα), j(ω)iα∈S ∈ Rϕ” and so hk(X),k(Y ),k(ω)i satisﬁes ϕ. Since k(ω) = ω, we thus have k(X)= n<ω k(Xn). Hence, a ∈ k(X), and so X ∈ Ea. <ω Let E T:= {Ea : a ∈ [β] }. Claim 5.6. E is normal. That is, whenever a ∈ [β]<ω and f is a |a| |a| function with domain [κ] such that {s ∈ [κ] : f(s) ∈ max(s)} ∈ Ea, |b| κ there is b ⊇ a such that {s ∈ [κ] : f(πba(s)) ∈ s} ∈ Eb, where κ |b| |a| πba :[κ] → [κ] is the standard projection function. THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 15

Proof. Fix a and f, and suppose |a| X := {s ∈ [κ] : f(s) ∈ max(s)} ∈ Ea . Since the formula ϕ(X, κ, |a|, f) deﬁning X is bounded, and since k(κ)= β and k(|a|)= |a|, we have that k(X)= {s ∈ [β]|a| : k(f)(s) ∈ max(s)} .

Also, since X ∈ Ea, we have that k(f)(a) ∈ max(a). Let δ = k(f)(a), and let b = a ∪{δ}. Thus, |b| β b ∈{s ∈ [β] : k(f)(πba(s)) ∈ s} β |b| |a| where πba :[β] → [β] is the standard projection function. So, since |b| β |b| κ {s ∈ [β] : k(f)(πba(s)) ∈ s} = k({s ∈ [κ] : f(πba(s)) ∈ s}), we have |b| κ {s ∈ [κ] : f(πba(s)) ∈ s} ∈ Eb which shows that E is normal. <ω For each a ∈ [β] , the ultrapower Ult(V, Ea) of V by the ω1- complete ultraﬁlter Ea is well-founded. So, let

ja : V → Ma =∼ Ult(V, Ea) with Ma transitive, be the corresponding ultrapower embedding. As usual, we denote the elements of Ma by their corresponding elements in Ult(V, Ea). Claim 5.7. E is coherent. I.e., for every a ⊆ b in [β]<ω, |b| X ∈ Ea if and only if {s ∈ [κ] : πba(s) ∈ X} ∈ Eb .

<ω |a| Proof. Let a ⊆ b in [β] , and suppose X ∈ Ea. Thus, X ⊆ [κ] and |b| a ∈ k(X). We need to see that b ∈ k({s ∈ [κ] : πba(s) ∈ X}). Now notice that, since k is the identity on natural numbers, and k(κ)= β, |b| |b| k({s ∈ [κ] : πba(s) ∈ X})= {s ∈ [β] : πba(s) ∈ k(X)}. |b| Hence, since πba(b) = a, and a ∈ k(X), we have that b ∈ {s ∈ [β] : πba(s) ∈ k(X)}, as wanted. |b| Conversely, if {s ∈ [κ] : πba(s) ∈ X} ∈ Eb, we have that b ∈ |b| |b| k({s ∈ [κ] : πba(s) ∈ X}) = {s ∈ [β] : πba(s) ∈ k(X)}. Hence, πba(b)= a ∈ k(X), and therefore X ∈ Ea.

<ω For each a ⊆ b in [β] , let iab : Ma → Mb be given by

iab([f]Ea ) = [f ◦ πba]Eb |a| for all f :[κ] → V . By coherence, the maps iab are well-deﬁned and commute with the ultrapower embeddings ja (see [Kan03] 26). Let ME be the direct limit of <ω hhMa : a ∈ [β] i, hiab : a ⊆ bii. THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 16

For notational simplicity, whenever we write [a, [f]] ∈ [b, [g]] in ME , what we mean is that [f] = [f]Ea ∈ Ma,[g] = [g]Eb ∈ Mb, and

[ha, [f]Ea i]]E ∈E [hb, [g]Eb i]E . Let jE : V → ME be the corresponding limit elementary embedding, i.e., a jE (x) = [a, [cx]Ea ] <ω a |a| for some (any) a ∈ [β] , and where cx :[κ] →{x}. Let ka : Ma → ME be given by

ka([f]Ea ) = [a, [f]Ea ] .

It is easily checked that jE = ka ◦ ja and kb ◦ iab = ka, for all a ⊆ b, <ω |a| |a| a, b ∈ [β] . Thus, letting Id|a| :[κ] → [κ] be the identity function, we have <ω |a| ME = {jE (f)(ka([id|a|]Ea )) : a ∈ [β] and f :[κ] → V } .

∗ |a| Let ME := {[a, [f]] ∈ ME : f ∩ Vα :[α] → Vα, all α ∈ S}. Suppose ∗ [a, [f]], [b, [g]] ∈ ME . Then the following can be easily verified: (1) [a, [f]] ∈E [b, [g]] iff k(f)(a) ∈ k(g)(b) (2) [a, [f]] =E [b, [g]] iff k(f)(a)= k(g)(b) ∗ Claim 5.8. ME is well-founded and closed under ∈E . Proof. Well-foundedness follows from items (1) and (2) above, as any ∗ infinite ∈E -descending sequence in ME would yield an infinite ∈-descending sequence in Vβ+1. ∗ Now suppose [a, [f]] ∈E [b, [g]], with [b, [g]] ∈ ME . Then for some c ⊇ a, b, and some X ∈ Ec,

(f ◦ πca)(s) ∈ (g ◦ πcb)(s) |a| for every s ∈ X. Let Y = {πca(s): s ∈ X} ∈ Ea. Deﬁne h :[κ] → V by: h(s) = f(s) for all s ∈ Y , and h(s) = 0, otherwise. Then [h]Ea = ∗ [f]Ea , and [a, [f]] = [a, [h]] ∈ ME . ∗ ∗ By the last Claim, ME is well-founded and extensional. So, let M ∗ be the transitive collapse of ME . ∗ Claim 5.9. Vβ ⊆ M . Proof of claim. Since κ and α, for α ∈ S, belong to C(1), we have that |Vκ| = κ and |Vα| = α, all α ∈ S. Let f ∈ V be a bijection between 1 1 1 [κ] and Vκ such that f ↾ [α] is a bijection between [α] and Vα, all 1 α ∈ S. Let ϕ(x, y, z)beaΠ1 formula expressing that x = [u] , with u an ordinal, y = Vu, and z : x → Vu is a bijection. Thus, 1 M |=“h[α] ,Vα, f ∩ Vαiα∈S ∈ Rϕ” . Hence, 1 β Aβ |=“hk([κ] ),k(Vκ),k(f)i ∈ Rϕ” THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 17

1 and so k(f):[β] → Vβ is a bijection. Therefore, for every x ∈ Vβ there exists γ < β such that k(f)({γ})= x. Thus, letting D := {[{γ}, [f]] : γ < β}, we have just shown that the map i : hD, ∈E ↾ Di→hVβ, ∈i given by i([{γ}, [f]]) = k(f)({γ}) is onto. Moreover, if [{γ}, [f]] ∈E [{δ}, [f]], then for some X ∈ E{γ,δ}, we have (f ◦ π{γ,δ}{γ})(s) ∈ (f ◦ π{γ,δ}{δ})(s) for every s ∈ X. Letting ϕ be the bounded formula expressing this, and since Vα is closed under f, for every α, we have

M |=“hX ∩ Vα, (f ◦ π{γ,δ}{γ}) ∩ Vα, (f ◦ π{γ,δ}{δ}) ∩ Vαi ∈ Rϕ” .

Hence, in Aβ, for every s ∈ k(X),

(k(f) ◦ π{γ,δ}{γ})(s) ∈ (k(f) ◦ π{γ,δ}{δ})(s) . In particular, since {γ, δ} ∈ k(X), k(f)({γ}) ∈ k(f)({δ}) . A similar argument shows that i is one-to-one. Hence, i is an isomor- ∗ phism, and so i is just the transitive collapsing map. Since D ⊆ ME , to ∗ conclude that Vβ ⊆ M 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 ME. For this, it suﬃces to see that every ∈E-element of an element of D is =E-equal to an element of D. So, suppose [{γ}, [f]] ∈ D and ∗ [a, [g]] ∈E [{γ}, [f]], with [a, [g]] ∈ ME. Then k(g)(a) ∈ k(f)(γ), by (1) 1 above (just before Claim 5.8). Now k(f):[β] → Vβ is surjective and Vβ is transitive, so there is some δ < β such that k(f)(δ) = k(g)(a). Hence, by (2) above, [{δ}, [f]] =E [a, [g]].

Claim 5.10. ME is closed under ω-sequences, hence it is well-founded.

Proof of claim. Let hjE (fn)(kan ([id|an|]Ean ))in<ω be a sequence of elements of ME . On the one hand, the sequence hjE (fn)in<ω = jE (hfnin<ω) belongs to ME . On the other hand, kan ([Id|an|]Ean ) = [an, [Id|an|]Ean ] be- ∗ longs to ME , all n<ω. Since E is normal (Claim 5.6), as in [Kan03]

26.2 (a) we can show that the transitive collapse of [an, [Id|an|]Ean ] is precisely an. The sequence hanin<ω belongs to Vβ, because β has uncount- ∗ able coﬁnality. Hence, since Vβ ⊆ M , the preimage of hanin<ω ∈ Vβ ∗ ∗ under the transitive collapsing map of ME to M , is precisely the sequence hkan ([Id|an|]Ean )in<ω and belongs to ME . It now follows that the sequence hjE (fn)(kan ([id|an|]Ean ))in<ω is also in ME .

Let π : ME → N be the transitive collapsing isomorphism, and let jN : V → N be the corresponding elementary embedding, i.e., jN = π ◦ jE .

Claim 5.11. jN (κ) ≥ β. THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 18

1 Proof of claim. Let α < β. Let Id1 be the identity function on [κ] , 1 1 1 and let c[κ]1 :[κ] →{[κ] } and cκ :[κ] →{κ}. In M{α}, we have

1 1 1 [Id1]E{α} ∈ [c[κ] ]E{α} = [[cκ]E{α} ] = [j{α}(κ)] hence in ME , 1 1 k{α}([Id1]E{α} ) ∈ k{α}([j{α}(κ)] ) = [jE (κ)] and therefore, since π(k{α}([Id1]E{α} )) = {α}, in N we have

1 {α} ∈ [jN (κ)] that is, α < jN (κ).

Since β > κ, the last claim implies that the critical point of jN is less than or equal to κ. Thus, since Vβ ⊆ N (by Claim 5.9), jN witnesses that the critical point of jN is a β-strong cardinal. But this is in contradiction to our choice of β. This completes the proof of theorem 5.1. The boldface version of theorem 5.1, i.e., with parameters, also holds by essentially the same arguments. Namely, Theorem 5.12. The following are equivalent: (1) There exists a proper class of strong cardinals. (2) Σ2-PRP (3) Π1-PRP (4) Σ2-SWVP (5) Π1-SWVP (6) Σ2-WVP (7) Π1-WVP For the proof of (3) implies (1), in order to show that there exists a strong cardinal greater than a ﬁxed ordinal γ we need to consider the class of structures α Aα := hVα+1, ∈, α, {Rϕ}ϕ∈Π1 , hδiδ<γi where α Aα := hVα+1, ∈, α, {Rϕ}ϕ∈Π1 i is as in the proof of Theorem 5.1, and we have a constant δ for every δ<γ.

6. 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. THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 19

If j : V → M is an elementary embedding, with M transitive and critical point κ, and A is a class deﬁnable by a formula ϕ (possibly with parameters in Vκ), we deﬁne j(A) := {X ∈ M : M |= ϕ(X)} . Note that j(A)= {j(A ∩ Vα): α ∈ OR} M as j(A ∩ Vα)= {X ∈ M : M[|= ϕ(X)} ∩ Vj(α). Also note that if A is a class of structures of the same type τ ∈ Vκ, then by elementarity j(A) is also a subclass of M of structures of type τ.

6.1. Γn-strong cardinals.

Deﬁnition 6.1. For n ≥ 1, a cardinal κ is λ-Γn-strong if for every Γn-deﬁnable (without parameters) class A there is an elementary embedding j : V → M, with M transitive, crit(j) = κ, Vλ ⊆ M, and A ∩ Vλ ⊆ j(A). κ is Γn-strong if it is λ-Γn-strong for every ordinal λ.

Note that in the deﬁnition above A ∩ Vλ is only required to be contained in j(A) ∩ Vλ and not equal to it. The reason is that in the Σ2 case, if A is the class of non-strong cardinals (which is Σ2) and κ is the least strong cardinal, then κ∈ / A, but κ ∈ j(A). See however the equivalence given in Proposition 6.8. As with the case of strong cardinals, standard arguments show (cf. [Kan03] 26.7(b)) that κ is λ-Γn-strong if and only if for every Γn- deﬁnable (without parameters) class A there is an elementary embedding j : V → M, with M transitive, crit(j) = κ, Vλ ⊆ M, j(κ) > λ, and A ∩ Vλ ⊆ j(A).

Proposition 6.2. Every strong cardinal is Σ2-strong.

Proof. Let κ be a strong cardinal and let A be a class that is Σ2- (2) deﬁnable (even allowing for parameters in Vκ). Let λ ∈ C be greater than κ. Let j : V → M be elementary, with M transitive, crit(j)= κ, and Vλ ⊆ M. Let ϕ be a Σ2 formula deﬁning A. If a ∈ A ∩ Vλ, then

Vλ |= ϕ(a). Hence, since Vλ Σ1 M, M |= ϕ(a), and so a ∈ j(A) = {x : M |= ϕ(x)}.

(n+1) Proposition 6.3. If λ ∈ C , then a cardinal κ is λ-Πn-strong if and only if it is λ-Σn+1-strong. (n+1) Proof. Assume κ is λ-Πn-strong, with λ ∈ C , and let A beaΣn+1- definable class. Let ϕ(x) ≡ ∃yψ(x, y)beaΣn+1 formula, with ψ(x, y) being Πn, that defines A. Now define B as the class of all structures of (n) the form hVα, ∈, ai, where α ∈ C , a ∈ Vα, and Vα |= ϕ(a). Then B is Πn-definable. By our assumption, let j : V → M be an elementary embedding, with M transitive, crit(j)= κ, Vλ ⊆ M, and B∩Vλ ⊆ j(B). THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 20

We just need to show that A∩Vλ ⊆ j(A). So, suppose a ∈ A∩Vλ. Since (n+1) λ ∈ C , we have that Vλ |= ϕ(a). Let b ∈ Vλ be a witness, so that (n) Vλ |= ψ(a, b). For some α<λ in C we have that a, b ∈ Vα. Hence, Vα |= ϕ(a). So hVα, ∈, ai ∈ B ∩ Vλ, and therefore hVα, ∈, ai ∈ j(B). (n) Thus, M |= “α ∈ C , a ∈ Vα, and Vα |= ϕ(a)”. Hence, M |= ϕ(a), i.e., a ∈ j(A).

Corollary 6.4. A cardinal κ is Πn-strong if and only if it is Σn+1- strong. Proposition 6.5. Suppose that n ≥ 2 and λ ∈ C(n). Then the following are equivalent for a cardinal κ<λ:

(1) κ is λ-Σn-strong. (2) There is an elementary embedding j : V → M, with M transi- (n−1) tive, crit(j)= κ, Vλ ⊆ M, and M |=“λ ∈ C ”. (n−1) Proof. (1)⇒(2): Suppose κ is λ-Σn-strong. Let A = C . Since A is Πn−1-definable, hence also Σn-definable, by (1) there is an elementary embedding j : V → M with M transitive, crit(j) = κ, Vλ ⊆ M, and (n) (n−1) A ∩ Vλ ⊆ j(A). Since λ ∈ C , C ∩ λ is a club subset of λ. For every α<λ in C(n−1), α ∈ j(A), hence M |= “α ∈ C(n−1)” and so M |=“λ is a limit point of C(n−1)”, which yields M |=“λ ∈ C(n−1)”. (2)⇒(1): Let A be a class definable by a Σn formula ϕ, and let j : V → M be an elementary embedding with M transitive, crit(j)= κ, (n−1) (n) Vλ ⊆ M, and M |= “λ ∈ C ”. Let a ∈ A ∩ Vλ. Since λ ∈ C , (n−1) Vλ |= ϕ(a). And since Vλ ⊆ M and M |= “λ ∈ C ”, M |= ϕ(a), i.e., a ∈ j(A). The last proposition suggests the following definition and the ensuing characterization of Σn-strong cardinals in terms of extenders.

Deﬁnition 6.6. Given cardinals κ<λ, a Σn-strong (κ, λ)-extender + is a (κ, |Vλ| )-extender E (see Deﬁnition 2.3) such that M E |= “λ ∈ (n−1) C ”, where M E is the transitive collapse of the direct limit ultrapower ME of V by E. Proposition 6.7. If n ≥ 2 and λ ∈ C(n), then a cardinal κ<λ is λ-Σn-strong if and only if there exists a Σn-strong (κ, λ)-extender.

Proof. If E is a Σn-strong (κ, λ)-extender, then the extender embedding jE : V → M E , witnesses that κ is λ-Σn-strong (by Proposition 6.5). Conversely, suppose j : V → M is an elementary embedding, with (n−1) M transitive, crit(j)= κ, Vλ ⊆ M, and M |=“λ ∈ C ”. Note that (1) + since λ ∈ C , |Vλ| = λ. Let E be the (κ, λ )-extender derived from j. + <ω Namely, for every a ∈ [λ ] let Ea be defined by: |a| X ∈ Ea if and only if X ⊆ [κ] and a ∈ j(X). One can easily check that E satisfies conditions (1)−(6) of Definition 2.3 (n−1) (see [Kan03] 26.7). So we only need to check that M E |=“λ ∈ C ”. THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 21

Let jE : V → ME and kE : M E → M be the standard maps given by: a a |a| jE (x) = [a, [cx]] (any a), where cx :[κ] → {x}; and kE (π([a, [f]])) = |a| j(f)(a), for f :[κ] → V , where π : ME → M E is the transitive collapse isomorphism. The maps jE and kE are elementary and j = kE ◦ jE . Moreover, kE ↾ Vλ is the identity. Since M |= “λ ∈ C(n−1)”, for each µ < λ in C(n−1), we have (n−1) that M |= “µ ∈ C ”. So, since kE is elementary and is the (n−1) identity on Vλ, we have that ME |= “µ ∈ C ”. Hence, ME |= (n−1) (n−1) “λ is a limit point of C ”, which yields ME |=“λ ∈ C ”.

Similar characterizations may also be given for Πn-strong cardinals. Notice that (3) of the following proposition characterizes Πn-strong cardinals as witnessing “OR is Woodin” restricted to Πn-deﬁnable classes (see Deﬁnition 6.16).

Proposition 6.8. Suppose that n ≥ 1 and λ is a limit point of C(n). Then the following are equivalent for a cardinal κ<λ:

(1) κ is λ-Πn-strong. (2) There is an elementary embedding j : V → M, with M transi- (n) tive, crit(j)= κ, Vλ ⊆ M, and M |=“λ ∈ C ”. (3) For every Πn-deﬁnable class A there is an elementary embedding j : V → M with M transitive, crit(j) = κ, Vλ ⊆ M, and A ∩ Vλ = j(A) ∩ Vλ.

(n) Proof. (1)⇒(2): Suppose κ is λ-Πn-strong. Let A = C . Since A is Πn-deﬁnable, by (1) there is an elementary embedding j : V → M with M transitive, crit(j) = κ, Vλ ⊆ M, and A ∩ Vλ ⊆ j(A). Thus, for every α<λ in A, α ∈ j(A), hence M |= “α ∈ C(n)” and so M |=“λ is a limit point of C(n)”, which yields M |=“λ ∈ C(n)”. (2)⇒(3): Let A be a class deﬁnable by a Πn formula ϕ(x), and let j : V → M be an elementary embedding with M transitive, crit(j)= κ, (n) (n) Vλ ⊆ M, and M |= “λ ∈ C ”. Let a ∈ A ∩ Vλ. Since λ ∈ C , (n) Vλ |= ϕ(a). And since Vλ ⊆ M and M |= “λ ∈ C ”, M |= ϕ(a), i.e., a ∈ j(A). Conversely, suppose a ∈ j(A) ∩ Vλ, i.e., M |=“ϕ(a)”. Since (n) (n) M |=“λ ∈ C ”, Vλ |= ϕ(a). And since λ ∈ C , a ∈ A. (3)⇒(1) is immediate.

Corollary 6.9. Suppose that n ≥ 1 and λ is a limit point of C(n). A cardinal κ is λ-Πn-strong if and only if for every Πn-deﬁnable class A there is an elementary embedding j : V → M with M transitive, crit(j)= κ, Vλ ⊆ M, and A ∩ Vλ = j(A) ∩ Vλ.

Similarly as before we may also characterize Πn-strong cardinals in terms of extenders. Namely, THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 22

Deﬁnition 6.10. Suppose that n ≥ 1 and λ is a limit point of C(n). + Given a cardinal κ<λ, a Πn-strong (κ, λ)-extender is a (κ, |Vλ| )- (n) extender E (see Deﬁnition 2.3) such that M E |=“λ ∈ C ”, where M E is the transitive collapse of the direct limit ultrapower ME of V by E. Proposition 6.11. Suppose that n ≥ 1 and λ is a limit point of C(n). Then a cardinal κ is λ-Πn-strong if and only if there exists a Πn-strong (κ, λ)-extender.

It easily follows from the last proposition that being a Πn-strong cardinal is a Πn+1 property. Moreover, if κ is Πn-strong, then κ ∈ (n+1) C . Hence, if κ is Πn+1-strong, then there are many Πn-strong cardinals below κ, which shows that the Πn-strong cardinals, n > 0, form a hierarchy of strictly increasing strength. Similarly as in 3.3 we can prove the following.

Proposition 6.12. If κ is a Σn-strong cardinal, where n ≥ 2, then Σn(Vκ)−PRP holds.

Proof. Let n ≥ 2. Let κ be Σn-strong and let C be a deﬁnable, by a Σn formula with parameters in Vκ, proper class of structures in a ﬁxed countable relational language. Let S := C ∩ Vκ. Given any A ∈ C, let λ ≥ κ with A ∈ Vλ. Let j : V → M be an elementary embedding, with crit(j) = κ, Vλ ⊆ M, j(κ) >λ, and C ∩ Vλ ⊆ j(C). By elementarity, the restriction of j to S yields a homomorphism M h : S → (j(C) ∩QVj(κ)). Y Y M Since A∈C∩ Vλ, we have that A ∈ j(C). Moreover A ∈ Vλ ⊆ Vj(κ). Thus, letting M g : (j(C) ∩ Vj(κ)) →A be the projection map, weY have that g ◦ h : S →A is a homomorphism, as wanted. Y

6.2. The main theorem for Γn-strong cardinals. Using similar arguments as in theorem 5.1 we can now prove the main theorem of this section. Theorem 6.13. The following are equivalent for n ≥ 2:

(1) There exists a Σn-strong cardinal. (2) There exists a Πn−1-strong cardinal. (3) Σn-PRP (4) Πn−1-PRP (5) Σn-SWVP (6) Πn−1-SWVP THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 23

(7) Σn-WVP (8) Πn−1-WVP Proof. (1)⇒(3) is given by proposition 6.12. (1)⇔(2) is given by corollary 6.4, (3)⇒(4), (5)⇒(6), and (7)⇒(8) are immediate. The equivalences (3)⇔(5) and (4)⇔(6) are given by corollary 3.7. The equivalence of (5) and (7), and also of (6) and (8), is given by theorem 4.1. So, we only need to prove (4)⇒(2). The proof is analogous to the proof of Theorem 5.1. So, we shall only indicate the relevant diﬀerences. Theorem 5.1 proves the case n = 2 (see proposition 6.2). Thus, we shall assume in the sequel that n> 2. Let A be the class of all structures (n−1) α Aα := hVα+1, ∈,α,C ∩ α, {Rϕ}ϕ∈Π1 i (n−1) α where the constant α is the α-th element of C , and {Rϕ}ϕ∈Π1 is the Π1 relational diagram for Vα+1, i.e., if ϕ(x1,...,xn) is a Π1 formula (n−1) in the language of hVα+1, ∈,α,C ∩ αi, then α (n−1) Rϕ = {hx1,...,xni : hVα+1, ∈,α,C ∩ αi|=“ϕ(x1,...,xn)”} .

Then A is Πn−1-deﬁnable without parameters. For X ∈ A if and only if X = hX0,X1,X2,X3,X4i, where (n−1) (1) X2 belongs to C

(2) X0 = VX2+1 (3) X1 =∈↾ X0 (n−1) (4) X0 satisﬁes that X3 = C ∩ X2 (5) X4 is the Π1 relational diagram of hX0,X1,X2,X3i (n−1) (6) hX0,X1,X2,X3i|=“X2 is the X2-th element of C ”.

Note that the class C of ordinals α such that Aα ∈A is a closed and unbounded proper class. By Πn−1-PRP there exists a subset S of C such that for every β ∈ C there is a homomorphism jβ : α∈S Aα → Aβ. By enlarging S, if necessary, we may assume that κ := sup(S) ∈ S. Now fix some β ∈ C greater than κ, of uncountable cofinality,Q and assume, towards a contradiction, that no cardinal ≤ κ is β-Πn−1-strong. Let j = jβ. From this point, the proof proceeds as in 5.1. Namely, we define k : Vκ+1 → Vβ+1 by

k(X)= j(hX ∩ Vαiα∈S) and note that k(κ)= β. <ω For each a ∈ [β] , define Ea by |a| X ∈ Ea iff X ⊆ [κ] and a ∈ k(X) . |a| As in 5.1, Ea is an ω1-complete ultrafilter over [κ] . Moreover, E := <ω {Ea : a ∈ [β] } is normal and coherent. THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 24

<ω For each a ∈ [β] , the ultrapower Ult(V, Ea) is well-founded, by ω1-completeness. So, let

ja : V → Ma =∼ Ult(V, Ea) with Ma transitive, be the corresponding ultrapower embedding, and let ME be the direct limit of <ω hhMa : a ∈ [β] i, hiab : a ⊆ bii where the iab : Ma → Mb are the usual commuting maps. The corresponding limit embedding jE : V → ME is elementary. As in 5.1, ME is closed under ω-sequences, hence it is well-founded. Moreover, letting π : ME → N be the transitive collapsing isomorphism, and jN : V → N the corresponding elementary embedding, i.e., jN = π◦jE , we have that Vβ ⊆ N and jN (κ) ≥ β. Since β > κ, this implies that the critical point of jN is less than or equal to κ. The only additional argument needed, with respect to the proof of 5.1, is the following: Claim 6.14. N |=“β ∈ C(n−1)”. Proof. Since β is a limit point of C(n−1), it suﬃces to show that if γ < β belongs to C(n−1), then N |= “γ ∈ C(n−1)”. So, ﬁx some γ < β in C(n−1). Let f :[κ]1 → κ be such that f({x}) = x. It is well known that k{γ}([f]E{γ} ) = γ, where k{γ} : M{γ} → N is the standard map given by k{γ}([f]E{γ} )= π([{γ}, [f]E{γ} ]) (see [Kan03] 26.2 (a)). 1 (n−1) Now notice that the set X := {{x} ∈ [κ] : x ∈ C } ∈ E{γ}, 1 (n−1) because {γ} ∈ k(X) = {{x} ∈ [β] : x ∈ C }. Hence, M{γ} |= (n−1) (n−1) “[f] ∈ C ”, and therefore ME |= “[{γ}, [f]] ∈ C ”, which yields N |=“γ ∈ C(n−1)”, as wanted.

Thus, by proposition 6.8, jN witnesses that the critical point of jN is less than or equal to κ and is β-Πn−1-strong, in contradiction to our choice of β. In a similar way we may obtain the following parameterized version of theorem 6.13. For the proof of (4) implies (2), we need to consider the class of structures (n−1) α Aα := hVα+1, ∈,α,C ∩ α, {Rϕ}ϕ∈Π1 , hδiδ<γi where (n−1) α Aα := hVα+1, ∈,α,C ∩ α, {Rϕ}ϕ∈Π1 i is as in the proof of Theorem 6.13, and we have a constant δ for every δ<γ. Theorem 6.15. The following are equivalent for n ≥ 2:

(1) There exist a proper class of Σn-strong cardinals. (2) There exist a proper class of Πn−1-strong cardinal. (3) Σn-PRP THE WVP FOR DEFINABLE CLASSES OF STRUCTURES 25

(4) Πn−1-PRP (5) Σn-SWVP (6) Πn−1-SWVP (7) Σn-WVP (8) Πn−1-WVP

Recall that a cardinal κ is Woodin if for every A ⊆ Vκ there is α < κ such that α is <κ-A-strong, i.e., for every γ < κ there is an elementary embedding j : V → M with crit(j) = α, γ < j(α), Vγ ⊆ M, and A ∩ Vγ = j(A) ∩ Vγ. (See [Kan03] 26.14.) Definition 6.16. OR is Woodin if for every definable (with set parameters) class A there exists some α which is A-strong, i.e., for every γ there is an elementary embedding j : V → M with crit(j) = α, γ < j(α), Vγ ⊆ M, and A ∩ Vγ = j(A) ∩ Vγ . The statement “OR is Woodin” is first-order expressible as a schema, namely as “There exists α which is A-strong”, for each definable A. Or equivalently, by corollary 6.9, as the schema “There exists α which is Πn-strong”, for every n. Let us note that, by theorem 6.15, “OR is Woodin” is also equivalent to the schema asserting “There exist a proper class of α which are Πn-strong”, for every n. Thus, theorem 6.15 yields the following corollary, first proved by the second author in [Wil19], which gives the exact large-cardinal strength of WVP and SWVP. Corollary 6.17. The following are equivalent: (1) OR is Woodin (2) SWVP (3) WVP.

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ICREA (Institucio´ Catalana de Recerca i Estudis Avanc¸ats) and Departament de Matematiques` i Informatica,` Universitat de Barcelona. Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Catalonia. Email address: [email protected]

Department of Mathematics Miami University, 236 Bachelor Hall, Oxford, OH 45056 Email address: [email protected]