Huge Reflection

We study Structural Reflection beyond Vop\v{e}nka's Principle, at the level of almost-huge cardinals and higher, up to rank-into-rank embeddings. We identify and classify new large cardinal notions in that region that correspond to some form of what we call Exact Structural Reflection ($\mathrm{ESR}$). Namely, given cardinals $\kappa<\lambda$ and a class $\mathcal{C}$ of structures of the same type, the corresponding instance of $\mathrm{ESR}$ asserts that for every structure $A$ in $\mathcal{C}$ of rank $\lambda$, there is a structure $B$ in $\mathcal{C}$ of rank $\kappa$ and an elementary embedding of $B$ into $A$. Inspired by the statement of Chang's Conjecture, we also introduce and study sequential forms of $\mathrm{ESR}$, which, in the case of sequences of length $\omega$, turn out to be very strong. Indeed, when restricted to $\Pi_1$-definable classes of structures they follow from the existence of $I1$-embeddings, while for more complicated classes of structures, e.g., $\Sigma_2$, they are not known to be consistent. Thus, these principles unveil a new class of large cardinals that go beyond $I1$-embeddings, yet they may not fall into Kunen's Inconsistency.


Introduction
Given a class 1 C of structures 2 of the same type and a cardinal κ, the principle of Structural Reflection 3 SR holds at κ for C if for every structure A in C, there exists some B ∈ C ∩ V κ and an elementary embedding of B into A. Different forms of SR have been investigated in [1,2,3,4,5,12], yielding canonical characterizations of large cardinals in different regions of the large cardinal hierarchy.For example, results in [1] and [3] use Magidor's classical characterization of supercompact cardinals from [13] to show that the existence of such a cardinal is equivalent to the validity of the principle SR for all classes of structures definable by Π 1 -formulas without parameters.
The principles of structural reflection considered so far correspond to large cardinals up to Vopěnka's Principle, stating that every proper class of structures of the same type contains a structure that is elementary embeddable into another structure in the given class.The validity of this principle can be shown to be equivalent to the existence of cardinals witnessing SR for every class of structures (see [1]).In this paper, we shall study principles of structural reflection that correspond to large cardinal notions stronger than Vopěnka's Principle, up to rank-into-rank embeddings, and beyond.These principles are given by variations of the following exact form of SR: This problem was brought to our attention by Nai-Chung Hou, a student of the first author.It can be fixed by replacing the notion of Πn(Vκ)-correctness with the weaker concept of Πn(Vκ)-upwards correctness.Through this modification, the proofs of all of the main theorems of the paper hold as before, and only three minor results (Propositions 5.6 and 7.2, and Corollary 6.5 of the published version) cannot be saved.The authors would like to thank Nai-Chung Hou for pointing out this problem.Part of this research was supported by the Generalitat de Catalunya (Catalan Government) under grant SGR 270-2017, and by the Spanish Government under grant MTM2017-86777-P.In addition, this project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 842082 (Project SAIFIA: Strong Axioms of Infinity -Frameworks, Interactions and Applications). 1 We work in ZFC.So, proper classes are always definable, with set parameters. 2 Throughout this paper, the term structure refers to structures for countable first-order languages.The cardinality (respectively, the rank ) of a structure is defined as the cardinality (respectively, the rank) of its domain. 3A detailed discussion of this principle can be found in [2].
Theorem 1.2.The following statements are equivalent for every singular cardinal κ: (i) κ is a limit of supercompact cardinals.
In order to state an analogous result for more complicated classes of structures, we have to introduce a weak form of the notion of C (n) -extendibility from [1].Recall that a cardinal κ is λ-extendible for some ordinal λ > κ if there is an ordinal η and an elementary embedding j : V λ → V η with crit(j) = κ and j(κ) > λ.Following [1], for every n < ω, we let C (n) denote the Π n -definable closed unbounded class of all of ordinals α such that V α is a Σ n -elementary substructure of V .Given cardinals κ < λ and n < ω, the cardinal κ is λ-C (n) -extendible if there is an elementary embedding j : V λ → V ν for some cardinal ν with crit(j) = κ, j(κ) > λ and j(κ) ∈ C (n) .In addition, we say that κ is C (n) -extendible if it is λ-C (n) -extendible for all (equivalently, for a proper class of) λ > κ (see [1,Section 3]).
Using this notion, the above characterization of singular limits of supercompact cardinals now generalizes in the following way: Theorem 1.4.For every n > 0, the following statements are equivalent for every singular cardinal κ: (i) κ is a limit of [κ, ∞)-C (n) -extendible cardinals.
In particular, Theorem 1.4 shows that a singular cardinal κ is a limit of extendible cardinals if and only if Π 2 (κ) ic -ESR(κ, λ) holds for a proper class of cardinals λ.
In combination with results from [1] and [3], the methods developed in the proof of Theorem 1.4 also allow us to conclude that exact structural reflection for classes of structures closed under isomorphisms holding at singular cardinals does not imply the existence of large cardinals stronger than Vopěnka's Principle.In fact, Vopěnka's Principle can be characterized through the validity of principles of the form Π n (V κ ) ic -ESR(κ, λ).
Theorem 1.5.Over the theory ZFC, the following schemes of sentences imply each other: (i) Vopěnka's Principle.
(ii) For every class C of structures of the same type that is closed under isomorphic images, there is a cardinal κ with the property that ESR C (κ, λ) holds for all λ > κ. (iii) For every natural number n > 0, there exists a proper class of cardinals κ with the property that Π n (V κ ) ic -ESR(κ, λ) holds for all λ > κ.
In contrast to the above results, both the validity of the principle Π ic 1 -ESR at a regular cardinal and the validity of the principle Π 1 -ESR at some cardinal turn out to imply the existence of large cardinals stronger than Vopěnka's Principle, e.g.almost huge cardinals.The large cardinal properties introduced below will allow us to capture the strength of these forms of exact structural reflection.Their definition is motivated by results in [12] that provide a characterization of shrewd cardinals (introduced by Rathjen in [14]) through a variation of Magidor's classical characterization of supercompactness in [13] and similar characterizations of n-hugeness in [9, Section 6].In the following, we will say that a set M is Π n (P )-upwards correct for some natural number n > 0 and a class P if all Π n -formulas with parameters in M ∩ P that hold in M also hold in V .Definition 1.6.Given a natural number n > 0, an infinite cardinal κ is weakly n-exact for a cardinal λ > κ if for every A ∈ V λ+1 , there exists a transitive, Π n (V κ+1 )-upwards correct set M with V κ ∪ {κ} ⊆ M , a cardinal λ ′ ∈ C (n−1) greater than λ and an elementary embedding j : M → H λ ′ with j(κ) = λ and A ∈ ran(j).If we further require that j(crit(j)) = κ, then we say that κ is weakly parametrically n-exact for λ.
Observe that if κ is weakly parametrically 1-exact for λ, then κ and λ are both inaccessible.The following result shows how weakly n-exact cardinals are connected to principles of exact structural reflection for Π n -definable classes of structures.
Theorem 1.7.The following statements are equivalent for all cardinals κ and all natural numbers n > 0: (i) κ is the least regular cardinal such that (iv) κ is the least cardinal that is weakly n-exact for some λ > κ. (v) κ is the least cardinal that is weakly parametrically n-exact for some λ > κ.
In the case of Σ n+1 -definable classes of structures, the large cardinal principles corresponding to the different forms of exact structural reflection are the following: Definition 1.8.Given a natural number n, an infinite cardinal κ is n-exact for some cardinal λ > κ if for every A ∈ V λ+1 , there exists a cardinal κ ′ ∈ C (n) greater than κ , a cardinal λ ′ ∈ C (n+1) greater than λ, an elementary submodel X of H κ ′ with V κ ∪ {κ} ⊆ X, and an elementary embedding j : X → H λ ′ with j(κ) = λ and A ∈ ran(j).If we further require that j(crit(j)) = κ holds, 4 then we say that κ is parametrically n-exact for λ.
Note that, if m ≤ n < ω and κ is (parametrically) n-exact for λ, then κ is also (parametrically) m-exact for λ.Moreover, standard arguments show that, if κ is parametrically 0-exact for λ, then both κ and λ are inaccessible cardinals.In addition, it is easily seen that, given 0 < n < ω, if κ is (parametrically) n-exact for λ, then it is also weakly (parametrically) n-exact for λ.
The equivalence of the existence of n-exact cardinals with ESR principles for Σ n -definable classes of structures is given in the following theorem.
Theorem 1.9.The following statements are equivalent for all cardinals κ and all natural numbers n > 0: (i) κ is the least cardinal such that Σ n+1 -ESR(κ) holds.
(iii) κ is the least cardinal that is n-exact for some λ > κ.
(iv) κ is the least cardinal that is parametrically n-exact for some λ > κ.
The above results allow us to compare the large cardinal properties introduced in Definitions 1.6 and 1.8.More specifically, if κ is a cardinal satisfying the equivalent statements of Theorem 1.9 for some 0 < n < ω, then there exists a cardinal µ < κ satisfying the equivalent statements of Theorem 1.7 for the same natural number n (Lemma 7.1).This implication should be compared with the corresponding statements for the principle SR, showing that SR for Π n -definable classes of structures is equivalent to SR for Σ n+1 -definable classes ([1, Section 4]).
Exact cardinals are very strong, consistency-wise.In Section 8 we give lower and upper bounds for their consistency strength, and we also prove they imply the existence of wellknown large cardinals in the upper ranges of the large-cardinal hierarchy.Recall that a cardinal κ is almost huge (see [11]) if it is the critical point of an elementary embedding j : V → M , with M transitive and closed under sequences of length less than j(κ).Given such an embedding j, we then say that κ is almost huge with target j(κ).If κ is either parametrically 0-exact for λ, or weakly parametrically 1-exact for λ, then many cardinals smaller than κ are almost huge with target κ (Corollary 8.2).As for upper bounds, while every huge cardinal (with target some λ) is weakly parametrically 1-exact (for the same λ), the least huge cardinal κ is not 1-exact for any λ > κ (Propositions 8.4 and 8.6).A strong consistency upper bound is provided by an I3-embedding (see [11, §24]), for if j : V δ → V δ is such an embedding, then in V δ a proper class of cardinals are parametrically n-exact for unboundedly-many λ, for every n (Proposition 8.7).A much lower upper bound, namely an almost 2-huge cardinal, is given in Proposition 8.9 for the consistency of weakly parametrically n-exact cardinals, all n > 0.
Finally, in Section 9, we show how the principle ESR(κ, λ) can be strengthened to encompass increasing sequences of cardinals of length at most ω, instead of a single cardinal λ, in order to obtain principles of structural reflection that are much stronger, implying the existence of many-times huge cardinals or even I3-embeddings.The formulation of these stronger sequential ESR principles is motivated by the observation that the principle Π 1 -ESR(κ, λ) directly implies the instance (λ, κ) ։ (κ, <κ) of Chang's Conjecture, i.e. every structure A in a countable language with domain λ has an elementary substructure B of cardinality κ with |B ∩ κ| < κ.The definition of our sequential ESR principles will then directly imply that higher versions of Chang's Conjecture hold for the respective cardinals.
We then also strengthen, accordingly, the notions of weakly exact and exact cardinals to obtain large cardinal properties that correspond to the new sequential ESR principles and show that much of the theory developed for ESR(κ, λ) can be generalized to this stronger context.In particular, we obtain exact equivalences for the least cardinals witnessing the sequential forms of ESR and the corresponding sequential forms of weakly exact and exact cardinals (Theorems 9.8 and 9.11).As for determining the position of these large cardinals in the large cardinal hierarchy, we show, on the one hand, that the existence of a weakly 1exact or a 0-exact cardinal for a sequence of cardinals of length n + 1 implies the existence of smaller n-huge cardinals.On the other hand, every n-huge cardinal is weakly parametrically n-exact for some sequence of cardinals of length n (Proposition 9.3).Also, if κ is the critical point of an I1-embedding (see [11, §24]), then it is weakly parametrically 1-exact for a sequence of cardinals of length ω.
Many questions remain, and some of them are addressed in the last section of the article.Most interesting is the problem of determining the exact strength of the sequential forms of ESR.We know that these principles, in the case of sequences of length ω, are very strong, so much so that even when restricted to Σ 2 -definable classes of structures we don't know them to be consistent.This makes the study of such principles both challenging and exciting, for they appear to constitute a new class of large-cardinal principles that go beyond I1-embeddings, yet they may not fall into Kunen's Inconsistency.

Isomorphism-Closed classes
We start by studying instances of the principle ESR for classes of structures closed under isomorphic copies.Notice that if a class C of structures of the same type is Σ n -definable (with or without parameters) for some n > 0, then the closure of C under isomorphic copies is also Σ n -definable (with the same parameters, if any).
Proposition 2.1.Given uncountable cardinals κ < λ, the following statements are equivalent for every class C of structures of the same type that is closed under isomorphic copies: there exists an elementary embedding of a structure Proof.Assume that (i) holds and fix a structure B in C whose cardinality is contained in the interval [cof(λ), λ ].Then we can pick an injection i from the domain of B into V λ such that the set is unbounded in λ.Let B 0 denote the isomorphic copy of B induced by i.Then B 0 is a structure in C of rank λ and our assumptions yield an elementary embedding of a structure A in C of rank κ into B 0 .But this allows us to conclude that the cardinality of A is contained in the interval [cof(κ), κ ], and there exists an elementary embedding of A into B. Now, assume that (ii) holds and fix a structure B in C of rank λ.Then the cardinality of B is contained in the interval [cof(λ), λ ] and our assumption yields an elementary embedding of a structure A in C into B whose cardinality is contained in the interval [cof(κ), κ ].Pick an injection i from the domain of A into V κ with the property that the set is unbounded in κ, and let A 0 denote the isomorphic copy of A induced by i.Then A 0 is a structure in C of rank κ and there exists an elementary embedding of A 0 into B. Corollary 2.2.Let κ < λ be inaccessible cardinals and let C be a class of structures of the same type that is closed under isomorphic copies.Then ESR C (κ, λ) holds if and only if for every structure B ∈ C of cardinality λ, there exists an elementary embedding of a structure A ∈ C of cardinality κ into B.
Corollary 2.3.Let C be a class of structures of the same type that is closed under isomorphic copies and let κ < µ < λ be infinite cardinals with the property that ESR C (κ, λ) holds.
, both statements follow directly from Proposition 2.1.

Low complexities
In this section, we study exact structural reflection for Σ 1 -definable classes of structures.In the case of classes closed under isomorphic copies, these principles are provable in ZFC.Proposition 3.1.If κ is an uncountable cardinal, then the principle Σ 1 (V κ ) ic -ESR(κ, λ) holds for every cardinal λ > κ with cof(κ) ≤ cof(λ).
Proof.Fix a Σ 1 -formula ϕ(v 0 , v 1 ) and z ∈ V κ such that C = {A | ϕ(A, z)} is a class of structures of the same type and pick a structure B in C whose cardinality is contained in the interval [cof(λ), λ ].Let B 0 be an isomorphic copy of B in H + λ and pick an elementary substructure X of H + λ of cardinality κ with V κ ∪ {κ, B 0 } ⊆ X.Let π : X → M denote the induced transitive collapse.Since π(z) = z, Σ 1 -absoluteness now implies that ϕ(π(B 0 ), z) holds and hence π(B 0 ) is an element of C.Moreover, our construction ensures that π(B 0 ) has cardinality at most κ and, since B 0 has cardinality at least cof(λ) ≥ cof(κ) and π(cof(κ)) = cof(κ), we know that π(B 0 ) has cardinality at least cof(κ).Finally, using the inverse collapse π −1 , it is easy to see that there exists an elementary embedding of π(B 0 ) into B. By Proposition 2.1, the above computations yield the desired conclusion.
In contrast with the previous Proposition, the principle ESR C (κ, λ) for some κ < λ and all Σ 0 -definable (without parameters) classes C of structures of the same type (so, no closure under isomorphic copies required), has considerable large-cardinal strength and fails in Gödel's constructible universe L. Lemma 3.2.If Σ 0 -ESR(κ) holds for some uncountable cardinal κ, then a # exists for every real a.
Proof.Let L denote the first-order language that extends the language L ∈ of set theory by a binary predicate symbol Ė, a constant symbol ċ and a unary function symbol ḟ .Define C to be the class of all L-structures of the form ν, ∈, E, α, f with the property that ν is an ordinal and f : ν, ∈ → ran(f ), E is an order-isomorphism.Then it is easy to see that C is definable by a Σ 0 -formula without parameters.Now, fix a real a and a cardinal λ > κ such that ESR C (κ, λ) holds.Pick a bijection b : L λ [a] → λ and set By our assumption, there exists a binary relation R on κ, a function f : κ → κ and α < κ such that is a structure in C with the property that there exists an elementary embedding i of A into B. Since our construction ensures that λ, E is well-founded and κ, R embeds into λ, E , it follows that κ, R is well-founded too.Moreover, elementarity implies that κ, R is extensional.Let π : κ, R → M, ∈ denote the corresponding transitive collapse and set Then j is an elementary embedding of transitive structures.Now, note that elementarity implies that ran(f is an order-isomorphism.But this shows that is also an order-isomorphism and hence we can conclude that M ∩ Ord = κ.In particular, elementarity implies that Finally, since j(π(α)) = κ > π(α), we know that j : L κ [a] → L λ [a] is a non-trivial elementary embedding.But then |crit(j)| < κ and the proof of [10,Theorem 18.27] shows that a # exists.
The next result provides an upper bound for the consistency strength of the assumption of Lemma 3.2.In particular, it shows that this assumption does not imply the existence of an inner model with a measurable cardinal.Its proof is based on arguments contained in the proof of [16,Theorem 2.3].Lemma 3.3.If δ is a Ramsey cardinal, then the set of inaccessible cardinals κ < δ with the property that Σ 1 (V κ )-ESR(κ) holds in V δ is unbounded in δ.
Claim.In V δ , the principle ESR C (κ, λ) holds for every class C of structures of the same type that is definable by a Σ 1 -formula with parameters in V κ .
Proof of the Claim.Assume, towards a contradiction, that there exists a Σ 1 -formula ϕ(v 0 , v 1 ) with the property that for some z ∈ V κ , the class C = {A ∈ V δ | ϕ(A, z)} consists of structures of the same type and there exists B ∈ C of rank λ such that for all A ∈ C of rank κ, there is no elementary embedding of A into B. Using elementarity, we now know that, in M , there exist z 0 ∈ V min(I) and B 0 ∈ V κ+1 \ V κ with the property that the class C 0 = {A ∈ M | ϕ(A, z 0 )} consists of structures of the same type, B 0 ∈ C 0 and for all A ∈ C 0 of rank min(I), there is no elementary embedding of A into B 0 .Since we have , the elementarity of π −1 implies that C * = {A ∈ V δ | ϕ(A, z 0 )} consists of structures of the same type, π −1 (B 0 ) ∈ C * and for all A ∈ C * of rank κ, there is no elementary embedding of A into π −1 (B 0 ).But this yields a contradiction, because the upwards absoluteness of Σ 1 -formulas implies that B 0 is a structure in C * of rank κ and π −1 induces an elementary embedding of B 0 into π −1 (B 0 ).
The above claim completes the proof of the lemma.

The Π n -case for isomorphism-closed classes
We now show that the structural reflection principles introduced in Definition 1.1 become very strong when they hold for more complex classes of structures.In particular, the validity of these principles for Π 1 -definable classes C of structures closed under isomorphic copies at singular cardinals already implies non-trivial fragments of supercompactness, and the corresponding principles for a regular cardinal will turn out to imply the existence of many almost huge cardinals.
Recall that a cardinal κ is λ-supercompact if there is a transitive class M closed under λ-sequences and an elementary embedding j : V → M with crit(j) = κ and j(κ) > λ.
Proof.Fix a Π 1 -formula ϕ(v 0 , v 1 ) and an element z of V µ such that C = {A | ϕ(A, z)} is a class of structures of the same type that is closed under isomorphic copies.Pick a structure B in C whose cardinality is contained in the interval [cof(λ), λ ] and whose domain is a subset of λ .Now, fix an elementary embedding j : V → M with crit(j) = µ, j(µ) > λ and λ M ⊆ M .Then the closure properties of M ensure that B is an element of M and the elementary embedding of B into j(B) induced by j is also contained in M .Moreover, Π 1 -downwards absoluteness for transitive classes implies that ϕ(B, z) holds in M .Thus, M satisfies that there exists a structure A whose cardinality is contained in the interval [cof(κ), j(µ)) such that ϕ(A, z) holds, and there exists an elementary embedding e : A → j(B), as this is witnessed by B and j ↾ B : B → j(B).Since j(z) = z and j(cof(κ)) = cof(κ), the elementarity of j yields an elementary embedding of a structure A of the given type into B such that ϕ(A, z) holds and the cardinality of A is contained in the interval [cof(κ), µ).This shows that there is an elementary embedding of a structure in C whose cardinality is contained in the interval [cof(κ), κ ] into B. By Proposition 2.1, this proves the lemma.
Lemma 4.3.Let n > 0 be a natural number and let κ < λ be infinite cardinals such that κ is singular and cof(κ ) and let j : V λ +1 → V η be an elementary embedding with crit(j) = δ, j(µ) ≥ λ + 1, and j(ν) ∈ C (n) .Then the fact that λ < j(µ) ≤ j(ν) implies that B is an element of V j(ν) and the elementary embedding of B into j(B) induced by j is contained in V η .Moreover, since j(ν) ∈ C (n) , by Π n+1 -downwards absoluteness for V j(ν) , we have that ϕ(B, z) holds in V j(ν) .Thus, V η satisfies that there exists a structure A whose cardinality is contained in the interval [cof(κ), j(µ)) such that ϕ(A, z) holds in V j(ν) , and there exists an elementary embedding e : A → j(B), as this is witnessed by B and j ↾ B : B → j(B).Since j(z) = z and j(cof(κ)) = cof(κ), the elementarity of j yields an elementary embedding of a structure A of the given type into B such that ϕ(A, z) holds in V ν and the cardinality of A is contained in the interval [cof(κ), µ).But since ν ∈ C (n+1) , we know that ϕ(A, z) also holds in V .This shows that there is an elementary embedding of a structure in C whose cardinality is contained in the interval [cof(κ), κ ] into B. By Proposition 2.1, this proves the lemma.
The following observation shows that n) , and since every true Σ n+1 statement, with parameters in V κ is true in V ν , the assumption that κ is λ-C (n) -extendible easily yields that κ ∈ C (n+1) .Then κ itself witnesses the [κ, λ)-C (n)extendibility of κ.
(ii): As shown in [15], a cardinal κ is C (n) -extendible if and only if it is C (n) + -extendible, i.e., for a proper class of λ ∈ C (n) there exists an elementary embedding j : Corollary 4.5.Let n > 0 be a natural number, let κ be a singular cardinal and let λ > κ be a cardinal with cof(κ) ≤ cof(λ). (i Proof.The first statement follows directly from Lemma 4.3.For the second statement, notice that our assumption implies that κ is a limit of ( λ + 1)-C (n) -extendible cardinals δ with the property that C (n+1) ∩ (δ, κ) = ∅.By Proposition 4.4, this implies that κ is a limit of cardinals δ that are [δ, λ + 1)-C (n) -extendible and therefore we can use the first part to derive the desired conclusion.Finally, since [1,Proposition 3.4] shows that all C (n)extendible cardinals are elements of C (n+2) , we can apply the second part of the corollary to prove the third statement.
We are now ready to show that Vopěnka's Principle can be characterized through principles of exact structural reflection.
Proof of Theorem 1.5.First, assume that (i) holds and fix a natural number n > 0. Since [3, Corollary 6.9] shows that our assumption implies the existence of a proper class of C (n)extendible cardinals, there exists a proper class of cardinals κ of countable cofinality that are limits of C (n) -extendible cardinals.The third part of Corollary 4.5 now implies that for every such cardinal κ and every λ > κ, the principle Π n+1 (V κ ) ic -ESR(κ, λ) holds.This shows that (iii) holds in this case.Now, assume that (ii) holds and let C 0 be a proper class of structures of the same type.Let C denote the class of all structures of the given type that are isomorphic to a structure in C 0 and define C = {|A| | A ∈ C}.By our assumptions, there exists a cardinal κ with the property that ESR C (κ, λ) holds for all λ > κ.If C is a proper class, then there exists a structure B ∈ C 0 of cardinality greater than κ and, since the principle ESR C (κ, |B|) holds, we can use Proposition 2.1 to find a structure A ∈ C 0 of cardinality at most κ and an elementary embedding of A into B. In particular, we know that Vopěnka's Principle for C holds in this case.In the other case, namely if C is a set, then we can find distinct A, B ∈ C 0 that are isomorphic and hence Vopěnka's Principle for C also holds in this case.This allows us to conclude that (i) holds.
This concludes the proof of the theorem, because (iii) obviously implies (ii).
Similar results hold also for Σ n (V κ )-definable classes closed under isomorphic copies, assuming κ is a singular limit of supercompact cardinals, in the case n = 2, or a singular limit of [κ, ∞)-C (n−2) -extendible cardinals, in the case n > 2.
Corollary 4.6.Let κ be a singular cardinal and let λ > κ be a cardinal with cof(κ) ≤ cof(λ). (i Proof.(i) Assume that κ is a limit of supercompact cardinals and fix z } is a class of structures of the same type that is closed under isomorphic copies.Pick a structure B in C whose cardinality is contained in the interval [cof(λ), λ ].We can use our assumption to find cardinals µ < κ and θ ≥ λ such that z ∈ V µ and there exists a transitive class M containing B and closed under θsequences, and an elementary embedding j : V → M with crit(j) = µ, j(µ) > θ and the property that ϕ(B, z) holds in M .In this situation, we can repeat the proof of Lemma 4.1 to find a structure A in C whose cardinality is contained in the interval [cof(κ), κ ] and an elementary embedding of A into B.
(ii) Now, assume that κ is a limit of [κ, ∞)-C (n) -extendible cardinals.Fix a Σ n+2 -formula ϕ(v 0 , v 1 ) and z ∈ V κ that define a class C of structures of the same type closed under isomorphic copies.If we now pick a structure B ∈ C whose cardinality is contained in the interval [cof(λ), λ ], then there is a cardinal λ < θ ∈ C (n+2) , a cardinal ν ∈ C (n+1) ∩ [κ, θ) and an elementary embedding j : V θ → V η for some η such that z ∈ V crit(j) , j(κ) ≥ θ and j(ν) ∈ C (n) .Using the fact that θ ∈ C (n+2) , we can now continue as in the proof of Lemma 4.3 to obtain the desired elementary embedding.
We shall next prove several results that will allow us to derive high lower bounds for the consistency strength of the principle Π ic 1 -ESR.In particular, these results will show that passing from Σ 1 -definable to Π 1 -definable classes of structures drastically increases the strength of the principle ESR.
Given a set z and a natural number n > 0, we let W n (z) denote the class of all structures (in the language L ∈ of set theory extended by five constant symbols and two unary function symbols) of the form D, E, a, b, c, d, e, f, g with the property that the relation E is wellfounded and extensional, and, if π : D, E → M, ∈ denotes the corresponding transitive collapse, then the following statements hold: (vi) The map g • π −1 restricts to a bijection between π(d) and D. Then it is easy to see that the class W n (z) is closed under isomorphic copies and is definable by a Π n -formula with parameter z.Lemma 4.7.Let µ < λ be cardinals such that ESR Wn(z) (µ, λ) holds for some element z of V µ and some natural number n > 0. Given a cardinal λ ′ ∈ C (n) greater than λ , there exist
The next results show that the assumptions of Corollary 4.2 are close to optimal.Lemma 4.9.Let µ < λ be cardinals and let α < µ be an ordinal.
(i) If Π 1 ({α}) ic -ESR(κ, λ) holds for a proper class of cardinals λ, then the interval A combination of the above results now yield short proofs of Theorems 1.2 and 1.4 stated in the Introduction.
Proof of Theorem 1.2.Let κ be a singular cardinal.If κ is a limit of supercompact cardinals, then Corollary 4.6 shows that Σ 2 (V κ ) ic -ESR(κ, λ) holds for a proper class of cardinals λ.In the other direction, if Π 1 (κ) ic -ESR(κ, λ) holds for a proper class of cardinals λ, then we can apply Corollary 4.10 to show that κ is a limit of supercompact cardinals.
We will eventually show that the strength of the principle Π ic 1 -ESR(κ) further increases significantly if κ is a regular cardinal.More specifically, we will show (see Corollary 8.2 below and Theorem 1.7 stated in the Introduction) that this assumption implies the existence of an almost huge cardinal.The next lemma is the starting point of this analysis.It will allow us to show that the least regular cardinal µ satisfying Π ic n -ESR(µ) coincides with the least cardinal ν satisfying Π n (V ν )-ESR(ν).Lemma 4.11.Given a natural number n > 0, assume that • M is a transitive, Π n (V κ+1 )-upwards correct set with V κ ∪ {κ} ⊆ M , and Then the following statements hold: we know that crit(j) ≤ θ and this allows us to apply Proposition 4.8 to show that θ < µ < ν.Fix a Π n -formula ϕ(v 0 , v 1 ) with the property that the class C = {A | ϕ(A, z)} consists of structures of the same type.Assume, towards a contradiction, that there exists B ∈ C of rank ν with the property that for all A ∈ C of rank µ, there exists no elementary embedding of A into B. Since ν ≤ λ < λ ′ , we know that B ∈ H λ ′ and the fact that λ ′ ∈ C (n) implies that ϕ(B, z) holds in H λ ′ and for every structure A of the given type and rank µ, either ϕ(A, z) fails in H λ ′ or H λ ′ contains no elementary embedding of A into B. The elementarity of j then yields a structure B 0 ∈ M of the given type and rank µ such that ϕ(B 0 , z) holds in M and for every structure A of the given type and rank θ, either ϕ(A, z) fails in M or M contains no elementary embedding of A into B 0 .Our setup then ensures that B 0 is an element of C of rank µ and the embedding j gives rise to an elementary embedding i of B 0 into j(B 0 ).But this yields a contradiction to the elementarity of j, because B 0 and i are both contained in H λ ′ , and the fact that M is correct about the sentence ϕ(B 0 , z), and λ ′ ∈ C (n) , implies that ϕ(B 0 , u) holds in H λ ′ .
(ii) Set ν = j(µ) > µ and fix a Π n -formula ϕ(v 0 , v 1 ).Assume, towards a contradiction, that there exists z ∈ V µ with the property that C = {A | ϕ(A, z)} is a class of structures of the same type and there exists B ∈ C of rank ν such that for all A ∈ C of rank µ, there is no elementary embedding of A into B. Then the fact that ν ≤ λ ≤ λ < λ ′ ∈ C (n) implies that this statement also holds in H λ ′ , and hence, in M , there exists z 0 ∈ V crit(j) with the property that C 0 = {A | ϕ(A, z 0 )} is a class of structures of the same type and there exists B ∈ C 0 of rank µ such that for all A ∈ C 0 of rank crit(j), there is no elementary embedding of A into B. Since M is Π n (V κ+1 )-upwards correct, λ ′ ∈ C (n) and j(z 0 ) = z 0 , we know that ϕ(B, z) and ϕ(j(B), z 0 ) hold in H λ ′ .So, since µ ∈ (crit(j), κ], we can now proceed as in the proof of (i) to derive a contradiction.Lemma 4.12.Given a natural number n > 0, let µ be a regular cardinal with the property that Π ic n -ESR(µ, λ) holds for some cardinal λ > µ.Then there exists an inaccessible cardinal δ ≤ µ with the property that Π n (V δ )-ESR(δ, ρ) holds for an inaccessible cardinal ρ with δ < ρ ≤ λ.

The Π n -case for arbitrary classes
In order to study principles of exact structural reflection for Π n -definable classes of structures that are not necessarily closed under isomorphic copies, we analyse connections between the validity of these principles and the existence of weakly n-exact cardinals.The following variation of Definition 1.6 will allow us to state our results more precisely: Definition 5.1.Given cardinals κ < λ, a set z ∈ V κ and a natural number n > 0, we say that κ is weakly n-exact for λ and z if for every A ∈ V λ+1 , there exists a transitive, 1) greater than λ and an elementary embedding j : M → H λ ′ with j(κ) = λ, j(z) = z and A ∈ ran(j).
Proposition 5.2.If a cardinal κ is weakly parametrically n-exact for some cardinal λ, then κ is weakly n-exact for λ and all z ∈ V κ .
greater than λ and an elementary embedding j : M → H λ ′ with j(κ) = λ, j(z) = z and B ∈ ran(j).Pick A ∈ M with j(A) = B. Since all Π n -formulas with parameters in H λ ′ are downwards absolute from V to H λ ′ , we know that ϕ(B, z) holds in H λ ′ and the structure B has rank λ in H λ ′ .But this means that, in M , the statement ϕ(A, z) holds and A has rank κ.The fact that M is Π n (V κ+1 )-upwards correct implies that the statement ϕ(A, z) holds in V and hence we can conclude that A is a structure in C of rank κ.Moreover, we know that the embedding j induces an elementary embedding of A into B. The second part of the proposition follows directly from Proposition 5.2.
For each natural number n > 0 and every set z, let E n (z) denote the class of structures D, E, a, b, c (in the language of set theory with three additional constant symbols) with the property that rnk(D) ⊆ D, E is a well-founded and extensional relation on D and, if Note that the class E n (z) is definable by a Π n -formula with parameter z.Lemma 5.4.Let κ < λ be cardinals, let z ∈ V κ and let n > 0 be a natural number with the property that ESR En(z) (κ, λ) holds.Given B ∈ V λ+1 and a cardinal λ ′ ∈ C (n) greater than λ, 5 there exists a transitive, Π n (V κ+1 )-upwards correct set M with V κ ∪ {κ} ⊆ M and an elementary embedding j : M → H λ ′ with j(κ) = λ, j(z) = z and B ∈ ran(j).In particular, the cardinal κ is weakly n-exact for λ and z.
Proof.Let X be an elementary submodel of H λ ′ of cardinality λ with V λ ∪ {λ, B} ⊆ Y .Pick a bijection f : X → V λ with f ↾ λ = id λ and let R be the induced binary relation on V λ .Since the transitive collapse of V λ , R is the composition of f −1 and the transitive collapse of X, ∈ and all Σ n -formulas using parameters from the transitive part 5 Since the class C (1) consists of all cardinals µ with the property that Hµ = Vµ, the given assumption ensures that λ ′ is greater than λ . of X are absolute between V and the transitive collapse of X, it follows that the structure V λ , R, f (B), f (λ), f (z) is an element of E n (z) of rank λ.
By our assumptions, we can find a structure D, E, a, b, c of rank κ in E n (z) such that there exists an elementary embedding Let π : D, E → M, ∈ denote the corresponding transitive collapse.Set A = π(a) and Then M is a transitive set with V κ ∪ {κ} ⊆ M , A ∈ M ∩ V κ+1 with the property that M is Π n (V κ+1 )-upwards correct and j is an elementary embedding with j(κ) = λ, j(A) = B and j(z) = z.
Arguments contained in the proofs of the above results also allow us to prove the following parametrical version of Corollary 5.5 that will be needed later on.Lemma 5.6.Let n > 0 be a natural number, and let κ be weakly parametrically n-exact for some cardinal λ > κ.If λ < λ ′ ∈ C (n) and B ∈ V λ+1 , then there exists a transitive, Π n (V κ+1 )-upwards correct set M with V κ ∪ {κ} ⊆ M and an elementary embedding j : M → H λ ′ satisfying j(crit(j)) = κ, j(κ) = λ and B ∈ ran(j).
In the remainder of this section, we prove that for all natural numbers n > 0, exact structural reflection for Π n -definable classes at cardinals in C (n) is strictly stronger than exact structural reflection for Σ n -definable classes at such cardinals.This will follow from Corollary 5.5 and the next Lemma.Lemma 5.7.Let n > 0 be a natural number and let κ ∈ C (n) be weakly parametrically nexact for some cardinal λ > κ.Then the set of cardinals µ < κ such that Σ n (V µ )-ESR(µ, κ) holds is stationary in κ.
Proof.Fix a closed unbounded subset K of κ.By Lemma 5.6, there exists a transitive, Π n (V κ+1 )-upwards correct set M with V κ ∪ {κ} ⊆ M , cardinals κ < λ < λ ′ ∈ C (n) and an elementary embedding j : M → H λ ′ with j(crit(j)) = κ, j(κ) = λ and K ∈ ran(j).Then crit(j) ∈ K. Assume, towards a contradiction, that Σ n (V crit(j) )-ESR(crit(j), κ) fails.Then there exists a Σ n -formula ϕ(v 0 , v 1 ), z ∈ V crit(j) and a structure B of rank κ with the property that the class C = {A | ϕ(A, z)} consists of structures of the same type, ϕ(B, z) holds and for all A ∈ C of rank crit(j), there is no elementary embedding of A into B. Since κ is inaccessible and therefore every elementary embedding of a structure of rank less than κ into a structure of rank κ is an element of V κ , and moreover κ ∈ C (n) , there is a Σ n -formula with parameters in V κ ∪ {V κ } that states that there exists a structure B of rank κ with the property that ϕ(B, z) holds and for every structure A ∈ V κ of rank crit(j) such that ϕ(A, z) holds in V κ , there is no elementary embedding from A into B in V κ .Since M is Π n (V κ+1 )upwards correct and V κ ∪ {V κ } ⊆ M , this yields a structure B 0 of rank κ in M with the property that ϕ(B 0 , z) holds in M and for every structure A of rank crit(j) such that ϕ(A, z) holds in V κ , there is no elementary embedding of A into B 0 in V κ .Moreover, since M is Π n (V κ+1 )-upwards correct and κ is an inaccessible cardinal in C (n) , it follows that for every structure A of rank crit(j) such that ϕ(A, z) holds in M , there is no elementary embedding of A into B 0 in M .Then, in H λ ′ , for every structure A of rank κ such that ϕ(A, z) holds H λ ′ , there is no elementary embedding of A into j(B 0 ) in V λ .Since κ < λ < λ ′ ∈ C (n) , it now follows that ϕ(B 0 , z) holds in H λ ′ and the map j ↾ B 0 : B 0 → j(B 0 ) is an element of H λ ′ .But this yields a contradiction, because j ↾ B 0 is an elementary embedding of B 0 into j(B 0 ) in H λ ′ .
Proof.Pick a Σ n+1 -formula ϕ(v 0 , v 1 ) with the property that the class C = {A | ϕ(A, z)} consists of structures of the same type and fix B ∈ C of rank λ.By our assumptions, there exists a cardinal κ < κ ′ ∈ C (n) , a cardinal λ < λ ′ ∈ C (n+1) , an elementary submodel X of H κ ′ with V κ ∪ {κ} ⊆ X and an elementary embedding j : X → H λ ′ with j(κ) = λ, j(z) = z and B ∈ ran(j).Pick A ∈ X with j(A) = B. Then our setup ensures that ϕ(A, z) holds and hence A is a structure in C of rank κ.Moreover, the map j induces an elementary embedding of A into B.These computations yield the first part of the proposition.The second part follows directly from a combination of the first part and Proposition 6.2.
For each natural number n > 0 and every set z, we let D n (z) denote the class of structures D, E, a, b, c (in the language of set theory with three additional constant symbols) with the property that for some cardinal θ ∈ C (n) greater than rnk(D) , there exists an elementary submodel X of H θ with V rnk(D) ∪ {rnk(D)} ⊆ X and an isomorphism π : D, E → X, ∈ with π(b) = z and π(c) = rnk(D).Note that the class D n (z) is definable by a Σ n+1 -formula with parameter z.Lemma 6.4.Let κ < λ be cardinals, let z ∈ V κ and let n > 0 be a natural number with the property that ESR Dn(z) (κ, λ) holds.Then κ is n-exact for λ and z.
Proof.Fix A ∈ V λ+1 , λ ′ ∈ C (n+1) greater than λ and an elementary submodel Y of H λ ′ of cardinality λ with V λ ∪ {A, λ} ⊆ Y .Pick a bijection f : Y → V λ and let R be the binary relation on V λ induced by f and ∈.Then λ ′ and f −1 witness that V λ , R, f (A), f (z), f (λ) is a structure of rank λ in D n (z).By our assumptions, there exists a structure D, E, a, b, c of rank κ in D n (z) and an elementary embedding Then j is an elementary embedding with j(κ) = λ, j(z) = z and A = j(π(a)) ∈ ran(j).

Proofs of the main theorems
We shall now give a proof of Theorems 1.7 and 1.9.
Proof of Theorem 1.7.Fix a cardinal κ and a natural number n > 0.
The next two claims, together with Proposition 5.3 and Corollary 5.5, will allow us to conclude that all statements (i)-(v) listed in the theorem are equivalent.
Claim.If κ is the least regular cardinal with the property that Π ic n -ESR(κ) holds, then κ is weakly parametrically n-exact for some λ > κ.
Claim.If κ is the least cardinal that is weakly n-exact for some cardinal λ, then κ is regular.
By the first claim above, the cardinal that satisfies (i) of the Theorem is greater than or equal to the cardinal that satisfies (v), which, by Proposition 5.3, is greater than or equal to the cardinal that satisfies (iii).Moreover, the cardinal satisfying (iii) is obviously greater than or equal to the cardinal that satisfies (ii), and an application of Corollary 5.5 then shows that the cardinal satisfying (ii) is greater than or equal to the cardinal that satisfies (iv).Our second claim then shows that the cardinal satisfying (iv) is regular and this allows us to use Corollary 5.5 again to conclude that it is greater than or equal to the cardinal satisfying (i).This shows that all of these cardinals are equal.
Proof of Theorem 1.9.Fix a cardinal κ and a natural number n > 0.
The next two claims, together with Proposition 6.3, will allow us to conclude that all statements (i)-(iv) listed in the theorem are equivalent.
Claim.If κ is the least cardinal with the property that Σ n+1 -ESR(κ) holds, then κ is parametrically n-exact for some λ > κ.
Proof of the Claim.Fix D ∈ V λ+1 .Using Lemma 6.4, we find a cardinal κ < κ ′ ∈ C (n) , cardinals κ < λ < λ ′ ∈ C (n+1) , an elementary submodel X of H κ ′ with V κ ∪ {κ} ⊆ X and an elementary embedding j : X → V λ ′ with j(κ) = λ and D, κ ∈ ran(j).Set µ = j(crit(j)) ≤ κ ∈ X and ν = j(µ).Assume, towards a contradiction, that Σ n+1 -ESR(µ, ν) fails.Then we can find a Σ n+1 -formula ϕ with the property that the class C = {A | ϕ(A)} consists of structures of the same type and B ∈ C of rank ν with the property that for all A ∈ C of rank µ, there is no elementary embedding of A into B. Since λ ′ ∈ C (n+1) , this statement also holds in H λ ′ and therefore the elementarity of j allows us to find B 0 ∈ X of rank µ with the property that ϕ(j(B 0 )) holds in H λ ′ and for all A ∈ V µ+1 \ V µ such that ϕ(A) holds, there is no elementary embedding of A into j(B 0 ) in H λ ′ .But this yields a contradiction, because our setup ensures that ϕ(B 0 ) holds in V and the fact that V κ ⊆ X implies that j induces an elementary embedding of B 0 into j(B 0 ) that is an element of H λ ′ .In this situation, the minimality of κ implies that κ = µ and ν = λ.In particular, we can conclude that κ is parametrically n-exact for λ.
Since Proposition 6.3 shows that Σ n+1 (V κ )-ESR(κ) holds whenever κ is parametrically n-exact for some cardinal λ > κ and therefore all statements listed in the theorem imply that Σ n+1 -ESR(κ) holds, the above claim allows us to conclude that all of the listed statements are equivalent.
In the remainder of this section, we show that, for all n > 0, exact structural reflection for Σ n+1 -definable classes is strictly stronger than exact structural reflection for Π n -definable classes.In combination with the equivalences provided by Theorems 1.7 and 1.9, the following lemma shows that, in general, the principle Π n (V κ )-ESR(κ) does not imply the principle Σ n+1 -ESR(κ).This statement should be compared with the results of [1, Section 4], showing that the validity of the principle SR for Π n -definable classes of structures is equivalent to the validity of this principle for Σ n+1 -definable classes.
Lemma 7.1.Let n > 0 be a natural number and let κ be parametrically n-exact for some cardinal λ > κ.Then the set of cardinals µ < κ with the property that Π n (V µ )-ESR(µ, κ) holds is stationary in κ.In particular, there exists a cardinal µ < κ that is weakly parametrically n-exact for some cardinal ν > µ.
Proof.Fix a closed unbounded subset K of κ.Using our assumptions, we can find a cardinal κ < κ ′ ∈ C (n) , a cardinal λ < λ ′ ∈ C (n+1) , an elementary submodel X of H κ ′ with V κ ∪ {κ} ⊆ X and an elementary embedding j : X → H λ ′ with j(κ) = λ, j(crit(j)) = κ and K ∈ ran(j).Then crit(j) is an element of K. Now, assume towards a contradiction, that there is a Π n -formula ϕ(v 0 , v 1 ) and z ∈ V crit(j) such that the class C = {A | ϕ(A, z)} consists of structures of the same type and there exists B ∈ C of rank κ such that for all A ∈ C of rank crit(j), there is no elementary embedding from A into B. Since κ < κ ′ ∈ C (n) , these statements hold in H κ ′ and we can find B ∈ C ∩X of rank κ with the property that for every A ∈ C of rank crit(j), there is no elementary embedding of A into B. In this situation, since λ < λ ′ ∈ C (n+1) , elementarity implies that j(B) is an element of C and for every A ∈ C of rank κ, there is no elementary embedding of A into j(B).But this yields a contradiction, because the fact that V κ is a subset of X implies that j induces an elementary embedding of B into j(B).
The above computations yield the first part of the lemma.The second part follows directly from a combination of the first part with Theorem 1.7.

The strength of exact cardinals
In this section, we measure the strength of the principles of exact structural reflection introduced above by positioning exact and weakly exact cardinals in the hierarchy of large cardinals.We start by deriving lower bounds for their consistency strength by showing that the existence of such cardinals implies the existence of many almost huge cardinals below them.
Recall that a cardinal κ is almost huge if there exists a transitive class M and a nontrivial elementary embedding j : V → M with crit(j) = κ and <j(κ) M ⊆ M .We then say that a cardinal κ is almost huge with target λ if there exists an embedding j witnessing the hugeness of κ with j(κ) = λ.
The following standard argument will allow us to prove these implications: Lemma 8.1.Let κ < λ be cardinals with the property that there exists a non-trivial elementary embedding j : V κ → V λ with j(crit(j)) = κ.Then crit(j) is almost huge with target κ.
Since normality allows us to conclude that [a → ot(a)] Uγ = γ, we know that By [11,Theorem 24.11], this shows that µ is almost huge with target κ.Corollary 8.2.Let κ < λ be cardinals with the property that κ is either parametrically 0-exact for λ or weakly parametrically 1-exact for λ.Then the set of cardinals µ < κ with the property that µ is almost huge with target κ is stationary in κ.
Proof.Let C be a closed unbounded subset of κ.By definition, both of the listed assumption imply the existence of a non-trivial elementary embedding j : V κ → V λ with j(crit(j)) = κ and C ∈ ran(j).Then crit(j) is an element of C and Lemma 8.1 shows that crit(j) is almost huge with target κ.Corollary 8.3.Let κ be a cardinal that is parametrically 0-exact for some cardinal λ > κ.Then κ is almost huge with target λ.
Proof.By definition, there exist a cardinal κ ′ > κ, a cardinal λ < λ ′ ∈ C (1) , an elementary submodel X of H κ ′ with V κ ∪ {κ} ⊆ X and an elementary embedding j : X → H λ ′ with j(κ) = λ and j(crit(j)) = κ.Then Lemma 8.1 implies that crit(j) is almost huge with target κ.Since the system of filters witnessing this statement is contained in H κ ′ , the model X also contains such a system.But then the elementarity of j implies that, in H λ ′ , there is a system of ultrafilters witnessing that κ is almost huge with target λ.Since λ < λ ′ ∈ C (1) , this statement also holds in V .
Recall that a cardinal κ is huge if there exists a transitive class M and a non-trivial elementary embedding j : V → M with crit(j) = κ and j(κ) M ⊆ M .We then say that a cardinal κ is huge with target λ if there exists an embedding j witnessing the hugeness of κ with j(κ) = λ.It is well-known that κ is huge with target λ if and only if there exists a κ-complete normal ultrafilter U over P(λ) such that {x ∈ P(λ) | otp (x) = κ} ∈ U (see [11,Theorem 24.8]).
Proof.Let M be an inner model with λ M ⊆ M and let j : V → M be an elementary embedding with crit(j) = κ and j(κ) = λ.Fix A ∈ V λ+1 and let N be an elementary submodel of H λ + of cardinality λ with V λ ∪ {A, λ} ⊆ N .We then have N ∈ M and, since Then j 0 is an elementary embedding that is an element of M .Thus, in M , there exists a transitive, Π 1 (V j(κ)+1 )-upwards correct set K with V j(κ) ∪{j(κ)} ⊆ K (namely N) and an elementary embedding k : K → H j(λ + ) with k(crit(k)) = j(κ), k(j(κ)) = j(λ) and j(A) ∈ ran(k) (namely j 0 ).Hence, the elementarity of j implies that, in V , there exists a transitive, Π 1 (V κ+1 )-upwards correct set K with V κ ∪ {κ} ⊆ K and an elementary embedding k : This result also allows us to show that the consistency strength of huge cardinals is strictly larger than the consistency strength of weakly 1-exact cardinals.
Proof.Let j : V → N be an elementary embedding with critical point κ such that j(κ) = λ and N is closed under λ-sequences.Note that H λ + ∈ V N j(λ) .Fix a Π 1 -formula ϕ(v 0 , v 1 ) and z ∈ V κ with the property that, in V N j(λ) , the class {A | ϕ(A, z)} consists of structures of the same type.Pick a structure B of rank λ with the property that ϕ(B, z) holds in V N j(λ) .Since H λ + ⊆ V N j(λ) , Σ 1 -absoluteness implies that ϕ(B, z) also holds in V .By our assumptions, we can now apply Proposition 8.4 to find a structure A of rank κ with the property that ϕ(A, z) holds in V and an elementary embedding i of A into B.But then A is contained in V N j(λ) , ϕ(A, z) holds in V N j(λ) and the map i is an element of V N j(λ) .These computations show that the principle Π 1 (V κ )-ESR(κ, λ) holds in V N j(λ) .Using the elementarity of j, we can now conclude that, in V λ , there is an inaccessible cardinal ρ < κ with the property that Π 1 (V ρ )-ESR(ρ, κ) holds .
We now show that the implication given by Proposition 8.4 is optimal.Proposition 8.6.If κ is the least huge cardinal, then κ is not 1-exact for any cardinal λ > κ.
Proof.Assume, towards a contradiction, that there exists a cardinal κ < κ ′ ∈ C (1) , cardinals λ < λ ′ ∈ C (2) , an elementary submodel X of H κ ′ with V κ ∪ {κ} ⊆ X and an elementary embedding j : X → H λ ′ with j(κ) = λ.Note that the statement "There exists a huge cardinal smaller than λ" can be formulated by a Σ 2 -formula with parameter λ and, since λ < λ ′ ∈ C (2) , this statement holds in H λ ′ .But then the elementarity of j and the fact that κ ′ ∈ C (1) allow us to conclude that there exists a huge cardinal smaller than κ, a contradiction.
The next proposition gives a consistency upper bound for the existence of a parametrically exact cardinal.Recall that an I1-embedding is a non-trivial elementary embedding j : V δ+1 → V δ+1 for some limit ordinal δ.Also, an I2-embedding is an elementary embedding j : V → M for some transitive class M such that V δ ⊆ M for some limit ordinal δ > crit(j) satisfying j(δ) = δ.Finally, an I3-embedding is a non-trivial elementary embedding j : V δ → V δ , for some limit ordinal δ (see [11, §24]).Note that, if j : V δ → V δ is an I3-embedding with critical point κ, then V δ is a model of ZFC and the sequence j m (κ) | m < ω is cofinal in δ.

Beyond huge reflection
In this section, we introduce a generalization of the principle ESR C (κ, λ) to sequences of cardinals in order to obtain principles of structural reflection that imply the existence of even stronger large cardinals.The following definition is motivated by the formulation of Chang's Conjecture.In order to determine the large cardinal strength of the above principles, we consider the following sequential versions of n-exact and weakly n-exact cardinals: Definition 9.2.Let 0 < η ≤ ω and let λ = λ m | m < η be a strictly increasing sequence of cardinals with supremum λ.
We show next that the large cardinal notions introduced above are located in the uppermost regions of the large cardinal hierarchy.Recall that, given a natural number n > 0, a cardinal κ is n-huge if there exists a transitive class M and an elementary embedding j : V → M with crit(j) = κ and j n (κ) M ⊆ M .Proposition 9.3.Let n > 0 be a natural number.
(i) If κ is an n-huge cardinal, witnessed by an elementary embedding j : V → M , then κ is weakly parametrically 1-exact for the sequence j m+1 (κ) | m < n .(ii) If κ is a cardinal and λ = λ m | m ≤ n is a sequence of cardinals such that κ is either weakly 1-exact for λ or 0-exact for λ, then some cardinal less than κ is n-huge.
(i) If κ is either weakly 1-exact for λ or 0-exact for λ, then there exists an I3-embedding j : V λ → V λ .(ii) If κ is either weakly parametrically 1-exact for λ or parametrically 0-exact for λ, then the set of critical points of I3-embeddings is stationary in κ.
In the remainder of this section, we show how the validity of the principle ESR( λ) is connected to the existence of cardinals that are exact or weakly exact for certain sequences of cardinals.
Proof.First, assume that (ii) holds.Fix a Π n -formula ϕ(v 0 , v 1 ) with the property that C = {A | ϕ(A, z)} is a suitable class of structures and pick a structure B in C of type 1) and an elementary embedding j : M → H λ ′ such that A, λ ∈ ran(j), j(z) = z, j(ρ) = λ and j(λ i ) = λ i+1 for all i < η.Elementarity then implies that ρ is the supremum of the sequence λ i | i < η .Moreover, the fact that λ ′ ∈ C (n−1) implies that Π n -statements are downwards absolute from V to H λ ′ , and this allows us to conclude that ϕ(B, z) holds in In this situation, the elementarity j and the fact that j(λ i ) = λ i+1 holds for all i < η cause ϕ(A, z) to hold in M and, by the Π n (V ρ+1 )-upwards correctness of M , this shows that A is a structure of type λ i | i < η in C. Finally, since A has rank ρ and V ρ is a subset of M , the map j induces an elementary embedding of A into B.This shows that (i) holds in this case.Next, assume that (i) holds and we shall prove (iii).So fix a cardinal λ < λ ′ ∈ C (n) and A ∈ V λ+1 .Define L to be the first-order language extending L ∈ by predicate symbols P = Ṗi | i < η and constant symbols Ȧ, λ, ż and λi | i < η .Define C to be the class of all L-structures D, E, P , a, b, c, d with the property that E is a well-founded and extensional relation on D and, if P = P i | i < η , d = d i | i < η and π : D, E → M, ∈ is the induced transitive collapse, then the following statements hold: • rnk(D) is a cardinal and V rnk(D) ∪ {rnk(D)} ⊆ M .
• rnk(P i ) | i < η is a strictly increasing sequence of cardinals with supremum rnk(D).
Then C is definable by a Π n -formula with parameter z.Now, let X be an elementary substructure of V λ ′ of cardinality λ with V λ ∪ {A, λ} ⊆ X. Pick a bijection f : X → V λ and let R denote the induced binary relation on V λ .These choices ensure that the transitive collapse of V λ , R is Π n (V λ+1 )-upwards correct and this allows us to conclude that . By our assumptions, there exists an elementary embedding i of a structure and j is an elementary embedding with j(ρ) = λ, j(z) = z and A ∈ ran(j).Moreover, given i < η, we now have λ i = rnk(P i ) = π(d i ) and this allows us to conclude that j(λ This shows that (iii) holds in this case.Since (iii) obviously implies (ii), this concludes the proof of the lemma.
In the case of sequences of finite length, we can now generalize Theorem 1.7 to principles of the form Π n -ESR( λ).
Theorem 9.8.The following statements are equivalent for every cardinal κ and all natural numbers n, η > 0: (i) κ is the least cardinal such that there exists a strictly increasing sequence λ = λ i | i < η + 1 of cardinals with λ 0 = κ and property that Π n -ESR( λ) holds.(ii) κ is the least cardinal such that there exists a strictly increasing sequence λ = λ i | i < η + 1 of cardinals with λ 0 = κ and the property that Π n (V κ )-ESR( λ) holds.(iii) κ is the least cardinal that is weakly n-exact for some strictly increasing sequence of cardinals greater than κ of length η. (iv) κ is the least cardinal that is weakly parametrically n-exact for some strictly increasing sequence of cardinals greater than κ of length η.
Proof of the Claim.Assume, towards a contradiction, that A ∈ V λ+1 witnesses that κ is not weakly parametrically n-exact for λ i+1 | i < η .Using Lemma 9.6, we can find a transitive, Π n (V ρ+1 )-upwards correct set M with V ρ ∪{ρ} ⊆ M and an elementary embedding j : M → V λ ′ with A, κ ∈ ran(j) and j(λ i ) = λ i+1 for all i < η.Our setup then ensures that j(crit(j)) = κ and, since κ ∈ ran(j) and j(κ) = λ 1 > κ, this implies that j(crit(j)) < κ.Given i < η + 2, set µ i = j i (crit(j)).Then µ i+1 ≤ λ i for all i < η + 1.Since µ 1 < κ, the minimality of κ yields a Π n -formula ϕ(v) with the property that the class C = {A | ϕ(A)} consists of structures of the same type and there exists a structure B of type µ i+2 | i < η in C such that for every structure A of type µ i+1 | i < η in C, there exists no elementary embedding of A into B. By the choice of λ ′ , these statements are absolute between V and V λ ′ .Elementarity now implies that, in M , there exists a structure B 0 of type µ i+1 | i < η with the property that ϕ(B 0 ) holds and for every structure A of type µ i | i < η such that ϕ(A) holds, there is no elementary embedding of A into B 0 .Since M is Π n (V ρ+1 )-upwards correct and V λ ′ is sufficiently elementary in V , it follows that both ϕ(B 0 ) and ϕ(j(B 0 )) hold in V λ ′ .But we can now use the elementarity of j to derive a contradiction, because j(B 0 ) is a structure of type µ i+2 | i < η , B 0 is a structure of type µ i+1 | i < η and the map j induces an elementary embedding of B 0 into j(B 0 ) that is an element of V λ ′ .
With the help of the above claim, we can apply Lemma 9.6 and Corollary 9.7 to conclude that all statements listed in the theorem are equivalent.Note that the above proof cannot be directly generalized to sequences of cardinals of length ω, because, if κ is the least cardinal with the property that Π n -ESR( λ) holds for some strictly increasing sequence λ = λ i | i < ω of cardinals with λ 0 = κ and supremum λ, we assume that κ is not weakly parametrically n-exact for λ i+1 | i < ω and we repeat the above construction to obtain a transitive, Π n (V λ+1 )-upwards correct set M with V λ ∪ {λ} ⊆ M and an elementary embedding j : M → V λ ′ with j(crit(j)) < κ and j(λ i ) = λ i+1 for all i < ω, then we do not know wheter the sequence j i+1 (crit(j)) | i < ω is contained in the range of j and this stops us from repeating the above minimality argument.
Proof.Assume that (ii) holds.Pick a Σ n+1 -formula ϕ(v 0 , v 1 ) with the property that the class C = {A | ϕ(A, z)} consists of suitable structures and fix a structure B in C of type λ i+1 | i < η .Since B ∈ V λ+1 , there exists a cardinal ρ, a cardinal κ ′ with ρ < κ ′ ∈ C (n) , a cardinal λ ′ with λ < λ ′ ∈ C (n+1) , an elementary submodel X of V κ ′ with V ρ ∪ {ρ} ⊆ X, and an elementary embedding j : X → H λ ′ with B ∈ ran(j), j(z) = z, j(ρ) = λ and j(λ i ) = λ i+1 for all i < η.Pick A ∈ X such that j(A) = B. Since λ ′ ∈ C (n+1) , we know that ϕ(B, z) holds in V λ ′ .The elementarity of j then implies that ϕ(A, z) holds in V κ ′ and, since κ ′ ∈ C (n) , we may then conclude that A ∈ C.Moreover, since j(λ i ) = λ i+1 holds for all i < η, elementarity allows us to conclude that A has type λ i | i < η .Finally, elementarity also implies that A has rank ρ and, since V ρ is a subset of X, the map j induces an elementary embedding of A into B. Next, assume that (i) holds and fix A ∈ V λ+1 .Let L be the first-order language extending L ∈ by predicate symbols P = Ṗi | i < η and constant symbols Ȧ, λ, ż and λi | i < η .Define C to be the class of all L-structures D, E, P , a, b, c, d for which there exists θ ∈ C (n)  and an isomorphism τ between D, E and an elementary substructure X of H θ with the property that, if P = P i | i < η and d = d i | i < η , then the following statements hold: • rnk(D) is a cardinal and V rnk(D) ∪ {rnk(D)} ⊆ X.
• rnk(P i ) | i < η is a strictly increasing sequence of cardinals with supremum rnk(D).Then the class C is then Σ n+1 -definable with parameter z.
As above, we can now generalize Theorem 1.9 to the principle Σ n+1 -ESR( λ) for finite sequences of cardinals λ.
Theorem 9.11.The following statements are equivalent for every cardinal κ and all natural numbers n, η > 0: (i) κ is the least cardinal such that there exists a strictly increasing sequence λ = λ i | i < η + 1 of cardinals with λ 0 = κ and the property that Σ n+1 -ESR( λ) holds.(ii) κ is the least cardinal such that there exists a strictly increasing sequence λ = λ i | i < η + 1 of cardinals with λ 0 = κ and the property that Σ n+1 (V κ )-ESR( λ) holds.(iii) κ is the least cardinal that is n-exact for some strictly increasing sequence of cardinals greater than κ of length η. (iv) κ is the least cardinal that is parametrically n-exact for some strictly increasing sequence of cardinals greater than κ of length η.
Proof of the Claim.Assume, towards a contradiction, that A ∈ V λ+1 witnesses that κ is not parametrically n-exact for λ i+1 | i < η .By Lemma 9.9, a cardinal κ ′ with ρ < κ ′ C (n) , a cardinal λ ′ with λ < λ ′ ∈ C (n+1) , an elementary submodel X of H κ ′ with V ρ ∪ {ρ} ⊆ X and an elementary embedding j : X → H λ ′ with A, κ ∈ ran(j) and j(λ i ) = λ i+1 for all i < η.We then know that j(crit(j)) = κ and, since κ ∈ ran(j) and j(κ) > κ, this allows us to conclude j(crit(j)) < κ.Given i < η + 2, set µ i = j i (crit(j)).Since µ 1 < κ, the minimality of κ yields a Σ n+1 -formula ϕ(v) with the property that the class C = {A | ϕ(A)} consists of structures of the same type and there exists a structure B of type µ i+2 | i < η in C such that for every structure A of type µ i+1 | i < η in C, there exists no elementary embedding of A into B. Our set up now ensures that, in X, there exists a structure B 0 of type µ i+1 | i < η with the property that ϕ(B 0 ) holds and for every structure A of type µ i | i < η such that ϕ(A) holds, there is no elementary embedding of A into B 0 .Then both ϕ(B 0 ) and ϕ(j(B 0 )) hold in V λ ′ , and j induces an elementary embedding of B 0 into j(B 0 ) that is an element of V λ ′ .Since B 0 has type µ i+1 | i < η = j( µ i | i < η ) and j(B 0 ) has type µ i+2 | i < η = j( µ i+1 | i < η ), we can now use the elementarity of j to derive a contradiction.
A combination of this claim and Corollary 9.10 now yields the desired equivalences.

Open questions and concluding remarks
We close this paper by discussing some questions raised by the above results.First, recall that a cardinal κ is superhuge if there is a proper class of cardinals λ with the property that κ is huge with target λ.In order to study principles of structural reflection related to superhugeness, Proposition 8.4 suggests to study cardinals κ that are weakly 1-exact for a proper class of cardinals λ.By Corollary 5.5, this property is equivalent to the assumption that the principle Π 1 -ESR(κ, λ) holds for a proper class of cardinals λ.Therefore, it is natural to ask whether a variation of Theorem 1.7 can be proven for these cardinals.
Question 10.1.Are the following statements equivalent for every cardinal κ and every natural number n > 0?
(i) κ is the least cardinal that is weakly parametrically n-exact for a proper class of cardinals λ. (ii) κ is the least cardinal with the property that Π n -ESR(κ, λ) holds for a proper class of cardinals λ.
Our next question deals with the exact position of n-exact and weakly n-exact cardinals in the large cardinal hierarchy.By Corollary 8.2 and Proposition 8.9, these notions are properly contained in the interval given by almost hugeness and almost 2-hugeness.Moreover, Corollary 8.5 shows that hugeness is strictly stronger than weak 1-exactness.Finally, Proposition 8.6 implies that, if κ is the least huge cardinal, then Σ 2 -ESR(κ, λ) fails for all λ > κ.These results leave open the precise relationship between hugeness and exactness, and motivate the following question:

Date:
January 2, 2024.2010 Mathematics Subject Classification.03E55, 03E65, 18A10, 18A15.This is an update of the published version of the paper (Ann.Pure Appl.Logic 174, No.1, Paper No. 103171, 2023) that corrects a problem in the definition of weakly exact cardinals (see Definition 1.6 below).

Definition 9 . 1 .
Let 0 < η ≤ ω and let L be a first-order language containing unary predicate symbols P = Ṗi | i < η .(i)Given a sequence µ = µ i | i < η of cardinals with supremum µ, an L-structure A has type µ (with respect to P ) if the universe of A has rank µ and rnk( Ṗ A i ) = µ i for all i < η. (ii) Given a class C of L-structures and a strictly increasing sequence λ = λ i | i < 1 + η of cardinals, we let ESR C ( λ) denote the statement that for every structure B in C of type λ i+1 | i < η , there exists an elementary embedding of a structure A in C of type λ i | i < η into B. (iii) Given a definability class Γ and a class P , we let Γ(P )-ESR( λ) denote the statement that ESR C ( λ) holds for every class C of structures of the same type that is Γdefinable with parameters in P .