aa r X i v : . [ m a t h . L O ] A ug C ( n ) -CARDINALS JOAN BAGARIA
Abstract.
For each natural number n , let C ( n ) be the closed and un-bounded proper class of ordinals α such that V α is a Σ n elementarysubstructure of V . We say that κ is a C ( n ) -cardinal if it is the criticalpoint of an elementary embedding j : V → M , M transitive, with j ( κ )in C ( n ) . By analyzing the notion of C ( n ) -cardinal at various levels ofthe usual hierarchy of large cardinal principles we show that, starting atthe level of superstrong cardinals and up to the level of rank-into-rankembeddings, C ( n ) -cardinals form a much finer hierarchy. The natural-ness of the notion of C ( n ) -cardinal is exemplified by showing that theexistence of C ( n ) -extendible cardinals is equivalent to simple reflectionprinciples for classes of structures, which generalize the notions of su-percompact and extendible cardinals. Moreover, building on results of[1], we give new characterizations of Vopeˇnka’s Principle in terms of C ( n ) -extendible cardinals. Introduction
For each natural number n , let C ( n ) denote the club (i.e., closed andunbounded ) proper class of ordinals α that are Σ n -correct in the universe V of all sets, meaning that V α is a Σ n -elementary substructure of V , andwritten V α (cid:22) n V . Observe that α is Σ n -correct in V if and only if it isΠ n -correct in V , i.e., V α is a Π n -elementary substructure of V . Notice alsothat if α is Σ n -correct and ϕ is a Σ n +1 sentence with parameters in V α thatholds in V α , then ϕ holds in V . And if ψ is a Π n +1 sentence with parametersin V α that holds in V , then it holds in V α . These basic facts will be usedthroughout the paper without further comment.The class C (0) is the class of all ordinals. But if V α (cid:22) V , then α isalready an uncountable strong limit cardinal: clearly α is a limit cardinalgreater than ω , and if β < α , then the sentence ∃ γ ∃ f ( γ an ordinal ∧ f : γ → V β is onto)is Σ in the parameter V β , and therefore it must hold in V α . Further, if α ∈ C (1) , then V α = H α . Thus, since H α (cid:22) V for every uncountablecardinal α , C (1) is precisely the class of all uncountable cardinals α such Date : August 27, 2019.2000
Mathematics Subject Classification.
Key words and phrases. C ( n ) -cardinals, Supercompact cardinals, Extendible cardinals,Vopenka’s Principle, Reflection.The author was supported by the Spanish Ministry of Education and Science un-der grant MTM2008-03389/MTM, and by the Generalitat de Catalunya under grant2009SGR-00187. Part of this work was done in November 2009 during the author’s stayat the Mittag-Leffler Institut, whose support and hospitality is gratefully acknowledged. For all standard set-theoretic undefined notions, see [4]. ( n ) -CARDINALS 2 that V α = H α . It follows that C (1) is Π definable, for α ∈ C (1) if and onlyif α is an uncountable cardinal and ∀ M ( M a transitive model of ZFC ∗ ∧ α ∈ M → M | = V α = H α ) . (Here ZFC ∗ denotes a sufficiently large finite fragment of ZFC.) The pointis that if α ∈ C (1) and M is a transitive model of ZFC ∗ that contains α ,then if in M we could find some transitive x ∈ V α \ H α , we would have that | x | ≥ α . But this contradicts the fact that in V the cardinality of x is lessthan α , because V α = H α .More generally, since the truth predicate | = n for Σ n sentences (for n ≥ n definable (see [5], Section 0.2), and since the relation x = V y is Π ,the class C ( n ) , for n ≥
1, is Π n definable: α ∈ C ( n ) if and only if α ∈ C ( n − ∧ ∀ ϕ ( x ) ∈ Σ n ∀ a ∈ V α ( | = n ϕ ( a ) → V α | = ϕ ( a )) . Let us remark that C ( n ) , for n ≥
1, cannot be Σ n definable. Otherwise,if α is the least ordinal in C ( n ) , then the sentence “there is some ordinal in C ( n ) ” would be Σ n and so it would hold in V α , yielding an ordinal in C ( n ) smaller than α , which is impossible.The classes C ( n ) , n ≥
1, form a basis for definable club proper classes ofordinals, in the sense that every Σ n club proper class of ordinals contains C ( n ) . For suppose C is a club proper class of ordinals that is Σ n definable,some n ≥
1. If α ∈ C ( n ) , then for every β < α , the sentence ∃ γ ( β < γ ∧ γ ∈ C )is Σ n in the parameter β and is true in V , hence also true in V α . This showsthat C is unbounded below α . Therefore, since C is closed, α ∈ C .By a similar argument one can show that every club proper class C ofordinals that is Σ n (i.e., Σ n definable with parameters) contains all α ∈ C ( n ) that are greater than the rank of the parameters involved in any given Σ n definition of C .Finally, note that since the least ordinal in C ( n ) does not belong to C ( n +1) , C ( n +1) ⊂ C ( n ) , all n .When working with non-trivial elementary embeddings j : V → M , with M transitive, one would like to have some control over where the image j ( κ )of the critical point κ goes. An especially interesting case is when one wants V j ( κ ) to reflect some specific property of V or, more generally, when onewants j ( κ ) to belong to a particular definable club proper class of ordinals.Now, since the C ( n ) , n ∈ ω , form a basis for such classes, the problem can bereformulated as follows: when can one have j ( κ ) ∈ C ( n ) , for a given n ∈ ω ?This prompts the following definition.Let us say that a cardinal κ is C ( n ) -measurable if there is an elementaryembedding j : V → M , some transitive class M , with critical point crit ( j ) = κ and with j ( κ ) ∈ C ( n ) .Observe that if j : V → M ∼ = U lt ( V, U ), M transitive, is the ultrapowerelementary embedding obtained from a non-principal κ -complete ultrafilter U on κ , then 2 κ < j ( κ ) < (2 κ ) + (see [5]). Hence, since V j ( κ ) (cid:22) V impliesthat j ( κ ) is a (strong limit) cardinal, j cannot witness the C (1) -measurabilityof κ . Nonetheless, by using iterated ultrapowers (see [4], 19.15), one has ( n ) -CARDINALS 3 that for every cardinal α > κ , the α -th iterated ultrapower embedding j α : V → M α ∼ = U lt ( V, U α ), where U α is the α -th iterate of U , has criticalpoint κ and j α ( κ ) = α . So, if κ is measurable, then for each n one can alwaysfind an elementary embedding j : V → M , M transitive, with j ( κ ) ∈ C ( n ) .We have thus shown the following. Proposition 1.1.
Every measurable cardinal is C ( n ) -measurable, for all n . A similar situation occurs in the case of strong cardinals.Let us say that a cardinal κ is C ( n ) -strong if for every λ > κ , κ is λ - C ( n ) -strong , that is, there exists an elementary embedding j : V → M , M transitive, with critical point κ , and such that j ( κ ) > λ , V λ ⊆ M , and j ( κ ) ∈ C ( n ) . Equivalently (see [5], 26.7), κ is λ - C ( n ) -strong if and only ifthere exists a ( κ, β )-extender E , for some β > | V λ | , with V λ ⊆ M E and λ < j E ( κ ) ∈ C ( n ) .Suppose now that j : V → M witnesses the λ -strongness of κ , with j ( κ )not necessarily in C ( n ) . Let E be the ( κ, j ( κ ))-extender obtained from j ,and let j E : V → M E be the corresponding λ -strong embedding (see [5]).Then in M E , E ′ := j E ( E ) is a ( j E ( κ ) , j E ( j ( κ )))-extender, which gives rise toan elementary embedding j E ′ : M E → M E ′ with critical point j E ( κ ). Stillin M E , let U be the standard j E ( κ )-complete ultrafilter on j E ( κ ) derivedfrom j E ′ , i.e., U = { X ⊆ j E ( κ ) : j E ( κ ) ∈ j E ′ ( X ) } and let j U : M E → M be the corresponding elementary embedding. Thenone can iterate j U α -times, for some α ∈ C ( n ) greater than 2 j E ( κ ) , so that if j α : M E → M α is the resulting elementary embedding, then j α ( j E ( κ )) = α .Letting k := j α ◦ j E , one has that k : V → M α is a λ -strong elementaryembedding with critical point κ and with k ( κ ) ∈ C ( n ) . We have thus provedthe following. Proposition 1.2.
Every λ -strong cardinal is λ - C ( n ) -strong, for all n . Hence,every strong cardinal is C ( n ) -strong, for every n . So for measurable or strong cardinals κ , the requirement that j ( κ ) belongsto C ( n ) for the corresponding elementary embeddings j : V → M does notyield stronger large cardinal notions. But as we shall see next the situationchanges completely in the case of superstrong embeddings, that is, when j issuch that V j ( κ ) ⊆ M . In the following sections we will analyze the notion of C ( n ) -cardinal at various levels of the usual large cardinal hierarchy, begin-ning with superstrong cardinals and up to rank-into-rank embeddings. Inalmost all cases we will show that the corresponding C ( n ) -cardinals form afiner hierarchy. The notion of C ( n ) -cardinal will prove especially useful in theregion between supercompact cardinals and Vopeˇnka’s Principle (VP). Therewe will establish new equivalences between the existence of C ( n ) -extendiblecardinals, restricted forms of VP, and natural reflection principles for classesof structures. Some important regions of the C ( n ) -cardinal hierarchy havenot yet been explored, e.g., C ( n ) -Woodin cardinals; some need further study,e.g., C ( n ) -supercompact cardinals; and there are still many open questions, ( n ) -CARDINALS 4 e.g., whether the C ( n ) -supercompact cardinals form a hierarchy, or the ex-act relationship between C ( n ) -supercompact and C ( n ) -extendible cardinals.Further work along these lines is already under way.Let us point out that the consistency of the existence of all the C ( n ) -cardinals to be considered in this paper, except for those in the last sec-tion, follows from the consistency of the existence of an E cardinal, thatis, a cardinal κ for which there exists a non-trivial elementary embedding j : V δ → V δ , some δ , with κ = crit ( j ). In V δ , κ is C ( n ) -superstrong, C ( n ) -extendible, C ( n ) -supercompact, C ( n ) - k -huge, and C ( n ) -superhuge, forall n, k ≥ C ( n ) -superstrong cardinals We shall see next that in the case of superstrong cardinals κ , the require-ment that j ( κ ) ∈ C ( n ) , for n >
1, produces a hierarchy of ever stronger largecardinal principles.
Definition 2.1.
A cardinal κ is C ( n ) -superstrong if there exists an elemen-tary embedding j : V → M , M transitive, with critical point κ , V j ( κ ) ⊆ M ,and j ( κ ) ∈ C ( n ) . Notice that if j : V → M witnesses that κ is C ( n ) -superstrong, then V κ is an elementary substructure of V j ( κ ) , and therefore κ ∈ C ( n ) . Thus, every C ( n ) -superstrong cardinal belongs to C ( n ) . Proposition 2.2. If κ = crit ( j ) , where j : V → M is an elementaryembedding, with M transitive and V j ( κ ) ⊆ M , then j ( κ ) ∈ C (1) . Hence,every superstrong cardinal is C (1) -superstrong.Proof. Since κ ∈ C (1) , M satisfies that j ( κ ) ∈ C (1) , i.e., M satisfies that j ( κ ) is a strong limit cardinal and V j ( κ ) = H j ( κ ) . But since ( V j ( κ ) ) M = V j ( κ ) , j ( κ ) is, in V , a strong limit cardinal with V j ( κ ) = H j ( κ ) , so j ( κ ) ∈ C (1) . (cid:3) Observe that for n ≥
1, the sentence “ κ is C ( n ) -superstrong” is Σ n +1 , for κ is C ( n ) -superstrong if and only if ∃ β ∃ µ ∃ E ( κ < β < µ ∧ µ ∈ C ( n ) ∧ E is a ( κ, β )-extender ∧ E ∈ V µ ∧ V µ | = “ j E ( κ ) ∈ C ( n ) ∧ V j E ( κ ) ⊆ M E ”) . Proposition 2.3.
For every n ≥ , if κ is C ( n +1) -superstrong, then thereis a κ -complete normal ultrafilter U over κ such that { α < κ : α is C ( n ) -superstrong } ∈ U . Hence, the first C ( n ) -superstrong cardinal κ , if it exists, is not C ( n +1) -superstrong.Proof. Suppose κ is C ( n +1) -superstrong, witnessed by a ( κ, β )-extender E with associated elementary embedding j E = j : V → M such that β = j ( κ )and V j ( κ ) ⊆ M . Since j ( κ ) ∈ C ( n +1) , V j ( κ ) | = “ κ is C ( n ) -superstrong” . ( n ) -CARDINALS 5 And since κ ∈ C ( n +1) , M | = “ j ( κ ) ∈ C ( n +1) ”. Hence, since V j ( κ ) = ( V j ( κ ) ) M ,and since “ κ is C ( n ) -superstrong” is a Σ n +1 statement, we have: M | = “ κ is C ( n ) -superstrong” . Now using a standard argument (see, e.g., [5], 5.14, 5.15, or 22.1) one canshow that the set { α < κ : α is C ( n ) -superstrong } belongs to the κ -completenormal ultrafilter U := { X ⊆ κ : κ ∈ j ( X ) } . (cid:3) The following Proposition gives an upper bound on the relative positionof C ( n ) -superstrong cardinals in the usual large cardinal hierarchy. Recallthat κ is λ -supercompact if there is an elementary embedding j : V → M ,with M transitive, crit ( j ) = κ , j ( κ ) > λ , and M closed under λ -sequences.Equivalently, κ is λ -supercompact if there exists a κ -complete, fine, andnormal ultrafilter over P κ ( λ ) (see [5], 22.7). κ is supercompact if it is λ -supercompact for all λ . Proposition 2.4. If κ is κ -supercompact and belongs to C ( n ) , then there isa κ -complete normal ultrafilter U over κ such that the set of C ( n ) -superstrongcardinals smaller than κ belongs to U .Proof. Let j : V → M be an elementary embedding coming from a κ -complete fine and normal ultrafilter V on P κ (2 κ ). Let j ∗ := j ↾ V κ +1 . So, j ∗ : V κ +1 → M j ( κ )+1 is elementary and j ∗ ∈ M . Hence M | = “ j ∗ : V κ +1 → V j ( κ )+1 is elementary”. Since κ ∈ C ( n ) , also M | = “ j ( κ ) ∈ C ( n ) ”. Thus, M | = “ κ is κ + 1- C ( n ) -extendible” (see Definition 3.2 below). Hence, if U isthe standard ultrafilter over κ derived from j , we have { α < κ : α is α + 1- C ( n ) -extendible } ∈ U . Now as in [5], Proposition 26.11 (a), one can show that if α is α + 1- C ( n ) -extendible, then α is C ( n ) -superstrong. (cid:3) C ( n ) -extendible cardinals Recall that a cardinal κ is λ –extendible if there is an elementary embed-ding j : V λ → V µ , some µ , with critical point κ and such that j ( κ ) > λ . And κ is extendible if it is λ –extendible for all λ > κ .The next lemma implies that every extendible cardinal is supercompact. Lemma 3.1 (M. Magidor [8]) . Suppose j : V λ → V µ is elementary, λ is alimit ordinal, and κ is the critical point of j . Then κ is < λ -supercompact.Proof. Fix γ < λ and define U γ = { X ⊆ P κ ( γ ) : j ′′ γ ∈ j ( X ) } . Note that this makes sense if j ( κ ) > γ , in which case it is easy to check that U γ is a κ -complete, fine, and normal measure. Otherwise, let j = j and j m +1 = j ◦ j m . If j m ( κ ) > γ for some m , then define U γ using j m instead of j . But such an m does exist, for otherwise δ := sup m ( j m ( κ )) ≤ γ < λ , andthen since j ( δ ) = δ we would have j ↾ V δ +2 : V δ +2 → V δ +2 is elementary withcritical point κ , contradicting Kunen’s Theorem ([6]; see also [5], 23.14). (cid:3) ( n ) -CARDINALS 6 Definition 3.2.
For a cardinal κ and λ > κ , we say that κ is λ - C ( n ) -extendible if there is an elementary embedding j : V λ → V µ , some µ , withcritical point κ , and such that j ( κ ) > λ and j ( κ ) ∈ C ( n ) .We say that κ is C ( n ) -extendible if it is λ - C ( n ) -extendible for all λ > κ . Proposition 3.3.
Every extendible cardinal is C (1) -extendible.Proof. Suppose κ is extendible and λ is greater than κ . Pick λ ′ ≥ λ in C (1) ,and let j : V λ ′ → V µ be an elementary embedding with crit ( j ) = κ and j ( κ ) > λ ′ . Since λ ′ is a cardinal and V λ ′ = H λ ′ , by elementarity of j we alsohave that µ is a cardinal and V µ = H µ . Hence µ ∈ C (1) . And since, againby elementarity, V µ | = j ( κ ) ∈ C (1) , it follows that j ( κ ) ∈ C (1) . (cid:3) Notice that if j : V λ → V µ has critical point κ , and κ, λ, µ ∈ C ( n ) , then j ( κ ) ∈ C ( n ) follows automatically.Clearly, if κ is C ( n ) -extendible, then κ ∈ C ( n ) . But more is true. Proposition 3.4. If κ is C ( n ) -extendible, then κ ∈ C ( n +2) .Proof. By induction on n . Every extendible cardinal is in C (3) (see [5],23.10), which takes care of the cases n = 0 and n = 1. Now suppose κ is C ( n ) -extendible and ∃ xϕ ( x ) is a Σ n +2 sentence, where ϕ is Π n +1 andhas parameters in V κ . If ∃ xϕ ( x ) holds in V κ , then since by the inductionhypothesis κ ∈ C ( n +1) , we have that ∃ xϕ ( x ) holds in V . Now suppose a issuch that ϕ ( a ) holds in V . Pick λ > κ with a ∈ V λ , and let j : V λ → V µ beelementary, with critical point κ and with j ( κ ) > λ . Then since j ( κ ) ∈ C ( n ) ,and since ϕ ( a ) is a Π n +1 sentence in the parameter a ∈ V j ( κ ) , we have that V j ( κ ) | = ϕ ( a ), and therefore by elementarity, V κ | = ∃ xϕ ( x ). (cid:3) Let us observe that for any given α < λ , the relation “ α is λ - C ( n ) -extendible” is Σ n +1 (for n ≥ ∃ µ ∃ j ( j : V λ → V µ ∧ j elementary ∧ crit ( j ) = α ∧ j ( α ) > λ ∧ j ( α ) ∈ C ( n ) ) . Hence, “ x is a C ( n ) -extendible cardinal” is a Π n +2 property of x . Proposition 3.5.
For every n ≥ , if κ is C ( n ) -extendible and κ +1 - C ( n +1) -extendible, then the set of C ( n ) -extendible cardinals is unbounded below κ .Hence, the first C ( n ) -extendible cardinal κ , if it exists, is not κ + 1 - C ( n +1) -extendible. In particular, the first extendible cardinal κ is not κ + 1 - C (2) -extendible.Proof. Suppose κ is C ( n ) -extendible and κ + 1- C ( n +1) -extendible, witnessedby j : V κ +1 → V j ( κ )+1 . Since j ( κ ) ∈ C ( n +1) , V j ( κ ) | = “ κ is C ( n ) -extendible” . Hence, for every α < κ , V j ( κ ) | = “ ∃ β > α ( β is C ( n ) -extendible)” , since this is witnessed by κ . By the elementarity of j , for every fixed α < κ ,there is β > α such that, V κ | = “ β > α ∧ β is C ( n ) -extendible” . And since, by Proposition 3.4, κ ∈ C ( n +2) , β is C ( n ) -extendible in V . (cid:3) ( n ) -CARDINALS 7 Proposition 3.6.
For every n , if there exists a C ( n +2) -extendible cardinal,then there exists a proper class of C ( n ) -extendible cardinals.Proof. By the last proposition, if κ is C ( n +2) -extendible, then the set of C ( n ) -extendible cardinals is unbounded below κ . Now the proposition follows eas-ily from the fact that if κ is C ( n +2) -extendible, then κ ∈ C ( n +4) (Proposition3.4), and the fact that being C ( n ) -extendible is a Π n +2 -property. (cid:3) Note however that the existence of a C ( n +1) -extendible cardinal κ doesnot imply the existence of a C ( n ) -extendible cardinal greater than κ . Forif λ is the least such C ( n ) -cardinal, then V λ is a model of ZFC plus “ κ is C ( n +1) -extendible”, because λ ∈ C ( n +2) (Proposition 3.4) and being C ( n +1) -extendible is a Π n +3 property of κ . And V λ also satisfies that “there is no C ( n ) -extendible cardinal greater than κ ”, because any such C ( n ) -extendiblecardinal would be C ( n ) -extendible in V , since κ ∈ C ( n +2) .The next proposition gives an upper bound on C ( n ) -superstrong cardinals. Proposition 3.7. If κ is κ + 1 - C ( n ) -extendible, then κ is C ( n ) -superstrong,and there is a κ -complete normal ultrafilter U over κ such that the set of C ( n ) -superstrong cardinals smaller than κ belongs to U .Proof. As in [5], Proposition 26.11 (a). (cid:3) Vopˇenka’s Principle
This section builds on results from [1], giving new and sharper character-izations of Vopˇenka’s Principle in terms of C ( n ) -extendible cardinals.Recall that Vopˇenka’s Principle (VP) states that for every proper class C of structures of the same type, there exist A = B in C such that A iselementarily embeddable into B .VP can be formulated in the first-order language of set theory as an axiomschema, i.e., as an infinite set of axioms, one for each formula with two freevariables. Formally, for each such formula ϕ ( x, y ) one has the axiom: ∀ x [( ∀ y ∀ z ( ϕ ( x, y ) ∧ ϕ ( x, z ) → y and z are structures of the same type) ∧∀ α ∈ OR ∃ y ( rank ( y ) > α ∧ ϕ ( x, y )) →∃ y ∃ z ( ϕ ( x, y ) ∧ ϕ ( x, z ) ∧ y = z ∧ ∃ e ( e : y → z is elementary))] . Henceforth, VP will be understood as this axiom schema.The theory ZFC plus VP implies, for instance, that the class of extendiblecardinals is stationary, i.e., every definable club proper class contains anextendible cardinal ([8]). And its consistency is known to follow from theconsistency of ZFC plus the existence of an almost-huge cardinal (see [5], or[4]). We will give below the exact equivalence in terms of C ( n ) -cardinals.Let us consider the following variants of VP, the first one apparently muchstronger than the second.We say that a class C is Σ n ( Π n ) if it is definable, with parameters, by aΣ n (Π n ) formula of the language of set theory. If no parameters are involved,then we use the lightface types Σ n (Π n ). ( n ) -CARDINALS 8 Definition 4.1. If Γ is one of Σ n , Π n , some n ∈ ω , and κ is an infinitecardinal, then we write V P ( κ, Γ) for the following assertion: For every Γ proper class C of structures of the same type τ such thatboth τ and the parameters of some Γ-definition of C , if any, belong to H κ , C reflects below κ , i.e., for every B ∈ C , there exists A ∈ C ∩ H κ that iselementarily embeddable into B . If Γ is one of Σ n , Π n , or Σ n , Π n , some n ∈ ω , we write V P (Γ) for thefollowing statement:
For every Γ proper class C of structures of the language of set theory withone (equivalently, finitely-many) additional 1-ary relation symbol(s), thereexist distinct A and B in C with an elementary embedding of A into B .VP for Σ classes is a consequence of ZFC. In fact, the following holds. Theorem 4.2. If κ is an uncountable cardinal, then every (not necessarilyproper) class C of structures of the same type τ ∈ H κ which is Σ definable,with parameters in H κ , reflects below κ . Hence, V P ( κ, Σ ) holds for everyuncountable cardinal κ .Proof. Fix an uncountable cardinal κ and a class C of structures of the sametype τ ∈ H κ , definable by a Σ formula with parameters in H κ .Given B ∈ C , let λ be a regular cardinal greater than κ , with B ∈ H λ , andlet N be an elementary substructure of H λ , of cardinality less than κ , whichcontains B and the transitive closure of { τ } together with the parametersinvolved in some Σ definition of C .Let A and M be the transitive collapses of B and N , respectively, andlet j : M → N be the collapsing isomorphism. Then A ∈ H κ , and j ↾ A : A → B is an elementary embedding. Observe that j ( τ ) = τ . So, since Σ formulas are upwards absolute for transitive models, and since M | = A ∈ C ,we have that A ∈ C . (cid:3) In contrast, Vopˇenka’s Principle for Π proper classes implies the existenceof very large cardinals. Theorem 4.3. (1) If V P (Π ) holds, then there exists a supercompact cardinal. (2) If V P ( Π ) holds, then there is a proper class of supercompact car-dinals.Proof. (1). Let C be the class of structures of the form h V λ +2 , ∈ , α, λ i , where λ is the least limit ordinal greater than α such that no κ ≤ α is < λ -supercompact.We claim that C is Π definable without parameters. For X ∈ C if andonly if X = h X , X , X , X i , where(1) X is an ordinal(2) X is a limit ordinal greater than X (3) X = V X +2 (4) X = ∈ ↾ X (5) And the following hold in h X , X i :(a) ∀ κ ≤ X ( κ is not < X -supercompact)(b) ∀ µ ( µ limit ∧ X < µ < X → ∃ κ ≤ X ( κ is < µ -supercompact)). ( n ) -CARDINALS 9 If there is no supercompact cardinal, then C is a proper class. So by V P (Π ), there exist structures h V λ +2 , ∈ , α, λ i 6 = h V µ +2 , ∈ , β, µ i in C and anelementary embedding j : h V λ +2 , ∈ , α, λ i → h V µ +2 , ∈ , β, µ i . Since j must send α to β and λ to µ , j is not the identity, for otherwise thetwo structures would be equal. Hence by Kunen’s Theorem ([6]; see also [5],23.14) we must have λ < µ . By the way λ and µ are uniquely defined from α and β , respectively, this implies that while no κ ≤ α is < λ -supercompact,there is some κ ≤ β which is < λ -supercompact, and therefore α < β . So, j has critical point some κ ≤ α . It now follows by Lemma 3.1 that κ is < λ -supercompact. But this is impossible because h V λ +2 , ∈ , α, λ i ∈ C . (2). Fixing an ordinal ξ , to show that there is a supercompact cardinalgreater than ξ , we argue as above. The only difficulty now is to ensure that κ > ξ . But this can be achieved by letting C be the class of structures ofthe form h V λ +2 , ∈ , α, λ, { γ } γ ≤ ξ i , where α > ξ and λ is the least limit ordinalgreater than α such that no κ ≤ α is < λ -supercompact. The class C is nowΠ definable with ξ as an additional parameter. If there is no supercompactcardinal above ξ , then C is a proper class. So arguing as before we have anelementary embedding j between two different structures in C , which nowmust be the identity on the ordinals less than or equal to ξ , so that j hascritical point some κ with ξ < κ . A contradiction then follows as before. (cid:3) For any given Π class of structures C of the same type one may wonderhow much supercompactness is needed to guarantee that VP holds for C .An upper bound is given in the next Proposition.Let us say that a limit ordinal λ captures a proper class C if the class ofordinal ranks of elements of C , intersected with λ , is unbounded in λ . I.e.,for every α less than λ there exists A ∈ C of rank strictly between α and λ .Note that if C is Π n , then every λ in C ( n +1) greater than the rank of theparameters involved in a Π n definition of C captures C . For if α < λ , thenthe assertion that there is a structure in C of rank greater than α can bewritten as a Σ n +1 sentence with parameter α and the parameters of somedefinition of C . And since this sentence is true in V , and λ ∈ C ( n +1) , itis also true in V λ . Notice also that, by a similar argument, every cardinalin C (2) belongs to the Π definable class Lim ( C (1) ) of all limit points of C (1) and, moreover, it captures all Π proper classes. However, the leastordinal λ in Lim ( C (1) ) that captures all Π proper classes is strictly lessthan the least ordinal µ in C (2) . The point is that, fixing an enumeration h ϕ n ( x ) : n < ω i of all Π formulas that define proper classes, the sentence ∃ λ ∃ x ( λ ∈ Lim ( C (1) ) ∧ x = V λ ∧∀ n ( V λ | = ∀ α ∃ β > α ∃ a ( rk ( a ) > β ∧ | = ϕ n ( a ))))is Σ in the parameter h ϕ n ( x ) : n < ω i , and so it is reflected by µ , therebyproducing a λ < µ in Lim ( C (1) ) that captures all Π proper classes. Proposition 4.4.
Let C be a Π proper class of structures of the same type.If there exists a cardinal κ that is < λ -supercompact, for some λ ∈ Lim ( C (1) ) greater than κ that captures C , then VP holds for C . ( n ) -CARDINALS 10 Proof.
Since λ captures C , in V λ there exist elements of C of arbitrarily highrank. So, since λ ∈ Lim ( C (1) ), we can find δ < λ such that V δ = H δ ,and B ∈ C ∩ V δ of rank greater than κ . Let j : V → M be an elementaryembedding with critical point κ , with j ( κ ) > δ , and M closed under δ -sequences. Since B ∈ M and C is Π definable, M | = “ B ∈ C ”. And since M is closed under δ -sequences, the elementary embedding j ↾ B : B → j ( B )belongs to M . Thus, M | = “ ∃ A ∈ C ∃ e ( rank ( A ) < j ( κ ) ∧ e : A → j ( B ) is elementary) , since this is witnessed by B and j ↾ B .By elementarity, the same must hold in V , namely, ∃ A ∈ C ∃ e ( rank ( A ) < κ ∧ e : A → B is elementary) , which is what we wanted. (cid:3) We give next a strong converse to Theorem 4.3.
Theorem 4.5 ([1]) . Suppose that C is a Σ (not necessarily proper) class ofstructures of the same type τ , and suppose that there exists a supercompactcardinal κ larger than the rank of the parameters that appear in some Σ definition of C , and with τ ∈ V κ . Then for every B ∈ C there exists A ∈ C∩ V κ that is elementarily embeddable into B .Proof. Fix a Σ formula ϕ ( x, y ) and a set b such that C = { B : ϕ ( B, b ) } ,and suppose that κ is a supercompact cardinal with b ∈ V κ . Fix B ∈ C ,and let λ ∈ C (2) be greater than rank( B ). Let j : V → M be an elementaryembedding with M transitive and critical point κ , such that j ( κ ) > λ and M is closed under λ -sequences. Thus, B and j ↾ B : B → j ( B ) are in M ,and also V λ ∈ M . Hence V λ (cid:22) M . Moreover, since j ( τ ) = τ , j ( B ) is astructure of type τ , and j ↾ B is an elementary embedding.Since V λ (cid:22) V , V λ | = ϕ ( B, b ). And since Σ formulas are upwards absolutebetween V λ and M , M | = ϕ ( B, b ).Thus, in M it is true that there exists X ∈ M j ( κ ) such that ϕ ( X, b ),namely B , and there exists an elementary embedding e : X → j ( B ), namely j ↾ B . Therefore, by elementarity, the same holds in V ; that is, thereexists X ∈ V κ such that ϕ ( X, b ), and there exists an elementary embedding e : X → B . (cid:3) The following corollaries give characterizations of Vopˇenka’s principle forΠ and Σ classes in terms of supercompactness. The equivalence of (2) and(3) in the next two corollaries was already proved in [1]. Corollary 4.6.
The following are equivalent: (1)
V P (Π ) . (2) V P ( κ, Σ ) , for some κ . (3) There exists a supercompact cardinal.Proof. (2) ⇒ (1) is immediate. (1) ⇒ (3) is given by Theorem 4.3, (1). And(3) ⇒ (2) follows from Theorem 4.5. (cid:3) The next corollary gives the parameterized version. The implication(1) ⇒ (3) is given by Theorem 4.3, (2). ( n ) -CARDINALS 11 Corollary 4.7.
The following are equivalent: (1)
V P ( Π ) . (2) V P ( κ, Σ ) , for a proper class of cardinals κ . (3) There exists a proper class of supercompact cardinals.
We shall give next a characterization of supercompactness in terms of anatural principle of reflection. Recall from Definition 4.1 that a cardinal κ reflects a class of structures C of the same type if for every B ∈ C thereexists A ∈ C ∩ H κ which is elementarily embeddable into B . Theorem 4.8 (Magidor [8]) . If κ is the least cardinal that reflects the Π proper class C of structures of the form h V λ , ∈i , then κ is supercompact.Proof. Let α < κ be such that there is an elementary embedding j : V α +1 → V λ +1 for some singular λ ∈ C (3) . Since j sends α to λ , it must have a criticalpoint β , which must be smaller than α , for if β = α , then α would be regularin V α +1 (being the critical point of the embedding), and so by elementarity λ would be regular in V λ +1 , hence regular in V , contrary to our choice of λ .By Lemma 3.1, β is < α -supercompact, and so V α | = “ β is supercompact”.By elementarity of j , V λ | = “ j ( β ) is supercompact”, hence, since λ ∈ C (3) ,and being supercompact is Π expressible, j ( β ) is supercompact.Thus j ( β ) ≥ κ , because j ( β ) reflects C , by Theorem 4.5, and κ is theleast cardinal that does this. Now suppose, aiming for a contradiction, that j ( β ) > κ . Then V j ( β ) | = “ κ reflects the class C ” . Hence, by elementarity of j , V β | = “ γ reflects the class C ”for some γ < β . And since γ is fixed by j , V j ( β ) | = “ γ reflects the class C ” . Making use of the fact that V j ( β ) (cid:22) Σ V , it follows that γ reflects the class C , thus contradicting the minimality of κ . (cid:3) The last two theorems yield the following characterizations of the firstsupercompact cardinal.
Corollary 4.9.
The following are equivalent: (1) κ is the first supercompact cardinal. (2) κ is the least cardinal that reflects all Σ definable, with parametersin V κ , classes of structures of the same type. i.e., κ is the leastordinal for which V P ( κ, Σ ) holds. (3) κ is the least cardinal that reflects the Π class of structures of theform h V λ , ∈i , λ an ordinal.Proof. If κ is a supercompact cardinal, then by Theorem 4.5 V P ( κ, Σ )holds, and therefore κ reflects the class of structures h V λ , ∈i , λ an ordinal.So by Theorem 4.8, (1), (2), and (3) are equivalent. (cid:3) The following parameterized version of the last Corollary has been pointedout by David Asper´o. ( n ) -CARDINALS 12 Corollary 4.10.
A cardinal κ reflects all Π (proper) classes of structuresof the same type if and only if either κ is a supercompact cardinal or a limitof supercompact cardinals.Proof. Clearly, the property of reflecting Π classes of structures is closedunder limits. So if κ is a supercompact cardinal or a limit of supercompactcardinals, then Theorem 4.5 implies that κ reflects all Π classes. The otherdirection can be proved as in Theorem 4.3 (2). (cid:3) We will prove next similar results for classes of higher complexity, forwhich we shall need C ( n ) -extendible cardinals. Theorem 4.11.
For every n ≥ , if κ is a C ( n ) –extendible cardinal, thenevery class C of structures of the same type τ ∈ H κ which is Σ n +2 definable,with parameters in H κ , reflects below κ . Hence V P ( κ, Σ n +2 ) holds.Proof. Fix a Σ n +2 formula ∃ xϕ ( x, y, z ), where ϕ is Π n +1 , such that for someset b ∈ V κ = H κ , C := { B : ∃ xϕ ( x, B, b ) } is a class of structures of the same type τ ∈ H κ .Fix B ∈ C and let λ ∈ C ( n +2) be greater than κ and the rank of B . Thus, V λ | = ∃ xϕ ( x, B, b ) . Let j : V λ → V µ be an elementary embedding with critical point κ , with j ( κ ) > λ , and j ( κ ) ∈ C ( n ) . Note that B and j ↾ B : B → j ( B ) are in V µ .Moreover, since j fixes τ , j ( B ) is a structure of type τ , and j ↾ B is anelementary embedding.As κ, λ ∈ C ( n +2) (see Proposition 3.4), it follows that V κ (cid:22) n +2 V λ . So wehave V λ | = “ ∀ x ∈ V κ ∀ θ ∈ Σ n +2 ( V κ | = θ ( x ) ↔| = n +2 θ ( x ))” . Hence, by elementarity, V µ | = “ ∀ x ∈ V j ( κ ) ∀ θ ∈ Σ n +2 ( V j ( κ ) | = θ ( x ) ↔| = n +2 θ ( x ))” , which says that V j ( κ ) (cid:22) n +2 V µ .Since j ( κ ) ∈ C ( n ) , we also have V λ (cid:22) n +1 V j ( κ ) , and therefore V λ (cid:22) n +1 V µ .It follows that V µ | = ∃ xϕ ( x, B, b ), because V λ | = ∃ xϕ ( x, B, b ).Thus, in V µ it is true that there exists X ∈ V j ( κ ) such that X ∈ C , namely B , and there exists an elementary embedding e : X → j ( B ), namely j ↾ B .Therefore, by the elementarity of j , the same is true in V λ , that is, thereexists X ∈ V κ such that X ∈ C , and there exists an elementary embedding e : X → B . Let A ∈ V κ be such an X , and let e : A → B be an elementaryembedding. Since λ ∈ C ( n +2) , A ∈ C , and we are done. (cid:3) The next theorem yields a strong converse to Theorem 4.11.The notion of C ( n ) -extendibility used in [1] has the following apparentlystronger form – let us call it C ( n )+ -extendibility: For λ ∈ C ( n ) , a cardinal κ is λ - C ( n )+ -extendible if it is λ - C ( n ) -extendible, witnessed by some j : V λ → V µ which, in addition to satisfying j ( κ ) > λ and j ( κ ) ∈ C ( n ) , also satisfies that µ ∈ C ( n ) . κ is C ( n )+ -extendible if it is λ - C ( n )+ -extendible for every λ > κ with λ ∈ C ( n ) . ( n ) -CARDINALS 13 Every extendible cardinal is C (1)+ -extendible (see the proof of Proposition3.3). We shall see below that the first C ( n ) -extendible cardinal is C ( n )+ -extendible, for all n . Theorem 4.12.
Suppose n ≥ . If V P (Π n +1 ) holds, then there exists a C ( n )+ -extendible cardinal.Proof. Suppose there are no C ( n )+ -extendible cardinals. Then the classfunction F on the ordinals given by: F ( α ) = the least λ ∈ C ( n +1) greater than α such that α isnot λ - C ( n )+ -extendible,is defined for all ordinals α .Let C = { η > ∀ α < η F ( α ) < η } . So C is a closed unbounded properclass of ordinals, and is contained in C ( n +1) because every η ∈ C is thesupremum of the set { F ( α ) : α < η } ⊆ C ( n +1) .We claim that C is Π n +1 definable, without parameters. It is sufficientto see that F is Π n +1 definable. We have: λ = F ( α ) if and only if(1) λ ∈ C ( n +1) (2) α < λ (3) ∀ β > λ ( β ∈ C ( n ) → V β | = ( α is not λ - C ( n )+ -extendible)), and(4) V λ | = ∀ λ ′ > α ( λ ′ ∈ C ( n +1) → ( α is λ ′ - C ( n )+ -extendible)).The point is that, for any α < λ ′ , the relation “ α is λ ′ - C ( n )+ -extendible” isΣ n +1 , for it holds if and only if ∃ µ ∃ j ( j : V λ ′ → V µ is elementary ∧ crit ( j ) = α ∧ j ( α ) > λ ′ ∧ j ( α ) , µ ∈ C ( n ) ) . So it holds in V if and only if it holds in V λ , for any λ ∈ C ( n +1) greaterthan λ ′ . And if it holds in V β , with β ∈ C ( n ) , then it holds in V . Moreover,since λ ∈ C ( n +1) , for every λ ′ < λ we have λ ′ ∈ C ( n +1) if and only if V λ | = λ ′ ∈ C ( n +1) .Since the conjunction of statements (1)-(4) above is Π n +1 , it follows that F , and therefore also C , is Π n +1 definable. Let ϕ be a Π n +1 formula thatdefines C .For each ordinal α , let λ α be the least limit point of C greater than α . Wehave that x = λ α if and only if x is an ordinal greater than α that belongsto C and is such that(1) V x | = ∀ β ∃ γ ( γ > β ∧ ϕ ( γ ))(2) V x | = ∀ β ( β > α → ∃ γ < β ∀ η ( γ < η < β → ¬ ϕ ( η ))) , which shows that the function α λ α is Π n +1 definable.Consider now the proper class C of structures A α of the form h V λ α , ∈ , α, λ α , C ∩ α + 1 i , where α ∈ C .We claim that C is Π n +1 definable. We have: X ∈ C if and only if X = h X , X , X , X , X i , where(1) X ∈ C (2) X = λ X (3) X = V X (4) X = ∈ ↾ X ( n ) -CARDINALS 14 (5) X = C ∩ X + 1We have already seen that (1) and (2) are Π n +1 expressible. And so are (3)and (4), as one can easily see. As for (5), note that X = C ∩ X + 1 holdsin V if and only if it holds in V X . So (5) is equivalent to V X | = ∀ x ( x ∈ X ↔ ϕ ( x ) ∧ x ∈ X + 1)which is Π n +1 expressible.So by V P (Π n +1 ) there exist α = β and an elementary embedding j : A α → A β . Since j must send α to β , j is not the identity. So j has critical point some κ ≤ α .We claim that κ ∈ C . Otherwise, γ := sup ( C ∩ κ ) < κ . Let δ be theleast ordinal in C greater than γ such that δ < λ α . So κ < δ ≤ α . Since δ is definable from γ in A α , and since j ( γ ) = γ , we must also have j ( δ ) = δ .But then j ↾ V δ +2 : V δ +2 → V δ +2 is a nontrivial elementary embedding,contradicting Kunen’s Theorem.Since λ α ∈ C ( n +1) , V λ α | = ϕ ( κ ). Hence by elementarity, V λ β | = ϕ ( j ( κ )).So since λ β ∈ C ( n +1) , it follows that j ( κ ) ∈ C .Note that since λ α ∈ C , we have κ < F ( κ ) < λ α . Thus, j ↾ V F ( κ ) : V F ( κ ) → V j ( F ( κ )) is elementary, with critical point κ .And since j ( κ ) ∈ C , F ( κ ) < j ( κ ). Moreover, by the elementarity of j , V λ β satisfies that j ( F ( κ )) belongs to C ( n +1) , and so since λ β ∈ C ( n +1) this is truein V . This shows that j ↾ V F ( κ ) witnesses that κ is F ( κ )- C ( n )+ -extendible.But this is impossible by the definition of F . (cid:3) The proof of the last theorem can be easily adapted to prove the pa-rameterized version: if
V P ( Π n +1 ) holds, then there is a proper class of C ( n ) -extendible cardinals. Fixing an ordinal ξ , to show that there is a C ( n ) -extendible cardinal greater than ξ , we argue as above. To ensure that κ > ξ ,we now let C be the class of structures of the form h V λ α , ∈ , α, λ α , C ∩ α + 1 , { γ } γ ≤ ξ i where α > ξ and α ∈ C . The class C is now Π n +1 definable with ξ as anadditional parameter. If there is no C ( n ) -extendible cardinal above ξ , then C is a proper class. So arguing as before we have an elementary embedding j between two different structures in C , which now must be the identity onthe ordinals less than or equal to ξ , and so j has critical point some κ with ξ < κ . A contradiction then follows as before.The following corollaries summarize the results above. The equivalencesof (2) and (4) in the next two corollaries were already proved in [1]. Corollary 4.13.
The following are equivalent for n ≥ : (1) V P (Π n +1 ) . (2) V P ( κ, Σ n +2 ) , for some κ . (3) There exists a C ( n ) -extendible cardinal. (4) There exists a C ( n )+ -extendible cardinal. ( n ) -CARDINALS 15 Proof. (1) ⇒ (4) follows from Theorem 4.12. (4) ⇒ (3) and (2) ⇒ (1) are im-mediate. And (3) ⇒ (2) follows from Theorem 4.11. (cid:3) In particular, since every extendible cardinal is C (1) -extendible (Proposi-tion 3.3), we have the following. Corollary 4.14.
The following are equivalent: (1)
V P (Π ) . (2) V P ( κ, Σ ) , for some κ . (3) There exists an extendible cardinal. (4)
There exists a C (1)+ -extendible cardinal. We finally obtain the following characterization of VP. The equivalenceof (2), (3), and (5) was already proved in [1].
Corollary 4.15.
The following are equivalent: (1)
V P (Π n ) , for every n . (2) V P ( κ, Σ n ) , for a proper class of cardinals κ , and for every n . (3) VP. (4)
For every n , there exists a C ( n ) -extendible cardinal. (5) For every n , there exists a C ( n )+ -extendible cardinal.Proof. Clearly, (3) implies (1) and (2) implies (3). All the other implicationsfollow immediately from Corollary 4.13. (cid:3)
Since the consistency of VP follows from the consistency of the existenceof an almost-huge cardinal (see [5], 24.18 and Section 6 below), an almost-huge cardinal gives an upper bound in the usual large cardinal hierarchy onthe consistency strength of C ( n ) -extendible cardinals, all n ≥ C ( n ) -extendible cardinals in terms ofreflection of classes of structures. Theorem 4.16. If n ≥ and κ is the least cardinal that reflects all Π n +1 proper classes of structures of the same type, then κ is C ( n )+ -extendible.Proof. Suppose otherwise. Then by 4.11 there is no C ( n ) -extendible, andtherefore no C ( n )+ -extendible, cardinal less than or equal to κ . For such acardinal would reflect all Σ n +2 (hence all Π n +1 ) classes of structures, con-tradicting the minimality of κ .Consider the class C of structures of the form h V ξ , ∈ , λ, α, C ( n ) ∩ ξ i , where α < λ < ξ , and(1) λ ∈ C ( n ) (2) ξ ∈ Lim ( C ( n ) )(3) the cofinality of ξ is uncountable(4) ∀ β < ξ ∀ µ ( ∃ j ( j : V λ → V µ ∧ crit ( j ) = α ∧ j ( α ) = β ) → ∃ j ′ ∃ µ ′ ( j ′ : V λ → V µ ′ ∧ µ ′ < ξ ∧ V ξ | = “ µ ′ ∈ C ( n ) ” ∧ crit ( j ′ ) = α ∧ j ′ ( α ) = β )) , and(5) λ witnesses that no ordinal less than or equal to α is λ - C ( n )+ -extendible. ( n ) -CARDINALS 16 Clearly, C is a Π n +1 definable proper class. So there exists an elementaryembedding j : h V ξ ′ , ∈ , λ ′ , α ′ , C ( n ) ∩ ξ ′ i → h V ξ , ∈ , λ, κ, C ( n ) ∩ ξ i with both h V ξ ′ , ∈ , λ ′ , α ′ , C ( n ) ∩ ξ ′ i and h V ξ , ∈ , λ, κ, C ( n ) ∩ ξ i in C , and with h V ξ ′ , ∈ , λ ′ , α ′ , C ( n ) ∩ ξ ′ i of rank less than κ . So ξ ′ < κ .Let α = crit ( j ). We claim that α ∈ C ( n ) . Otherwise, let γ := sup ( C ( n ) ∩ α ). So γ < α . Let δ ∈ C ( n ) be the least such that γ < δ < ξ ′ . Since δ isdefinable from γ in h V ξ ′ , ∈ , λ ′ , α ′ , C ( n ) ∩ ξ ′ i and j ( γ ) = γ , also j ( δ ) = δ . Hence j ↾ V δ +2 : V δ +2 → V δ +2 is elementary, contradicting Kunen’s Theorem.If j m ( α ) < ξ ′ for all m , then { j m ( α ) } m ∈ ω ∈ V ξ ′ , because ξ ′ has uncount-able cofinality, contradicting Kunen’s Theorem. So for some m we have j m ( α ) < ξ ′ ≤ j m +1 ( α ).We claim that there exists an elementary embedding k : V λ ′ → V µ , some µ ∈ C ( n ) , with crit ( k ) = α and k ( α ) = j m +1 ( α ). We prove this by inductionon i ≤ m . For i = 0 take k = j ↾ V λ ′ . Now suppose it true for i < m . Since j i +1 ( α ) < ξ ′ , by (4) above there exist j ′ and µ ′ such that j ′ : V λ ′ → V µ ′ is elementary, µ ′ < ξ ′ , V ξ ′ | = µ ′ ∈ C ( n ) , crit ( j ′ ) = α , and j ′ ( α ) = j i +1 ( α ).Notice that since ξ ∈ C ( n ) and, by the elementarity of j , V ξ | = j ( µ ′ ) ∈ C ( n ) ,it follows that j ( µ ′ ) ∈ C ( n ) . Composing j ′ with j we now have that k :=( j ◦ j ′ ) : V λ ′ → V j ( µ ′ ) is elementary, has critical point α , and k ( α ) = j i +2 ( α ),as desired.Note that since α, ξ ′ , ξ ∈ C ( n ) , it follows that j m +1 ( α ) ∈ C ( n ) . Thus, k witnesses that α is λ ′ - C ( n )+ -extendible, contradicting (5) above. (cid:3) Observe that if κ is the least C ( n ) -extendible cardinal, then by Theorem4.11 it reflects all Σ n + classes of structures. Hence, by the theorem above, κ is the least cardinal that does this, and therefore κ is C ( n )+ -extendible. Corollary 4.17.
The following are equivalent for each n ≥ : (1) κ is the least C ( n ) -extendible cardinal. (2) κ is the least cardinal that reflects all Σ n +2 definable, with parametersin V κ , classes of structures of the same type. I.e., κ is the leastordinal for which V P ( κ, Σ n + ) holds. (3) κ is the least cardinal that reflects all Π n +1 proper classes of struc-tures of type h V α , ∈ , A i , where A is a unary predicate.Proof. If κ is a C ( n ) -extendible cardinal, then by Theorem 4.11 V P ( κ, Σ n + )holds, and therefore κ reflects all Π n +1 proper classes of structures of type h V α , ∈ , A i , where A is a unary predicate. Now, the proof of Theorem 4.16shows that if κ is the least cardinal that reflects a particular Π n +1 definableproper class C of structures of the form h V ξ , ∈ , λ, α, C ( n ) ∩ ξ i , where α < λ <ξ , then κ is C ( n )+ -extendible, and therefore C ( n ) -extendible. Thus, since thetriple h λ, α, C ( n ) ∩ ξ i can be easily coded as a subset of V ξ , the equivalenceof (1), (2), and (3) follows immediately from Theorem 4.16. (cid:3) The following parameterized version also follows. ( n ) -CARDINALS 17 Theorem 4.18.
A cardinal κ reflects all Π n +1 (proper) classes of structuresof the same type if and only if either κ is a C ( n ) -extendible cardinal or alimit of C ( n ) -extendible cardinals.Proof. It is clear that the property of reflecting Π n + classes of structures isclosed under limits. So if κ is a C ( n ) -extendible cardinal or a limit of C ( n ) -extendible cardinals, then Theorem 4.11 implies that κ reflects all Π n + classes of structures. The other direction can be proved similarly as inTheorem 4.16. For suppose κ reflects all Π n +1 (proper) classes of structuresand is neither C ( n ) -extendible nor a limit of C ( n ) -extendible cardinals. Thenfor some ordinal η < κ there is no C ( n ) -extendible cardinal greater than η and less than or equal to κ .Consider the class C of structures of the form h V ξ , ∈ , λ, α, C ( n ) ∩ ξ, { η ′ } η ′ ≤ η i ,where η < α < λ < ξ , which satisfy (1)-(4) from the proof of Theorem 4.16and also(5) λ witnesses that no ordinal less than or equal to α and greater than η is λ - C ( n ) -extendible.Clearly, C is a Π n +1 definable proper class, with η as a parameter. So thereexists an elementary embedding j : h V ξ ′ , ∈ , λ ′ , α ′ , C ( n ) ∩ ξ ′ , { η ′ } η ′ ≤ η i → h V ξ , ∈ , λ, κ, C ( n ) ∩ ξ, { η ′ } η ′ ≤ η i with both structures in C , and with h V ξ ′ , ∈ , λ ′ , α ′ , C ( n ) ∩ ξ ′ , { η ′ } η ′ ≤ η i of rankless than κ . So ξ ′ < κ and η < crit ( j ). The rest of the proof proceeds nowas in Thereom 4.16. (cid:3) We finish this section with the following observation. Suppose n ≥ n +1 definable class of structures C , say via the Σ n +1 formula ϕ ( x ),let C ∗ be the class of structures of the form A ∗ = h V α , ∈ , α, A i , where α is theleast ordinal in C ( n ) such that V α | = ϕ ( A ). If A ∈ C , then such an α exists,because the set of ordinals α such that V α | = ϕ ( A ) is club. Conversely, if h V α , ∈ , α, A i ∈ C ∗ , then V α | = ϕ ( A ) and α ∈ C ( n ) , which implies that ϕ ( A )holds in V , and so A ∈ C . Thus, we have that A ∈ C if and only if A ∗ ∈ C ∗ . Now notice that C ∗ is Π n definable. This explains why, e.g., V P (Π n ) isequivalent to V P (Σ n +1 ), or why a cardinal reflects Π n classes if and only ifit reflects Σ n +1 classes.5. C ( n ) -supercompact cardinals Let us consider next the C ( n ) -cardinal form of supercompactness. Definition 5.1. If κ is a cardinal and λ > κ , we say that κ is λ - C ( n ) -supercompact if there is an elementary embedding j : V → M , with M transitive, such that crit ( j ) = κ , j ( κ ) > λ , M is closed under λ -sequences,and j ( κ ) ∈ C ( n ) .We say that κ is C ( n ) -supercompact if it is λ - C ( n ) -supercompact for every λ > κ . ( n ) -CARDINALS 18 If κ is C ( n ) -superhuge (see Section 6), then κ is C ( n ) -supercompact. Thus,it follows from Proposition 6.4 below that if κ is C ( n ) -2-huge, then there isa κ -complete normal ultrafilter U over κ such that { α < κ : V κ | = “ α is C ( n ) -supercompact” } ∈ U .The notion of λ - C ( n ) -supercompactness, unlike λ -supercompactness, can-not be formulated in terms of normal measures on P κ ( λ ). The problem isthat if j : V → M is an ultrapower embedding coming from such a measure,then 2 λ <κ < j ( κ ) < (2 λ <κ ) + (see [5], 22.11), and so j ( κ ) is not a cardinal.So, in order to formulate the notion of λ - C ( n ) -supercompactness in the first-order language of set theory we will make use of long extenders having assupport a sufficiently rich transitive set (see [9] for a presentation of short extenders of this kind).So suppose j : V → M witnesses that κ is λ - C ( n ) -supercompact. Let Y be a transitive subset of M which contains j ↾ λ , is closed under sequencesof length ≤ λ , and is closed under j . Let ζ be the least ordinal such that Y ⊆ j ( V ζ ). For each a ∈ [ Y ] <ω := { x ⊆ Y : x is finite } , let E a be definedby: X ∈ E a if and only if X ⊆ ( V ζ ) a and j − ↾ j ( a ) ∈ j ( X ) . Note that the function j − ↾ j ( a ) : j ( a ) → a is an ∈ -isomorphism that sends j ( x ) to x , for every x ∈ a .It is not difficult to check that the sequence E := h E a : a ∈ [ Y ] <ω i is an extender over V ζ with critical point κ and support Y , that is,(1) Each E a is a κ -complete ultrafilter over ( V ζ ) a , and E { κ } is not κ + -complete.(2) If a ⊆ b and X ∈ E a , then { f ∈ ( V ζ ) b : f ↾ a ∈ X } ∈ E b .(3) For every a , the set { f : a → range ( f ) : f is an ∈ -isomorphism } belongs to E a .(4) If F : ( V ζ ) a → V is such that { f : F ( f ) ∈ S ( range ( f )) } ∈ E a , thenthere is z ∈ Y such that { f ∈ ( V ζ ) a ∪{ z } : F ( f ↾ a ) = f ( z ) } ∈ E a ∪{ z } .(5) The ultrapower U lt ( V, E ) is well-founded.To check (5), observe that the map k : U lt ( V, E ) → M given by k ([ a, [ f ]]) = j ( f )( j − ↾ j ( a )) is an elementary embedding.If j E : V → M E ∼ = U lt ( V, E ) is the corresponding ultrapower embedding,then j = k ◦ j E . Moreover, Y ⊆ M E and k is the identity on Y (see [9] fordetails). Since Y was assumed to be closed under j , it easily follows that Y is also closed under j E . Hence, j ↾ Y = j E ↾ Y . For if y ∈ Y , then j ( y ) = k ( j E ( y )) = j E ( y ). In particular, κ = crit ( j E ) and j ( κ ) = j E ( κ ).We claim that j E witnesses the λ - C ( n ) -supercompactness of κ , for whichit only remains to check that M E is closed under λ -sequences. First notethat M E = { j E ( f )( j − ↾ j ( a )) : a ∈ [ Y ] <ω and f : ( V ζ ) a → V } . For if x = [ a, [ f ]] ∈ M E , then writing s for j − ↾ j ( a ) and noticing that k ( s ) = s , because s ∈ Y , we have k ( x ) = j ( f )( s ) = k ◦ j E ( f )( s ) = k ( j E ( f ))( k ( s )) = k ( j E ( f )( s ))and since k is one-to-one we have that x = j E ( f )( s ). I want to thank Ralf Schindler for illuminating discussions on long extenders. ( n ) -CARDINALS 19 Now fix j E ( f i )( j − ↾ j ( a i )), for i < λ . Let f = h f i : i < λ i , let c be j E ↾ λ , and let d = h j − ↾ j ( a i ) : i < λ i . Notice that since Y is closed under j and under λ sequences, d ∈ Y . Set b = { c, d } and let F : ( V ζ ) b → V bedefined as follows: if s = { s c , s d } ∈ ( V ζ ) b is such that s c and s d are functionswith the same ordinal α as their domain, then F ( s ) is the function g withdomain α such that g ( i ) = f ( s c ( i ))( s d ( i )), whenever s c ( i ) is an ordinal and s d ( i ) ∈ ( V ζ ) a sc ( i ) , and g ( i ) = 0 otherwise. Otherwise, F ( s ) = 0. Then,noticing that j − ↾ j ( b ) maps j ( c ) to j E ↾ λ and j ( d ) to d , we have: j E ( F )( j − ↾ j ( b ))( i ) = j E ( f )( j E ( i ))( j − ↾ j ( a i )) = j E ( f i )( j − ↾ j ( a i ))and so h j E ( f i )( j − ↾ j ( a i )) : i < λ i = j E ( F )( j − ↾ j ( b )) ∈ M E .Conversely, suppose Y is transitive and E = h E a : a ∈ [ Y ] <ω i is an exten-der over some V ζ with critical point κ and support Y . If j E : V → M E ∼ = U lt ( V, E ) is the corresponding ultrapower embedding, then crit ( j E ) = κ and Y ⊆ M E (see [9], Lemmas 1.4 and 1.5). Moreover, if [ a, [ f ]] ∈ M E , then j E ( f )( j − E ↾ j E ( a )) = [ a, [ c af ]]([ a, [ Id ( V ζ ) a ]]) = [ a, [ f ]]where c af : ( V ζ ) a → V is the constant function with value f , and Id ( V ζ ) a :( V ζ ) a → V is the identity function. Hence, M E = { j E ( f )( j − E ↾ j E ( a )) : a ∈ [ Y ] <ω and f : ( V ζ ) a → V } . So if Y is closed under j E and under λ sequences, then one can show, asabove, that M E is closed under λ sequences.Notice that for every µ ∈ C (1) and every κ, ζ, Y, E ∈ V µ , we have: E is anextender over V ζ with critical point κ and support Y if and only if V µ | = “ E is an extender over V ζ with critical point κ and support Y ”. Moreover,( j E ( κ )) V µ = j E ( κ ).Thus, for n ≥ κ is λ - C ( n ) -supercompact if and only if ∃ µ ∃ E ∃ Y ∃ ζ ( µ ∈ C ( n ) ∧ λ, E, Y ∈ V µ ∧ Y is transitive ∧ [ Y ] ≤ λ ⊆ Y ∧ V µ | = “ E is an extender over V ζ with critical point κ and support Y ∧ j E [ Y ] ⊆ Y ∧ j E ( κ ) > λ ∧ j E ( κ ) ∈ C ( n ) ”) . It follows that, for n ≥
1, “ κ is λ - C ( n ) -supercompact” is Σ n +1 expressible.Hence, “ κ is C ( n ) -supercompact” is Π n +2 expressible.Thus, for n ≥
1, if κ is C ( n ) -supercompact and α is any ordinal in C ( n +1) greater than κ , then V α | = “ κ is C ( n ) -supercompact”. Moreover, since forevery n , “ ∃ κ ( κ is C ( n ) -supercompact)” is Σ n +3 expressible, the first C ( n ) -supercompact cardinal does not belong to C ( n +3) . But we don’t know if, e.g.,the first C (1) -supercompact cardinal belongs to C (3) . We don’t know eitherif the C ( n ) -supercompact cardinals form a hierarchy in a strong sense, thatis, if the first C ( n ) -supercompact cardinal is smaller than the first C ( n +1) -supercompact cardinal, for all n .Every extendible cardinal is supercompact, and the first extendible ismuch greater than the first supercompact (see [5]), but we don’t know if, for n ≥
1, every C ( n ) -extendible cardinal is C ( n ) -supercompact, or if the first C ( n ) -extendible cardinal is actually greater than the first C ( n ) -supercompactcardinal. However, since every C ( n ) -extendible cardinal belongs to C ( n +2) ( n ) -CARDINALS 20 (Proposition 3.4), the first C ( n ) -supercompact cardinal is smaller than thefirst C ( n +1) -extendible cardinal, assuming both exist.So far, the only upper bound we know, in the usual large cardinal hi-erarchy, on the consistency strength of the existence of C ( n ) -supercompactcardinals, for n ≥
1, is the existence of an E cardinal (see Section 7).6. C ( n ) -huge and C ( n ) -superhuge cardinals Recall that a cardinal κ is m -huge , for m ≥
1, if it is the critical point ofan elementary embedding j : V → M with M transitive and closed under j m ( κ )-sequences, where j m is the m -th iterate of j . A cardinal is called huge if it is 1-huge. Definition 6.1.
We say that a cardinal κ is C ( n ) - m -huge ( n ≥ ) if it is m -huge, witnessed by j , with j ( κ ) ∈ C ( n ) . We say that κ is C ( n ) -huge if itis C ( n ) - -huge. In contrast with C ( n ) -supercompact cardinals, which do not admit a char-acterization in terms of ultrafilters, but only in terms of long extenders, C ( n ) - m -huge cardinals can be characterized in terms of normal ultrafilters.To wit: κ is C ( n ) - m -huge if and only if it is uncountable and there isa κ -complete fine and normal ultrafilter U over some P ( λ ) and cardinals κ = λ < λ < . . . < λ m = λ , with λ ∈ C ( n ) , and such that for each i < m , { x ∈ P ( λ ) : ot ( x ∩ λ i +1 ) = λ i } ∈ U . (See [5], 24.8 for a proof of the case n = 1, which also works for arbitrary n .) It follows that “ κ is C ( n ) - m -huge” is Σ n +1 expressible.Clearly, every huge cardinal is C (1) -huge. But the first huge cardinal isnot C (2) -huge. For suppose κ is the least huge cardinal and j : V → M witnesses that κ is C (2) -huge. Then since “ x is huge” is Σ expressible, wehave V j ( κ ) | = “ κ is huge” . Hence, since ( V j ( κ ) ) M = V j ( κ ) , M | = “ ∃ δ < j ( κ )( V j ( κ ) | = “ δ is huge”)” . By elementarity, there is a huge cardinal less than κ in V , which is absurd.A similar argument, using that for n ≥ κ is C ( n ) - m -huge” is Σ n +1 ex-pressible, for all m , shows that the first C ( n ) - m -huge cardinal is not C ( n +1) -huge, for all m, n ≥ κ is almost-huge if it is the critical point ofan elementary embedding j : V → M with M transitive and closed un-der γ -sequences, for every γ < j ( κ ). Thus, we say that a cardinal κ is C ( n ) -almost-huge if it is almost-huge, witnessed by an embedding j with j ( κ ) ∈ C ( n ) . C ( n ) -almost-huge cardinals can also be characterized in terms of normalultrafilters. To wit: κ is C ( n ) -almost-huge if and only if there exist aninaccessible λ ∈ C ( n ) greater than κ and a coherent sequence of normalultrafilters hU γ : κ ≤ γ < λ i over P κ ( γ ) such that the corresponding embed-dings j γ : V → M γ ∼ = U lt ( V, U γ ) and k γ,δ : M γ → M δ satisfy: if κ ≤ γ < λ ( n ) -CARDINALS 21 and γ ≤ α < j γ ( κ ), then there exists δ such that γ ≤ δ < λ and k γ,δ ( α ) = δ .(See [5], 24.11 for details.) It follows that for n ≥
1, “ κ is C ( n ) -almost-huge”is Σ n +1 expressible. Now similar arguments as in the case of C ( n ) -hugecardinals show that for n ≥
1, the first C ( n ) -almost-huge cardinal is not C ( n +1) -almost-huge.Clearly, if κ is C ( n ) -huge, then it is C ( n ) -almost-huge. Moreover, simi-larly as in Proposition 2.3, one can show that there is a κ -complete normalultrafilter U over κ such that the set of C ( n ) -almost-huge cardinals below κ belongs to U . Notice also that every C ( n ) -almost-huge cardinal is C ( n ) -superstrong, and therefore belongs to C ( n ) . Hence, since being C ( n ) -hugeis Σ n +1 expressible, the first C ( n ) -huge cardinal is smaller than the first C ( n +1) -almost-huge cardinal, provided both exist. Definition 6.2.
We say that a cardinal κ is C ( n ) -superhuge if and only iffor every α there is an elementary embedding j : V → M , with M transitive,such that crit ( j ) = κ , α < j ( κ ) , M is closed under j ( κ ) -sequences, and j ( κ ) ∈ C ( n ) . Clearly, κ is superhuge (see [5]) if and only if it is C (1) -superhuge. Proposition 6.3. If κ is C ( n ) -superhuge, then κ ∈ C ( n +2) .Proof. Similarly as in Proposition 3.4. (cid:3)
Note that κ is C ( n ) -superhuge if and only if for every α there is a κ -complete fine and normal ultrafilter U over some P ( λ ), with λ ∈ C ( n ) greaterthan α and κ , so that { x ∈ P ( λ ) : ot ( x ) = κ } ∈ U .Thus, “ κ is C ( n ) -superhuge” is Π n +2 expressible.Arguing similarly as in the case of C ( n ) -huge cardinals, one can easily seethat the first C ( n ) -superhuge cardinal is not C ( n +1) -superhuge. For suppose κ is the least C ( n ) -superhuge cardinal, and suppose, towards a contradiction,that it is C ( n +1) -superhuge. Let j : V → M be an elementary embeddingwith crit ( j ) = κ , V j ( κ ) ⊆ M , and j ( κ ) ∈ C ( n +1) . Then V j ( κ ) | = “ κ is C ( n ) -superhuge”. Hence, since ( V j ( κ ) ) M = V j ( κ ) ,( V j ( κ ) ) M | = “ ∃ δ ( δ is C ( n ) -superhuge)” . By elementarity, V κ | = “ ∃ δ ( δ is C ( n ) -superhuge)” . And since κ ∈ C ( n +2) , it is true in V that there exists a C ( n ) -superhugecardinal δ < κ , contradicting the minimality of κ .Clearly, every C ( n ) -superhuge cardinal is C ( n ) -supercompact. The follow-ing Proposition is the C ( n ) -cardinal version of similar results for m -huge andsuperhuge cardinals due to Barbanel-Di Prisco-Tan [2] (see also [5], 24.13). Proposition 6.4. (1) If κ is C ( n ) -superhuge, then it is C ( n ) -extendible. Moreover, thereis a κ -complete normal ultrafilter U over κ such that { α < κ : α is C ( n ) -extendible } ∈ U . (2) If κ is C ( n ) - -huge, then there is a κ -complete normal ultrafilter U over κ such that { α < κ : V κ | = “ α is C ( n ) -superhuge ” } ∈ U . ( n ) -CARDINALS 22 Proof. (1): Fix λ > κ . Let j : V → M be a witness to the C ( n ) -hugenessof κ with j ( κ ) > λ . Then M | = “ κ is λ - C ( n ) -extendible”. Since M | =“ j ( κ ) ∈ C ( n +2) ”, we have ( V j ( κ ) ) M | = “ κ is λ - C ( n ) -extendible”. Hence,since ( V j ( κ ) ) M = V j ( κ ) , V j ( κ ) | = “ κ is λ - C ( n ) -extendible”, and therefore κ is λ - C ( n ) -extendible.Notice that the argument above actually shows that ( V j ( κ ) ) M | = “ κ is C ( n ) -extendible”. Thus, if U is the standard κ -complete normal ultrafilterover κ derived from j , we have { α < κ : V κ | = “ α is C ( n ) -extendible” } ∈ U .Hence, since κ ∈ C ( n +2) , { α < κ : α is C ( n ) -extendible } ∈ U .(2): Let j : V → M witness that κ is C ( n ) -2-huge. Since M is closed under j ( κ )-sequences, the κ -complete fine and normal ultrafilter over P ( j ( κ ))derived from j that witnesses the hugeness of κ belongs to M . Hence, M | = “ κ is huge, witnessed by some embedding k with k ( κ ) = j ( κ )” . Thus, if U is the standard κ -complete normal ultrafilter over κ derived from j , we have A := { α < κ : α is huge, witnessed by an embedding k with k ( α ) = κ } ∈ U . Since M contains all the ultrafilters over P ( κ ), it follows that for each α ∈ A , M | = “ α is huge, witnessed by an embedding k with k ( α ) = κ ” . Hence, { β < κ : α is huge, witnessed by an embedding k with k ( α ) = β } ∈ U .Notice that since κ, j ( κ ) ∈ C ( n ) , V j ( κ ) | = “ κ ∈ C ( n ) ”. And since ( V j ( κ ) ) M = V j ( κ ) , the set C ( n ) ∩ κ is in U . So, { β < κ : α is C ( n ) -huge, witnessed by anembedding k with k ( α ) = β } ∈ U .Thus, for each α ∈ A , V κ | = “ α is C ( n ) -superhuge” . It follows that { α < κ : V κ | = “ α is C ( n ) -superhuge” } ∈ U . (cid:3) On elementary embeddings of a rank into itself
Finally, we will consider C ( n ) -cardinal forms of the very strong large car-dinal principles known as E i , for 0 ≤ i ≤ ω (see [7]). The principle E (also known in the literature as I3 (see [5], 24)) asserts the existence of anon-trivial elementary embedding j : V δ → V δ , with δ a limit ordinal. Letus call the critical point of such an embedding an E cardinal .If j : V δ → V δ witnesses that κ is E , then Kunen’s Theorem implies that δ = sup { j m ( κ ) : m ∈ ω } , where j m is the m-th iterate of j . It follows that δ ∈ C (1) , because all the j m ( κ ) are inaccessible cardinals (in fact, measurablecardinals) and therefore they all belong to C (1) . Moreover, V κ and V j m ( κ ) ,all m ≥
1, are elementary substructures of V δ . Therefore, V δ | = ZFC. Theorem 7.1. If κ is E , witnessed by j : V δ → V δ , then in V δ , κ (andalso all the cardinals j m ( κ ) , m ≥ ), are C ( n ) -superstrong, C ( n ) -extendible, C ( n ) -supercompact, C ( n ) - k -huge, and C ( n ) -superhuge, for all n, k ≥ .Proof. First notice that in V δ , κ and all iterates j m ( κ ), m ≥
1, belong to C ( n ) , for all n . ( n ) -CARDINALS 23 To see that κ is C ( n ) -superhuge in V δ , pick any α < δ . Then we can find m such that j m ( κ ) > α . Thus, j m : V δ → V δ , crit ( j m ) = κ , α < j m ( κ ), V δ is closed under j m ( κ )-sequences, and j m ( κ ) ∈ C ( n ) . Define U by: X ∈ U if and only if X ⊆ P ( j m ( κ )) ∧ j m “ j m ( κ ) ∈ j m ( X ) . One can easily check that U is a κ -complete fine and normal ultrafilter U over P ( j m ( κ )) with { x ∈ P ( j m ( κ )) : ot ( x ) = κ } ∈ U . Since U ∈ V δ , we can now,in V δ , define the ultrapower embedding k : V δ → M ∼ = U lt ( V δ , U ). Then crit ( k ) = κ , α < k ( κ ), M is closed under k ( κ )-sequences, and k ( κ ) ∈ C ( n ) (see [5], 24.8 for details). Since the same can be done for each α < δ , thisshows that in V δ κ is C ( n ) -superhuge. Hence κ is also C ( n ) -supercompact, C ( n ) -extendible, and C ( n ) -superstrong. The argument for showing that κ is C ( n ) - k -huge is similar to the one for C ( n ) -superhugeness, using the charac-terization of C ( n ) - k -hugeness in terms of ultrafilters (see Section 6). (cid:3) Thus, modulo ZFC, the consistency of the existence of an E cardinalimplies the consistency with ZFC of the existence of all C ( n ) -cardinals con-sidered in previous sections. Notice that a consequence of the Theoremabove (and Corollary 4.15) is that V δ satisfies VP.Let us now say that κ is a C ( n ) - E cardinal if it is E , witnessed by someembedding j : V δ → V δ , with j ( κ ) ∈ C ( n ) .Clearly, if κ is C ( n ) - E , then κ ∈ C ( n ) . In fact, κ is a limit point of C ( n ) .For suppose α is smaller than κ . Then V j ( κ ) satisfies that there exists some β ∈ C ( n ) greater than α , since κ is such a β . Hence, by the elementarity of j , V κ satisfies that some β greater than α belongs to C ( n ) . But since κ ∈ C ( n ) , β does indeed belong to C ( n ) .Every E cardinal is, evidently, C (1) - E . However, a simple reflectionargument shows that the least C ( n ) - E cardinal, for n ≥
1, is smaller thanthe first cardinal in C ( n +1) , and therefore it is not C ( n +1) - E . For suppose α ∈ C ( n +1) is less than or equal to the first C ( n ) - E cardinal κ . Then V α satisfies the Σ n +1 statement asserting the existence of a C ( n ) - E cardinal,because κ witnesses it in V . But if V α thinks some λ is a C ( n ) - E cardinal,then so does V , contradicting the minimality of κ . Proposition 7.2. If κ is C ( n ) - E , then it is C ( n ) - m -huge, for all m , andthere is a κ -complete normal ultrafilter U over κ such that { α < κ : α is C ( n ) - m -huge for every m } ∈ U . Proof.
Let j : V δ → V δ witness that κ is C ( n ) - E , with δ limit. Then as in[5], 24.8 one can show that the ultrafilter V over P ( λ ), where λ = j m ( κ ),defined by X ∈ V if and only if j ′′ λ ∈ j ( X )witnesses that κ is C ( n ) - m -huge. Let U be the usual κ -complete normalultrafilter over κ obtained from j . Since V ∈ V δ , V δ satisfies that κ is C ( n ) - m -huge, and so { α < κ : α is C ( n ) - m -huge for every m } ∈ U . (cid:3) Proposition 7.3.
Suppose j : V δ → V δ witnesses that κ is E , and δ is alimit ordinal. Then the following are equivalent for every n ≥ , (1) j m ( κ ) ∈ C ( n ) , all ≤ m < ω . ( n ) -CARDINALS 24 (2) δ ∈ C ( n ) .Proof. (1) implies (2) is immediate, since δ = sup { j m ( κ ) : m < ω } . And (2)implies (1) directly follows from the easily verifiable fact that V κ and V j m ( κ ) ,all m ≥
1, are elementary substructures of V δ . (cid:3) This suggests the following definitions.
Definition 7.4.
We say that κ is m - C ( n ) - E , where m ≥ , if it is C ( n ) - E ,witnessed by some j : V δ → V δ with j m ′ ( κ ) ∈ C ( n ) for all ≤ m ′ ≤ m . Andwe say that κ is ω - C ( n ) - E if it is C ( n ) - E , witnessed by j : V δ → V δ with δ ∈ C ( n ) . Clearly, κ is E if and only if it is C (1) - E if and only if it is ω - C (1) - E .Observe that if κ is m - C ( n ) - E , where 1 ≤ m < ω , witnessed by j : V δ → V δ with δ the minimal such, then δ C (2) . Otherwise, V δ would reflect theΣ statement: ∃ η ∃ k ( k : V η → V η ∧ crit ( k ) = κ ∧ ^ ≤ m ′ ≤ m ( k m ′ ( κ ) = j m ′ ( κ )))where κ and j m ′ ( κ ), all 1 ≤ m ′ ≤ m , are parameters, and so a witness to thestatement would yield a counterexample to the minimality of δ . It followsthat j cannot witness that κ is ω - C (2) - E .Note also, by the reflection argument given prior to Proposition 7.2, thatthe least ω - C ( n ) - E cardinal κ is smaller than the first cardinal in C ( n +1) .Hence, no cardinal less than or equal to κ is C ( n +1) - E , for n ≥ Proposition 7.5.
The least m - C ( n ) - E cardinal is not ( m + 1) - C ( n ) - E , forall m ≥ and n ≥ .Proof. Suppose κ is the least m - C ( n ) - E cardinal and suppose, aiming fora contradiction, that j : V δ → V δ is elementary, with crit ( j ) = κ and j m ′ ( κ ) ∈ C ( n ) , for all m ′ ≤ m + 1. Then, V j m +1 ( κ ) satisfies the sentence: ∃ i ∃ β ∃ µ ( i : V β → V β is elementary ∧ crit ( i ) = µ ∧ µ < j ( κ ) ∧ ^ ≤ m ′ ≤ m i m ′ ( µ ) = j m ′ ( κ ))because j , δ , and κ witness it and the sentence is Σ with parameters j m ′ ( κ ),all 1 ≤ m ′ ≤ m . Hence, by elementarity the following holds in V j m ( κ ) : ∃ i ∃ β ∃ µ ( i : V β → V β is elementary ∧ crit ( i ) = µ ∧ µ < κ ∧ ^ ≤ m ′ ≤ m i m ′ ( µ ) = j m ′ − ( κ ))where j ( κ ) = κ . And since j m ( κ ) ∈ C ( n ) , it also holds in V . But if µ witnesses it, then µ is m - C ( n ) - E , contradicting the minimality of κ . (cid:3) It is not hard to see that κ is E , witnessed by an elementary embedding j : V δ → V δ , if and only if κ is the critical point of an embedding k : V δ +1 → V δ +1 which is Σ elementary, i.e., it preserves truth for bounded formulas,with parameters. The main point is to notice that j extends uniquely to a Σ ( n ) -CARDINALS 25 elementary embedding k : V δ +1 → V δ +1 by letting k ( A ) := S α<δ j ( A ∩ V α ),for all A ⊆ V δ (see [7] or [3] for details).So it is only natural to consider the principles E i , for 1 ≤ i ≤ ω ([7]),which assert the existence of a non-trivial Σ i elementary embedding j : V δ +1 → V δ +1 , i.e., j preserves the truth of Σ i formulas, with parameters.Thus, E ω asserts that j is fully elementary. E and E ω are also known inthe literature as I2 and I1, respectively (see [5]).Observe that an embedding j : V δ +1 → V δ +1 is Σ i elementary if and onlyif its restriction to V δ is Σ i elementary. (Recall that a formula is Σ i ifit is a second order formula which begins with exactly i -many alternatingsecond-order quantifiers, beginning with an existential one, and the rest ofthe formula has only first-order quantifiers.) We shall later make use of thefact (folklore) that for each i ≥
1, the formula j : V δ → V δ is Σ i elementaryis Π i +1 expressible in V δ +1 , in the parameters j and δ , for it is equivalent to:For every A ⊆ V δ and every Σ i formula ∃ X ∀ X . . . ∃ X i ψ ( X , . . . , X i , Y ),where ψ has only first-order quantifiers, ∃ X ∀ X . . . ∃ X i ( h V δ , ∈ , X , . . . , X i , A i | = ψ ( X , . . . , X i , A ))if and only if ∃ X ∀ X . . . ∃ X i ( h V δ , ∈ , X , . . . , X i , A, j i | = ψ ( X , . . . , X i , [ α<δ j ( A ∩ V α ))) . Now, it was shown by Donald Martin that, for i odd, if j : V δ +1 → V δ +1 is Σ i elementary, then it is also Σ i +1 elementary (see [7]). So one only needsto consider the principles E i when i is even.Analogously as in the case of E , let us call the critical point κ of a Σ i elementary embedding j : V δ +1 → V δ +1 an E i cardinal . If, in addition, j ( κ ) ∈ C ( n ) , then we call κ a C ( n ) - E i cardinal . More generally, if j m ′ ( κ ) ∈ C ( n ) , for all 1 ≤ m ′ ≤ m , then we say that κ is an m - C ( n ) - E i cardinal . Andif δ ∈ C ( n ) , then we say that κ is ω - C ( n ) - E i . (Note that, by Proposition 7.3, δ ∈ C ( n ) if and only if j m ( κ ) ∈ C ( n ) for all m .)For each i ≤ ω and m, n ≥
1, the existence of an m - C ( n ) - E i cardinal canbe expressed as a Σ n +1 statement, namely ∃ δ ∃ j ∃ κ ( j : V δ +1 → V δ +1 is Σ i elementary ∧ crit ( j ) = κ ∧∀ ≤ m ′ ≤ m ( j m ′ ( κ ) ∈ C ( n ) )) . (Notice that in the case m = ω , j m ( κ ) = δ .) A reflection argument similarto the one given prior to Proposition 7.2 now yields that the least m - C ( n ) - E i cardinal is smaller than the first cardinal in C ( n +1) , for all m, i ≤ ω and n ≥
1, and therefore smaller than the least C ( n +1) - E cardinal.Trivially, if κ is m - C ( n ) - E i +1 , then it is m - C ( n ) - E i . In the case i = 0much more is true: arguing similarly as in [5], 24.4 one can show that thereis a normal ultrafilter U over κ such that the set of cardinals α < κ that are m - C ( n ) - E belongs to U . In the general case we have the following. Theorem 7.6.
Suppose j : V δ +1 → V δ +1 witnesses that κ is an m - C ( n ) - E i +2 cardinal, where i, n < ω and m ≤ ω . Then the set of m - C ( n ) - E i cardinals isunbounded below κ . ( n ) -CARDINALS 26 Proof.
Fix γ < κ . Then the following holds in V δ +1 : ∃ k ∃ β ∃ α ( k : V β → V β is Σ i elementary ∧ γ < crit ( k ) = α < j ( κ ) ∧∀ ≤ m ′ ≤ m ( k m ′ ( α ) = j m ′ ( κ )))because j , δ , and κ witness it.As we observed above, the formula “ k : V β → V β is Σ i elementary” isΠ i +1 expressible in V δ +1 in the variables k and β . So the last displayedstatement is Σ i +2 , with γ and h j ( κ ) , j ( κ ) , . . . , j m ( κ ) i as parameters. Thus,since j is Σ i +2 elementary, V δ +1 satisfies: ∃ k ∃ β ∃ α ( k : V β → V β is Σ i elementary ∧ γ < crit ( k ) = α < κ ∧∀ ≤ m ′ ≤ m ( k m ′ ( α ) = j m ′ − ( κ )))where j ( κ ) = κ . If k , β , and α witness the statement, then the embedding k : V β → V β witnesses that α is an m - C ( n ) - E i cardinal greater than γ . (cid:3) Similarly as in Proposition 7.5 one can show that the least m - C ( n ) - E i cardinal, where i ≤ ω , is not ( m + 1)- C ( n ) - E i , for all m ≥
1, and all n ≥ n , let κ ( n ) denote the first cardinal in C ( n ) , andfor i ≤ ω and 2 ≤ m ≤ ω , let κ ( n ) i and m - κ ( n ) i denote the least C ( n ) - E i cardinal and the least m - C ( n ) - E i cardinal, respectively. Then, assuming allthese cardinals exist, we have: κ ( n ) < κ ( n ) i < m - κ ( n ) i < m - κ ( n ) i +2 < m - κ ( n ) ω < ( m + 1)- κ ( n ) ω < ω - κ ( n ) ω < κ ( n +1) , for all i in all cases; all n in the case of the inequalities 1,3,4,and 6; all n ≥ n ≥ ≤ m ≤ ω in the case of inequalities 3,4, 5, 6, and 7; and all 2 ≤ m ≤ ω in the case of the second inequality.The first inequality is clear. Inequalities 2 and 5 follow from an argumentsimilar to the proof of Proposition 7.5. Inequalities 3,4, and 6 follow fromTheorem 7.6. And the last inequality can be shown by a reflection argumentsimilar to that given just before Proposition 7.2. References [1] Bagaria, J., Casacuberta, C., Mathias, A.R.D., and Rosick´y, J. (2010). Definableorthogonality classes are small. Submitted for publication.[2] Barbanel, J., Di Prisco, C. A., and Tan, I. B. (1984) Many times huge and superhugecardinals.
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