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  • Arch. Math. Logic (2014) 53:89118DOI 10.1007/s00153-013-0357-4 Mathematical Logic

    Elementary chains and C(n)-cardinals

    Konstantinos Tsaprounis

    Received: 29 November 2012 / Accepted: 17 September 2013 / Published online: 3 October 2013 Springer-Verlag Berlin Heidelberg 2013

    Abstract The C (n)-cardinals were introduced recently by Bagaria and are strongforms of the usual large cardinals. For a wide range of large cardinal notions, Bagariahas shown that the consistency of the corresponding C (n)-versions follows from theexistence of rank-into-rank elementary embeddings. In this article, we further studythe C (n)-hierarchies of tall, strong, superstrong, supercompact, and extendible cardi-nals, giving some improved consistency bounds while, at the same time, addressingquestions which had been left open. In addition, we consider two cases which werenot dealt with by Bagaria; namely, C (n)-Woodin and C (n)-strongly compact cardi-nals, for which we provide characterizations in terms of their ordinary counterparts.Finally, we give a brief account on the interaction of C (n)-cardinals with the forcingmachinery.

    Keywords C (n)-cardinals Woodin cardinals Strongly compact cardinals

    Mathematics Subject Classification (2010) 03E35 03E55

    1 Introduction

    The concept of reflection, which pervades the body of set theory, has a long historywith its origins tracing back to Gdel and, indeed, Cantor himself. Perhaps its mostrenowned formal manifestation is the well-known Reflection Principle, which is dueto Lvy and Montague from the early 1960s.

    K. Tsaprounis (B)Department of Logic, History and Philosophy of Science,University of Barcelona, 08001 Barcelona, Spaine-mail: kostas.tsap@gmail.com

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  • 90 K. Tsaprounis

    For any fixed natural number n, we let C (n) denote the collection of ordinalswhich are n-correct in the universe, that is, C (n) = { : V n V }. In otherwords, C (n) if and only if V reflects all true n-statements, with parame-ters in V . Already by the work of Lvy and Montague it follows that, for any nat-ural number n, C (n) is a closed and unbounded proper class. For example, C (0) =ON while C (1) is precisely the class of those uncountable cardinals for whichH = V; notably, such a local characterization is not available for the classes C (n),when n > 1.

    Blending this aspect of reflection with the usual large cardinal notions, Bagariarecently introduced the various hierarchies of C (n)-(large) cardinals (cf. [1]), andunderlined their significance by revealing intimate connections between such princi-ples and the concept of structural reflection for the set-theoretic universe. For instance,Bagaria showed that the existence of C (n)-extendible cardinals is equivalent to frag-ments of Vopenkas Principle, in a way which generalizes the ordinary notions ofsupercompactness and extendibility.

    To define theC (n)-cardinals, wework in the usual framework of elementary embed-dings j for the ordinary large cardinal postulates (i.e., where the critical point cp( j) = is the cardinal in question) and we require, in addition to the standard defini-tion for the postulate at hand and for any given n, that the image j () belongsto the class C (n), as the latter is computed in V . This modification gives rise tothe corresponding C (n)-(large) cardinal. For concreteness, let us give the definitionof C (n)-extendibility.

    Definition 1.1 ([1]) A cardinal is called -C (n)-extendible, for some > , if thereexists a > and an elementary embedding j : V V with cp( j) = , j () > and j () C (n); we say that is C (n)-extendible if it is -C (n)-extendible for all > .

    Bagaria studied the C (n)-versions of large cardinals such as measurable, strong,superstrong, supercompact, extendible, huge, superhuge (and more). It is shown in[1] that the consistency of all the aforementioned C (n)-cardinals follows from theexistence of an I3 embedding (see 24 in [7]).

    The structure of this article is as follows. In the rest of this section, we give somenecessary preliminaries and we also introduce the notions of C (n)-tall, C (n)-Woodin,and C (n)-strongly compact cardinals.

    In Sect. 2, we use the versatile tool of elementary chains in order to derive consis-tency upper bounds for the C (n)-versions of tall, strong, superstrong, supercompactand extendible cardinals.We also study the connection betweenC (n)-extendibility andC (n)-supercompactness.

    In Sect. 3, we consider the cases of C (n)-Woodin and of C (n)-strongly compactcardinals, and we obtain equivalent formulations for them in terms of their ordinarycounterparts.

    Finally, in Sect. 4, we briefly study the interaction ofC (n)-cardinals with the forcingmachinery.

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  • Elementary chains and C(n)-cardinals 91

    1.1 Preliminaries

    Our notation and terminology are mostly standard.1 ZFC denotes the familiar first-order axiomatization of ZermeloFraenkel set theory, together with the Axiom ofChoice. Given any function f and any S dom( f ), we write f S for the restrictionof the function to S and also, we write f S for the pointwise image, that is, thecollection { f (x) : x S}. For any X and Y, X Y is the collection of all functions fwith dom( f ) = X and range( f ) Y . The class of ordinal numbers will be denotedby ON; given an infinite ordinal , cof() stands for its cofinality.

    For any A ON, sup(A) denotes the supremum of A (in case A is a set), whileLim(A) denotes the collection of its limit points, that is, { : sup(A) = }. Given alimit ordinal with cof() > and some C , C is called club in if sup(C) = and Lim(C) C ; moreover, C is called -club in , for some regular < cof(),if sup(C) = and { Lim(C) : cof() = } C . Likewise, if I cof() isan ordinal interval, then C is called I -club in if it is -club, for all regular I .Finally, S is called stationary in if S C = for every club C .

    Partial orders will be denoted by blackboard capital letters such as P and Q. Wewrite p < q to mean than p is stronger than q. We use V P to stand for the universe ofP-names. Moreover, if G is P-generic over V and ON, we let V [G] = (V)V [G]and V[G] = {G : V P V}.

    Given a non-trivial elementary embedding j : V M , where M is a transitiveclass model of ZFC, we denote by cp( j) its critical point, that is, the least ordinalmoved by j . Most of the large cardinal notions that we study are standard; see [7] formore details. We now give a less popular definition, which is still interesting in itsown right.

    Definition 1.2 ([6]) A cardinal is called -tall, for some > , if there exists anelementary embedding j : V M , with M transitive, cp( j) = , M M andj () > ; we say that is tall if it is -tall for every > .

    Tall cardinals were introduced by Hamkins. In [6], it is shown that tallness embed-dings can be described by extenders, and that the existence of a tall cardinal is equicon-sistent with the existence of a strong cardinal.

    1.2 C (n)-cardinals

    Recall that, for every n 1, membership in C (n) is expressible by a n (but not byany n) formula. In particular, C (n+1) C (n), i.e., the inclusion is proper.

    All the background definitions on C (n)-cardinals may be found in [1]. A specialcase is that of C (n)+-extendibility, defined as follows.Definition 1.3 ([1]) A cardinal is called -C (n)+-extendible, for some > with C (n), if there exists a C (n) and an elementary embedding j : V V ,with cp( j) = , j () > and j () C (n); we say that is C (n)+-extendible if it is-C (n)+-extendible for all > with C (n).

    1 See [7] for an account on all undefined set-theoretic notions.

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  • 92 K. Tsaprounis

    For the C (n)-cardinals which will be of interest to us, it is shown in [1] that, forevery n 1, the statements is-C (n)-strong, isC (n)-superstrong, is-C (n)-supercompact, is -C (n)-extendible and is -C (n)+-extendible are alln+1-expressible. Consequently, the corresponding global versions aren+2-expressible.2

    Let us now introduce the C (n)-versions of tall, Woodin, and strongly compactcardinals.

    Definition 1.4 A cardinal is called -C (n)-tall, for some > , if there existsan elementary embedding j : V M , with M transitive, cp( j) = , j () >, M M and j () C (n); we say that is C (n)-tall if it is -C (n)-tall for all > .

    Either using ordinary extenders or, MartinSteel ones, for every n 1, the state-ment is -C (n)-tall isn+1-expressible; thus, isC (n)-tall isn+2-expressible.

    Definition 1.5 A cardinal is called C (n)-Woodin if for every f there exists a < with f , and there exists an elementary embedding j : V M , withM transitive, cp( j) = , Vj ( f )() M, j () = and j () C (n).

    Observe that the above definition is in accordancewith the local character ofWoodincardinals, i.e., we demand that j () = so that the various embeddings may bewitnessed by extenders in V . It is easy to see that, for fixed n, the statement isC (n)-Woodin is absolute for any V with > and C (n). Notice also that ifthe cardinal is C (n)-Woodin, then (is of course Woodin and) belongs to Lim(C (n)).As we shall see in 3.1, this is no coincidence. Finally, we consider the followingnotion.

    Definition 1.6 A cardinal is called -C (n)-compact, some > , if there existsan elementary embedding j : V M , with M transitive, cp( j) = , j () C (n)and such that, for every X M with |X | , there is a Y M so that X Y andM | |Y | < j (); we say that is C (n)-strongly compact if it is -C (n)-compact forall > .

    2 Consistency bounds

    Wefocus on theC (n)-versions of tall, superstrong, strong, supercompact and extendiblecardinals (in order of appearence). Our general method can be roughly described asfollows.

    Suppose that we are given such a C (n)-cardinal and an elementary embeddingj : V M witnessing this fact appropriately. Under the additional assumptionthat the image j () is a regular (or even inaccessible) cardinal in V , we shall con-struct various elementary chains of substructures of the model M , giving rise to factorelementary embeddings which have analogous strength to that of the initial j .

    2 It should be pointed out that, e.g., in the case of -C(n)-supercompactness, the expressibility can beattained using the machinery of MartinSteel extenders. For more details on such extenders see [8]; fortheir use in the context of (C(n)-) supercompactness, the reader may consult 5 in [1] or the Appendix of[10].

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  • Elementary chains and C(n)-cardinals 93

    The aim of these constructions is to ensure that the ordinals below j () whicharise as images of the large cardinal under embeddings of the sort in question, isa sufficiently rich subset of j (); e.g., stationary, -club for some < j (), etc.If j () is indeed an inaccessible cardinal, we then check that all the aforementionedfactor embeddings can be witnessed inside the model Vj () via derived extenders andwe, consequently, obtain corresponding consistency upper bounds for each individualC (n)-hierarchy.

    We begin by first considering the case of tallness, where we shall describe ourmethod in detail.

    2.1 Tallness

    Suppose that is -tall for some > , as witnessed by the elementary embeddingj : V M , i.e., M transitive, cp( j) = , j () > and M M . Suppose that,in addition, j () is a regular cardinal. We pick some limit ordinal (, j ()) andconsider the following elementary substructure:

    X ={

    j ( f )(x) : f V, f : V V, x V M} M.

    To check that X M , one uses the fact that is a limit ordinal and verifies theTarskiVaught criterion. Notice that X is, in fact, the Skolem hull of the range( j)together with V M inside M , with respect to the functions of the form f : V V .

    Starting with X0 = X and 0 = , we recursively build, for any < j (), anincreasing (under ) sequence of elementary substructures X M , together witha strictly increasing sequence of corresponding limit ordinals < j (), such thateach X is of the form

    X ={

    j ( f )(x) : f V, f : V V, x V M}.

    Our aim is to show that, at appropriate ordinals < j (), using the current sub-structure X of this chain, we can define an elementary embedding j which can benicely represented and which, at the same time, witnesses -tallness for the cardinal .

    So let 0 = and X0 as defined above. For any + 1 < j (), given and X ,we define

    +1 = sup(X j ())+

    and

    X+1 ={

    j ( f )(x) : f V, f : V V, x V M+1}.

    If < j () is limit and we have already defined and X for every < , we let = sup

  • 94 K. Tsaprounis

    X = , let us consider the current substructure X =

  • Elementary chains and C(n)-cardinals 95

    filled in all the ordinal holes below j () along our recursive construction of thesubstructures X . It then easily follows that

    cp(k ) = j () = sup(X j ()) = .

    In fact, in such constructions, X j () is an ordinal if and only if cp(k ) = j ()in which case, we call the embedding j an initial factor of j . Being slightly moregeneral, suppose that j : V M is an elementary embeddingwith cp( j) = and Mtransitive; moreover, suppose that X M with range( j) X, X [, j ()) = and X j () bounded in j (). Let : X = M0 be the Mostowski collapse andconsider, as above, the map j0 = j : V M0 with cp( j0) = , which formsa commutative diagram (with k = 1). Observe that the imposed requirements onthe elementary substructure X ensure that j0 is well-defined, M0 = V (i.e., j0 = id),cp( j0) = and j0() < j (). We now introduce the following notion.Definition 2.3 Such a j0 is called an initial factor of j , if cp(k) = j0().The following facts are easily verified.

    Fact 2.4 In the situation described above, j0 is an initial factor of j if and only ifX j () is an ordinal. In such a case, j0() = sup(X j ()).Fact 2.5 If j0 is an initial factor of j (via the collapse : X = M0 with k = 1),then V M0j0() = V Mj0() range(k) and therefore,

    {j ( f )(x) : f V, f : V V, x V Mj0()

    } range(k).

    Returning to our argument, we have that j () = > and so, in order toconclude that this embedding witnesses -tallness, we only need to check that M M . This essentially comes from the fact that the set of seeds that generate M is closedunder -sequences, a fact which, in turn, follows from M M and cof( ) > .Notice that

    M = X ={

    j ( f )(x) : f V, f : V V, x V M}.

    Now suppose that { j ( f)(x) : < } M , where for each < , x V M andf V . Since cof( ) > , we clearly have that x : < V M M . It is alsoobvious that

    j ( f) : < = j ( f : < ) M

    and, hence, we can compute in M the sequence j ( f)(x) : < by evaluatingpointwise the functions j ( f)s at the corresponding xs, i.e., j ( f)(x) : < M . We have thus proved the following.

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  • 96 K. Tsaprounis

    Proposition 2.6 Suppose that j : V M is a -tall embedding for , with j ()regular. Then, for any given (initial limit ordinal) 0 (, j ()) and for any < j ()with cof( ) > , the embedding j : V M arising from the elementary chainconstruction as above is an initial factor of j witnessing the -tallness of .

    In order to establish the closure under -sequences the only relevant informationwas the fact that M M and that cof( ) = cof( ) > . Thus, we may be slightlymore general and state the following.

    Corollary 2.7 Suppose that j : V M is a -tall embedding for , with j ()regular. Suppose that j0 : V M0 = X is an initial factor of j via the Mostowskicollapse of

    X ={

    j ( f )(x) : f V, f : V V, x V M} M,

    where (, j ()) is such that cof() > . Then, j0 is -tall for .Our next aim is to consider the class of all images h() below j (), where h is any

    (initial factor) tallness embedding for .

    Proposition 2.8 Suppose that j : V M witnesses the -tallness of , with j ()regular. Then, the collection

    D = {h() < j () : h is -tall for , for some < j ()}

    contains a [+, j ())-club.Proof Let us consider the collection D of all images j0() below j (), where j0 isany initial factor embedding arising from a Mostowski collapse of some elementarysubstructure X M of the form

    X ={

    j ( f )(x) : f V, f : V V, x V M},

    for some (, j ())with cof() > . We show that D , which is contained in D, isin fact a [+, j ())-club in j (). Clearly, D contains all the images j () arising frominitial factor embeddings coming from our elementary chain construction, for variousinitial limit ordinals 0 (, j ()) and various lengths < j () with cof( ) > .

    So suppose that [+, j ()) is regular and that ji () : i < is a strictlyincreasing sequence below j () where, for all i < , there is some i (, j ())with cof(i ) > , such that ji : V Mi is an initial factor of j arising via thecollapse i of the substructure

    Xi ={

    j ( f )(x) : f V, f : V V, x V Mi} M.

    Recall that Xi j () is an ordinal and ji () = sup(Xi j ()). Let us point out that,the fact that ji comes from the collapse of Xi , only implies that ji () i . On theother hand, it follows from Fact 2.5 that

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  • Elementary chains and C(n)-cardinals 97

    {j ( f )(x) : f V, x V Mji ()

    }=

    {j ( f )(x) : f V, x V Mi

    }

    and thus, we may as well assume that ji () = i , for every i < . Hence, for anyi < < , we have that Xi X which, in turn, gives that Xi X and so, anelementary chain is formed. We may now let = sup

    i , a fact which iscomputed correctly since M0 M0. This means that the set of seeds V M0j0() (which

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  • 98 K. Tsaprounis

    generate ME ) is closed under -sequences and so, by argumentswhichwe have alreadydescribed, it follows that ME ME , i.e., jE witnesses -tallness as well.

    Actually, in our case, M0 = ME which follows from general facts regarding thecommutative diagram of embeddings, for derived extenders. Notice also that, in thecurrent setting, all derived extenders belong to Vj ().

    Then, in the ZFC model Vj (), for any such extender E coming from a factorj0, Vj () | jE is -tall for . This follows from the inaccessibility of j () whichenables Vj () to faithfully verify that jE () = j0() and ME ME . Therefore, thecollection

    Ctall = {hE () < j () : hE is -tall extender embedding, < j ()} ,

    which is a class in j () definable in Vj (), contains a [+, j ())-club (in particular,Ctall is stationary in j ()). Also, again by inaccessibility, for each n , we have aclub C (n)j () j (), consisting of all ordinals below j () which are n-correct in thesense of Vj (). Hence, C

    (n)j () Ctall = , for every n , and the theorem is proved.

    Remark 2.10 We could have assumed that j () is inaccessible, right at the beginningof the section, without any change in the construction of the chains. Nevertheless, wepresented our construction under the minimal assumptions and this is accomplishedby just requiring the regularity of j ().

    Remark 2.11 It is important to note that Theorem 2.9 is really a theorem, and not atheorem schema, something that is indicated by the clause for every n . Thisis not problematic since we have a truth predicate for set structures like Vj (). Insubsequent sections, we shall frequently state similar (consistency) results for otherlarge cardinal notions. Throughout, we understand that clauses of the form for everyn , followed by some satisfaction in a set model, indicate a theorem instead of aschema. For example, the conclusion of Theorem 2.9 can be written (slightly more)formally as: n (Vj () | is C (n)-tall). This remark applies to all similarresults throughout this article. In other places of the text where we shall be lookingat satisfaction in class models (e.g., C (n)-cardinals in V ), we will explicitly indicateany result which is a countable schema of statements, one for each (meta-theoretic)natural number n N.

    Notice that Theorem 2.9 gives an (indirect) upper bound on the consistency strengthof the theory ZFC+ is C (n)-tall, for every natural number n, where the latter isindeed a countable schema of formulas. In turn, it gives rise to the following naturalquestion.

    Question 2.12 What is the consistency strength of the existence of a tallness embed-ding j for such that, additionally, j () is inaccessible?

    As we shall see, 1-extendibility is an adequate upper bound answering the previousquestion; indeed, it will be a sufficient assumption for results concerning the cases ofsuperstrongness and that of strongness as well (cf. 2.2 and 2.3). On the other hand,

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  • Elementary chains and C(n)-cardinals 99

    let us mention that for supercompactness (and extendibility), we shall be using theassumption almost hugeness (cf. 2.4), although the exact bounds for all these casesare not known.

    Having dealt with tall cardinals, the essential features of our method have becomeapparent and we now proceed with the rest of the large cardinals. Since many ofthe constructions will be in a similar spirit, we will skip several details and refer topreviously established facts when needed.

    2.2 Superstrongness

    Suppose that is a superstrong cardinal and let j : V M be a witnessingembedding, i.e., M is transitive, cp( j) = and Vj () M . In addition, suppose thatj () is regular. Bear in mind that in such a case, this is actually equivalent to requiringthat j () is inaccessible, since superstrongness already implies that j () C (1). Wecan thus forget about this distinction and assume that j () is inaccessible right away,proving the following.

    Proposition 2.13 Suppose that j : V M is superstrong for , with j () inacces-sible. Then, for each (initial limit ordinal) 0 (, j ()) and any limit < j (), theembedding j : V M arising from the elementary chain construction as before,is an initial factor of j and superstrong for .

    Proof Having fixed an initial limit ordinal 0 (, j ()), we recursively constructan elementary chain of substructures X of M as before, starting with seeds in V0 .At the same time, we produce a sequence of ordinals b below j ().

    At any limit < j (),4 we let : X = M be the Mostowski collapse, and welet j = j : V M with cp( j ) = , producing a commutative diagram ofembeddings (where k = 1 ). Once again, X j () is an ordinal and hence j isan initial factor of j such that

    cp(k ) = j () = sup(X j ()) = sup

  • 100 K. Tsaprounis

    Exactly as in the case of tallness, for each superstrong initial factor embeddingj0 : V M0, the derived (, j0())-extender E belongs to Vj () and then, thecorresponding extender embedding jE : V ME is superstrong for . More-over, one again checks that, in fact, M0 = ME . Finally, by inaccessibility, Vj () |

    jE is superstrong for for any such extender and so, in particular, Vj () | issuperstrong. This means that the collection Cs-strong defined as

    {hE () < j () : hE is a superstrong extender embedding for },

    is a definable class of Vj () and contains a club. Then, by considering the clubsC(n)j ()

    j () consisting of all ordinals below j () which are n-correct in the sense of Vj (),we get the following.

    Theorem 2.15 If j : V M is superstrong for with j () inaccessible, then, forevery n , Vj () | is C (n)-superstrong.

    This theorem gives an (indirect) upper bound on the consistency of the theoryZFC+ is C (n)-superstrong, for every natural number n, where the latter is again acountable schema.

    On the other hand, for any fixed natural number n, by Proposition 2.4 in [1], theexistence of a C (n) which is 2 -supercompact implies the existence of many C (n)-superstrong cardinals below . Since being 2 -supercompact implies the existenceof many 1-extendible cardinals below it, the following corollary is an improvement ofthe aforementioned consistency bound of Bagaria.

    Corollary 2.16 If is 1-extendible then there exists a normal ultrafilter U on suchthat

    { < : n (V | is C (n)-superstrong)} U .

    In particular, if ZFC + ( is 1-extendible) is consistent, then so is the theoryZFC + is C (n)-superstrong, for every natural number n.Proof Suppose that j : V+1 Vj ()+1 witnesses the 1-extendibility of . LetE be the derived (, j ())-extender from j and consider the extender embeddingjE : V ME . By standard arguments, jE is superstrong for , jE () = j (), andmoreover,

    Vj ()+1 | jE is superstrong for jE () = j () is inaccessible

    (where note that E Vj ()+1). Also, Cs-strong, the definable stationary subclass ofj () which was mentioned after Corollary 2.14, belongs to Vj ()+1. Additionally, forevery n , we may consider the club class C (n)j () j () consisting of the ordinalsthat are n-correct in Vj (); since P( j ()) Vj ()+1, the latter verifies the fact thatCs-strong is stationary in j (). Thus, for every n ,

    Vj ()+1 | (, )-extender E Vj () for some < j (), such thatVj () | jE is superstrong for and jE () C (n)

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  • Elementary chains and C(n)-cardinals 101

    that is, for every n ,

    Vj () | is C (n)-superstrong.

    Now, if we derive the usual normal ultrafilter U on from the initial embedding j ,it then follows that, for every n , the set

    Sn = { < : V | is C (n)-superstrong} U .

    By the completeness of U , we may now intersect all the Sns, and the conclusionfollows.

    Observe that, in the previous proof, since is itself inaccessible, V is a model ofthe theory ZFC+ is C (n)-superstrong, for every natural number n, in which thereis actually a proper class of s that satisfy the schema.

    Let us also make a remark on the connection between the cases of superstrongnessand tallness. Notice that if j : V M is superstrong for , we may as well assumethat j = jE , i.e., that it is an extender superstrong embedding with

    M = { j ( f )(x) : f V, f : V V, x Vj ()}.

    So, if j () is also regular (and thus, inaccessible), then the same embedding witnesses< j ()-tallness for . Thus, everything we did for tallness can be entirely done underthe context of a superstrong embedding with regular (inaccessible) target. We cantherefore state the following, which immediately follows from Theorem 2.9.

    Corollary 2.17 Suppose that j : V M is superstrong for , with j () inacces-sible. Then, for every n , Vj () | is C (n)-tall.

    Consequently, 1-extendibility is an adequate (consistency) upper bound both forthe C (n)-superstrongness and for the C (n)-tallness case.

    2.3 Strongness

    Suppose that is -strong, for some limit > , as witnessed by the embeddingj : V M , i.e., M transitive, cp( j) = , j () > and V M . In addition,suppose that j () is regular.

    We apply the same ideas in order to build an elementary chain of substructures of Mand then produce a -strong factor embedding from it. Two remarks are in order here.First, sinceas in the case of superstrongswe are not interested in closure undersequences for the factor embedding, the length at which we take collapses can be anylimit ordinal < j (). Additionally, since the crucial requirement for -strongnessis V M, we will start our chain by just throwing in all the seeds from V, i.e.,we let 0 = and define our first elementary substructure as

    X0 ={

    j ( f )(x) : f V, f : V V, x V0} M.

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  • 102 K. Tsaprounis

    From that point on, there two ways to proceed. On the one hand, we might as wellproceed with the rest of the chain as usual, using seeds x from V M for > 0 (wherethe s are defined as in the case of superstrongness). That would result in the desired-strong initial factor embedding after steps, for any limit ordinal < j (). In thiscase, there is really notmuchmore to say, as the construction and the subsequent resultsare totally parallel to the ones of 2.2 (in particular, we will get a full club containedin the collection of images h() < j (), where h is a -strong factor embeddingfor ).

    Alternatively, one can consider only ordinal seeds, i.e., for appropriate limit ordi-nals5 (, j ()), we take substructures of the form

    X ={

    j ( f )() : f V, f : V, < } M

    (note the modification in the domain of the functions). We omit more details on thisapproach as the results obtained are the same.

    Finally, if assume that j () is also inaccessible, we then get that for every n , Vj () | is -C (n)-strong. Furthermore, as in the case of tallness and super-strongness, 1-extendibility is an adequate upper bound for the schema asserting fullC (n)-strongness, for every natural number n. Such a conclusion is hardly surprising inthe light of Proposition 1.2 in [1], which asserts that every -strong cardinal is actually-C (n)-strong; this is shown in [1] by the method of iterated ultrapowers which, bythe way, will be relevant for us in Sect. 3.

    2.4 Supercompactness

    Suppose that is -supercompact, for some > with and M M . Inaddition, suppose that j () is regular.

    In this case, as opposed to the previous ones, we build our elementary chain usinga slightly different collection of seeds. Namely, we also include j which (belongsto M and) is used in order to obtain the closure under -sequences for the initial factorembedding that we are aiming for.

    We start by picking an initial limit ordinal 0 (, j ()) and by letting

    X0 ={

    j ( f )( j, x) : f V, f : P V V, x V M0} M,

    where note that the domain of the functions has beenmodified accordingly. For +1 with

  • Elementary chains and C(n)-cardinals 105

    producing the elementary chain X ()0 X () M . As before, at anylimit < j ()with cof( ) +, wemay pause and take the transitive collapse of thecurrent substructure, in order to produce a -supercompact initial factor embeddingj () with j

    () () = () , where () is the current ordinal of the produced sequence

    () : < j (). Thus, for any regular (, j ()), we have a corresponding classof ordinals in j (), namely, C = {() : cof( ) = +} which consists of the imagesof under the initial factor -supercompactness embeddings that are taken exactly atlimits of cofinality +. Clearly, C is +-club and, thus, stationary in j ().

    Now, we may let C s.c. =

    {C : (, j ()), regular} and then C s.c. is a sta-tionary subset of j () which is a disjoint union of stationary subsets. Of course, sincej () is inaccessible, we may also consider the clubs C (n)j () Vj () of ordinals thatare n-correct in the sense of Vj ().

    Then, for every n and every regular (, j ()), there is an initial factor-supercompact embedding j0, with j0() C (n)j (). We now use the characterizationof Theorem 2.20 in order to show that all these embeddings are witnessed by extendersinside Vj ().

    Fix some n and suppose that, for some regular (, j ()), j0 : V M0is an initial factor -C (n)-supercompact embedding arising from our construction,i.e., arising via a collapse of an elementary substructure X M of the form above,associatedwith a corresponding limit ordinal = sup(X j ()). As before, a countingargument shows that, for all < j (), we have that j0() < j ().

    We now describe how to produce the appropriate transitive Y M0. The idea issimple: we start with j0 (which belongs to M0) and we recursively close under allthe relevant properties, repeating +-many times (and taking unions at limit stages).Formally, we let Y0 = trcl({ j0}) and, given any Y for some < +, we letY+1 = trcl(Y [Y]

  • 106 K. Tsaprounis

    The last theorem gives an upper bound on the consistency strength of the theoryT=ZFC+ is C (n)-supercompact, for every natural number n, where the latteris again a countable schema. Namely, if the theory ZFC+ ( is almost huge)is consistent, then so is T. Moreover, we now show that this bound is sharp in thefollowing sense.

    Corollary 2.23 If the theory ZFC+ ( is almost huge) is consistent, then so isthe theory T + ( is not almost huge).Proof Let be the least almost huge cardinal, witnessed by the embedding j : V M . By Theorem 2.21, Vj () | T. Towards a contradiction, suppose that is the leastalmost huge cardinal in the sense of Vj (). Recall that the least almost huge cardinal isstrictly smaller than the least supercompact, provided that they both exist. Thus, since is supercompact in Vj (), we get that < . But this is a contradiction since isalmost huge is a 2-statement and j () is inaccessible, i.e., would have to be analmost huge cardinal below .

    It is a standard fact that if is < -supercompact and is supercompact, then is fully supercompact. A natural question is whether this generalizes to the case ofC (n)-supercompactness, for n > 0. We now address this question and show that if,as mentioned in Remark 2.22, the witnessing extenders are also superstrong, then itdoes.

    In this direction, we introduce the following notion, which will be of particularinterest and importance in the context of (C (n)-) extendibility.

    Definition 2.24 A cardinal is called jointly -supercompact and -superstrong,for some , , if there is an elementary embedding j : V M with Mtransitive, cp( j) = , j () > , M M and Vj () M .

    Note that the clause -superstrong justmeans usual superstrongness (i.e., Vj () M). For the global version(s) of this notion, the absence of one of the two parametersindicates universal quantification on the parameter missing; for instance, is jointlysupercompact and -superstrong, for some fixed , if and only if it is jointly-supercompact and -superstrong, for every . On the other hand, the absenceof both parameters is intended to mean that the same is quantified for both of them,i.e., is jointly supercompact and superstrong if and only if it is jointly-supercompactand -superstrong, for every .7

    We stress the fact that the above notion transcends supercompactness in the sensethat if is the least supercompact, then it is not jointly -supercompact and-superstrong, for any . In fact, as we shall see in 2.5, global joint (C (n)-) super-compactness and -superstrongness is equivalent to (C (n)-) extendibility. Having saidthat, and given an analogous result of Bagaria for C (n)-extendibles (cf. Proposition3.4 in [1]), the following lemma (schema) is hardly surprising.

    Lemma 2.25 If is jointly C (n)-supercompact and -superstrong, then C (n+2).

    7 Similar remarks apply to the corresponding C(n)-version of this notion, which is straightforward to state;we leave these to the reader.

    123

  • Elementary chains and C(n)-cardinals 107

    Proof The proof is an induction in the meta-theory. Alternatively, the lemma followsfrom Proposition 3.4 in [1], and Proposition 2.30 below.

    Recalling the n+2-expressibility of C (n)-supercompactness, the previous lemmaimmediately implies the following.

    Corollary 2.26 Suppose that is jointly C (n)-supercompact and -superstrong, andsuppose that < is < -C (n)-supercompact. Then, is C (n)-supercompact.

    Before concluding this section, let us briefly return to the model Vj () obtained inthe proof of Theorem 2.21. Clearly, the s below which are C (n)-supercompact forevery n, are unbounded in . For any such < and for any < , the very sameextenders belonging to V witness the -C (n)-supercompactness of either in Vor in Vj (). In particular, Vj () thinks that is a limit of n-correct ordinals and thus,for every n , we have that Vj () | Lim(C (n)). In fact, we now give a smalllist of the properties that enjoys inside that particular model Vj (). Regarding part(iii) below, recall Definition 1.3.

    Proposition 2.27 Suppose that is almost huge, as witnessed by the embeddingj : V M. Let U be the usual normal measure on derived from j. Then,for any n , the following hold in the (ZFC) model Vj ():

    (i) Lim(C (n)).(ii) is C (n)-supercompact and

    { < : is C (n)-supercompact

    } U .

    (iii) is C (n)+-extendible and{ < : is C (n)+-extendible

    } U .

    Proof Parts (i) and (ii) follow from the preceding discussion. For (iii), fix some n andsome (, j ()) so that Vj () | C (n); now, by the closure of M and theinaccessibility of j (), j V M and then, M thinks that j V : V Vj ()is elementary, has critical point , ( j V)() > , j () < j ( j ()) and moreover,that j (Vj ()) | j () C (n).

    Let us temporarily fix a formula(, , ) asserting that there exists a-extendibi-lity embedding h for with = h(). From the point of view of M , we have justargued that for every (, j ())with Vj () | C (n), there exists a < j ( j ())such that (, , ) holds and, moreover, such that j (Vj ()) | C (n). But now,by the usual reflection of the normal measure, we have that the set of ordinals < such that, for all (, ), it is true that

    V | C (n) < j ()((, , ) Vj () | C (n)),

    belongs to U . Let us call this set A. Fix any A and fix any (, ) withV | C (n). Furthermore, fix a < j () witnessing that A, i.e., such that(, , ) holds and, moreover, such that Vj () | C (n).

    123

  • 108 K. Tsaprounis

    Since < j (), by the inaccessibility of the latter we have that the witnessing-extendibility embedding for belongs to Vj (), that is,

    Vj () | ((, , ) C (n)).

    Therefore, by elementarity, for any such A and any fixed (, ) with V |

    C (n), we have that there exists a < such that

    V | (, , ) C (n),

    i.e., such extendibility embeddings for may be witnessed inside V . Note how wehave successively bounded the s, first below j () and now below , ensuring thatthey are also in the relative C (n) of these structures. This discussion shows that theset

    B = { < : V | > ( C (n) ((, , ) C (n)))}

    includes A and hence, it also belongs to U . Consequently, by reversing the reflectionargument of the measure, we obtain that

    Vj () | > ( C (n) ((, , ) C (n))).

    But then, if we pick any (, j ()) with Vj () | C (n) and consider anywitnessing extendibility embedding h : V V for in Vj (), we get that, in thesense of the latter structure, all three: , and are in C (n). Hence, h() C (n) aswell. This shows that is C (n)+-extendible in Vj (). In turn, once more by a reflectionargument,

    { < : V | is C (n)+-extendible} U

    and then, it follows that, in Vj (), { < : is C (n)+-extendible} U . Therefore, the assumption of almost hugeness is sufficient for the consistency of

    all the C (n)-cardinals considered so far, in a strong sense: for any n , the modelVj () thinks that the cardinal is C (n)-supercompact, C (n)-superstrong and, clearly,C (n)-strong and C (n)-tall as well. In addition, we just showed that it is also C (n)+-extendible. Evidently, if we consider the least almost huge cardinal, the correspondingversions of Corollary 2.23 are obtained for all these cases.

    2.5 More extendibility

    Given the established consistency results, we now lookmore closely atC (n)-extendibi-lity and its connection with C (n)-supercompactness. We begin with the followingtheorem (schema).

    Theorem 2.28 Suppose that is +1-C (n)-extendible, for some n > 0 and some = > with cof() > . Then, is jointly -C (n)-supercompact and -superstrong.

    123

  • Elementary chains and C(n)-cardinals 109

    Proof Fix some n > 0 and some = > with cof() > . Now let j : V+1 Vj ()+1 be an elementary embedding that witnesses the + 1-C (n)-extendibility of , i.e., cp( j) = , j () > + 1 and j () C (n). Let us consider E , the ordinary(, j ())-extender derived from j ; that is, E is of the form Ea : a [ j ()]

  • 110 K. Tsaprounis

    Proposition 2.30 If is jointly C (n)-supercompact and -superstrong, then it is C (n)-extendible.

    Proof Fix some (meta-theoretic) n 1 and suppose that is jointly C (n)-supercomp-act and -superstrong. Moreover, fix some > with C (n+2). Recall that = |V|. Now let j : V M be an elementary embedding which witnessesthe fact that is jointly -C (n)-supercompact and -superstrong, i.e., M transitive,cp( j) = , j () > , M M, j () C (n) and Vj () M .

    Recall that, by Lemma 2.25, C (n+2). Thus, by elementarity, we have thatM | j () C (n+2) and M | j () C (n+2). Now note that, by the closure of thetarget model, the restricted embedding j V : V V Mj () belongs to M and thiswitnesses the fact that

    M | is < -C (n)-extendible.

    Moreover, since j () C (n), we have that Vj () | C (n+1) and so, since Vj () M and M | j () C (n+2), we consequently get that M | C (n+1). It nowfollows that the .Let = with cof() > and let E be the (, j ())-extender derived fromj. Then, the extender embedding jE : V ME is -supercompact for withjE () = j ().

    123

  • Elementary chains and C(n)-cardinals 111

    Proof Fix > and j : V M , a -supercompact embedding for . Let = with cof() > and let E be the (, j ())-extender derived from j . Then,recalling the proof of Theorem 2.28 one easily checks that a similar idea goes through;namely, we consider the (in this case, full) third factor embedding kE , arguing thatkE V MEjE () = id and V

    MEjE ()

    = V Mj (). Moreover, for all , jE () = j ()and also, since cof( j ()) > (computed in V ), we get that jE = j V Mj ()and jE ME as well. To conclude, we show closure under -sequences for MEusing the arguments of 2.4, noting that the assumption cof() > implies thatcof( j ()) > , which is exactly what is needed for these arguments to work in thecurrent setting.

    Observe once more that if j happens to have some degree of superstrongness,then this is carried over to the extender embedding; e.g., in the above proof as itstands, if j was also -superstrong, then the same would be true for jE . Moreover,since the arising extender embedding is such that jE () = j (), the previous proofworks for C (n)-supercompactness as well, i.e., if j : V M witnessed the -C (n)-supercompactness of , then jE would also be -C (n)-supercompact for .

    3 Woodinness and strong compactness

    We treat theC (n)-versions ofWoodin and strongly compact cardinals together, for tworeasons: firstly, they were not considered in [1] where the rest of the C (n)-cardinalswere introduced; but most importantly, because they both admit similar constructionsusing iterated ultrapowers, completely characterizing them in terms of their ordinarycounterparts.

    3.1 C (n)-Woodins

    Recalling Definition 1.5, we start by showing that C (n)-Woodins form a large cardinalhierarchy of increasing strength.

    Lemma 3.1 For n 1, if is C (n+1)-Woodin then there are unboundedly many C (n)-Woodins below . Hence, if is the least C (n)-Woodin then it is not C (n+1)-Woodin.

    Proof Fix some n 1 and suppose that is C (n+1)-Woodin. Fix some < andsome f with f = , and let < and j : V M witness C (n+1)-Woodinness with respect to f , i.e., f , cp( j) = , Vj ( f )() M, j () = and j () C (n+1). Clearly, < < j () < and, moreover, the following is a truen+1-statement in the parameter :

    ( C (n) V | some C (n)-Woodin above ),

    since it holds for any C (n) above . Therefore, it must hold in Vj (). But then,and since j () C (n+1), any < j () which is a witness to this statement has to ben-correct and, consequently, there exists some C (n)-Woodin cardinal above (andbelow ).

    123

  • 112 K. Tsaprounis

    The reader has probably noticed that the case n = 0 is conspicuously missing fromthis lemma. The following proposition explains why.

    Proposition 3.2 If is Woodin then it is C (1)-Woodin.

    Proof Suppose that is Woodin and fix some function f . We further fix < with f and a (, )-extender E V (for some < ), such that j =jE : V ME has cp( j) = , j () < , j () = and Vj ( f )() ME .Working in ME , since j () is measurable, let U ME be a normal, j ()-completemeasure on j () and let jU : ME M be the ultrapower embedding. Then, 2 j () and >(2 j ()

    )M. We iterate the ultrapower construction

    inside M , starting with U and repeating for -many steps. Let j : M M be theresulting embedding with cp( j) = j (), where M is (taken to be) transitive. Again,by known facts, we have that j( j ()) = andwe let h : V M be the composedelementary embedding, i.e., h = j j , with cp(h) = and with h() = C (n).We now check that h is -compact for .

    Suppose that X M , and assume that |X | = (in particular, is a cardinal).By the representation of iterated utrapowers we may assume that each z X is ofthe form j( f )(1 , . . . , m ), where f : [ j ()]m M belongs to M and, for each1 i m, m belongs to the critical sequence : < (where 0 = j ()).Thus, X = { j( fi )( i ) : i < } where each i is a finite tuple from the criticalsequence. Now, since cof() > , there exists some ordinal < such that, for everyi < , max( i ) < . Notice that []

  • Elementary chains and C(n)-cardinals 115

    4 C(n)-cardinals and forcing

    In this section, we briefly explore the interaction of C (n)-cardinals with the forcingmachinery. It will be convenient to work with flat pairing functions.11 We first give afolklore result regarding names constructed by such functions.

    Lemma 4.1 (Folklore) Let P be a poset with rank(P) = , and suppose thatP-names are constructed using a flat pairing function. Then, for any G P-genericover V and for any , we have that V [G] = V[G].Proof Let f be a fixed flat pairing function; we build the universe V P recursively,as follows. We initially let V P0 = . Given V P for some ON, we let V P+1 =V P P(V P P), where we use f in order to compute (the pairs in) the set V P P.For limit , we let V P =

  • 116 K. Tsaprounis

    V [G] = V [G] , all the xi s may be assumed to belong to V . Therefore, as C (n),we get that V | p y(y, x1, . . . , xk) and then, since P V ,

    V [G] | y(y, (x1)G, . . . , (xk)G).

    The same argument going backwards, shows that if V [G] = V [G] satisfies a n-formula then the same is true for V [G].

    For (ii), we recall that membership inC (1) is1-expressible and hence, the forwardimplication is true anyway. For the converse, assume that C (1), i.e., is anuncountable (strong limit) cardinal such that V = H . But if V = V [G] , it easilyfollows that (V)V [G] = (H)V [G]. Moreover, remains uncountable in V [G] andthus, V [G] | C (1).

    For (iii), suppose that C (2) and that P does not change V . Fix G P-genericover V and fix some < so that V [G] | C (2). By part (ii), C (1) andV [G] | C (1). Now let be a 2-formula, whose parameter, if any, belongsto V, and suppose that holds in V . Then, V [G] = V | and so, since isC (1) in the extension, V [G] | . Going downwards, we now get V [G] | by thecorrectness of in V [G] and, then, V | by assumption on P.

    For n > 1, the lack of a (local) combinatorial characterization of membership inthe class C (n) is a serious obstacle in showing the converse of (i), or of generalizing(ii) of Lemma 4.2. Note that (iii) is only a partial result in this direction. The converseof (iii) does not necessarily hold since the following situation is consistent.

    Relative to assumptions at the level of hyper-measurability, it is consistent thatGCH holds at successor cardinals but fails at limits.13 Suppose we are in such a modelV and consider any C (2). Now force the GCH at by the canonical poset to addone subset to +. This does not change V . Then, and, in fact, all cardinals below itwhich were 2-correct in V , are no longer 2-correct in the extension, although allC (1)s below are preserved.

    Let us finally turn to the preservation of (some of) the C (n)-large cardinals consid-ered so far, under appropriate forcings.

    Lemma 4.3 Suppose that is C (n)-tall (superstrong, supercompact, extendible), forsome n 1, and let P be a poset with |P| < . Then, remains C (n)-tall (resp.superstrong, supercompact, extendible) in V P.

    Proof Assume that P V and consider the case of supercompactness. Fix an n 1and a (cardinal) > and let j : V M witness the -C (n)-supercompactness of in V . We let G be P-generic over V and, by standard arguments, the embedding liftsto j : V [G] M[G] with V [G] | M[G] M[G]. Moreover, by Lemma 4.2(i),V [G] | j () C (n) and hence remains -C (n)-supercompact in the extension. Fortallness, the argument is essentially the same. For superstrongness, considering again aflat pairing function,we have that V [G] j () = Vj ()[G] and hence, V [G] j () M[G]by virtue of Vj () M M[G].

    13 See [3]. More precisely, given a P3()-hyper-measurable cardinal , there is a model of ZFC in which2 = ++ for every limit cardinal , whereas GCH holds everywhere else.

    123

  • Elementary chains and C(n)-cardinals 117

    Finally, by Corollary 2.31, extendibility is a straightforward combination of thecases of supercompactness and superstrongness.

    We now consider preservation of C (n)-tall and C (n)-supercompact cardinals undersufficiently distributive forcing notions.

    Lemma 4.4 Suppose that is C (n)-tall and let P be a -distributive forcing. Then, remains C (n)-tall in V P.

    Proof Fix any > max {, |P|} and let j : V M = { j ( f )(x) : f V, f :V V, x V Mj ()} be a -C (n)-tallnness embedding for . By standard facts,j lifts through any -distributive forcing (see, e.g., 15 in [4]). Thus, and usingLemma 4.2, remains -C (n)-tall in the extension. Lemma 4.5 Suppose that is C (n)-supercompact and supppose that P is a . Then, remains -C (n)-supercompactin V P.

    Proof Fix some n and some > max {, |P|} and let j : V M witness the -C (n)-supercompactness of . Now consider the substructure

    X0 ={

    j ( f )( j, x) : f V, f : P V V, x V Mj ()} M,

    which gives rise to a (not necessarily initial) factor -supercompact embedding j0 :V M0, via the Mostowski collapse 0 : X0 = M0 as usual. Since j0() =j (), j0 is moreover -C (n)-supercompact for .Let G be P-generic over V . Again, using the representation of M0 and the distrib-

    utivity of P, standard arguments show that j0 lifts through the forcing, witnessing the -C (n)-supercompactness of in V [G].

    We conclude with a (somewhat informal) question.

    Question 4.6 Suppose that is C (n)-extendible. Under what types of forcing notionsis the (local or global) C (n)-extendibility of preserved? For instance, can we forcethe GCH while preserving such cardinals?14

    Acknowledgments The author is grateful to Joel D. Hamkins for many helpful discussions; the initialidea of using the tool of elementary chains in the context of C(n)-cardinals is due to him. Further gratitudegoes to the anonymous referee for many valuable comments and suggestions. The results in this article arepart of the authors doctoral dissertation; the latter was composed under the supervision of Joan Bagaria,whose guidance and support are warmly acknowledged.

    References

    1. Bagaria, J.: C(n)-cardinals. Arch. Math. Logic 51(34), 213240 (2012)2. Bagaria, J., Hamkins, J.D., Tsaprounis, K., Usuba, T.: Superstrong and other large cardinals are never

    Laver indestructible. Preprint (2013), available at http://arxiv.org/abs/1307.3486

    14 For n {0, 1}, i.e., for ordinary extendibility, the answer is affirmative: the canonical (class-length)iteration which forces global GCH preserves every extendible cardinal of the universe (see [9]).

    123

    http://arxiv.org/abs/1307.3486

  • 118 K. Tsaprounis

    3. Cummings, J.: A model in which GCH holds at successors but fails at limits. Trans. Am. Math.Soc. 329(1), 139 (1992)

    4. Cummings, J.: Iterated forcing and elementary embeddings. In: Foreman,M., Kanamori, A. Handbookof Set Theory, pp. 775883. Springer, New York (2010)

    5. Hamkins, J.D.: Canonical seeds and Prikry trees. J. Symb. Logic 62(2), 373396 (1997)6. Hamkins, J.D.: Tall cardinals. Math. Logic Q. 55(1), 6886 (2009)7. Kanamori, A.: The Higher Infinite. Springer, Berlin (1994)8. Martin, D., Steel, J.: A proof of projective determinacy. J. Am. Math. Soc. 2(1), 71125 (1989)9. Tsaprounis, K.: On extendible cardinals and the GCH. Arch. Math. Logic 52(56), 593602 (2013)

    10. Tsaprounis, K.: Large cardinals and resurrection axioms. Ph.D. dissertation, University of Barcelona,Spain (2012)

    123

    Elementary chains and C(n)-cardinalsAbstract1 Introduction1.1 Preliminaries1.2 C(n)-cardinals

    2 Consistency bounds2.1 Tallness2.2 Superstrongness2.3 Strongness2.4 Supercompactness2.5 More extendibility

    3 Woodinness and strong compactness3.1 C(n)-Woodins3.2 C(n)-strongly compacts

    4 C(n)-cardinals and forcingAcknowledgmentsReferences