Structural reflection, shrewd cardinals and the size of the continuum
aa r X i v : . [ m a t h . L O ] J a n STRUCTURAL REFLECTION, SHREWD CARDINALS AND THE SIZEOF THE CONTINUUM
PHILIPP L ¨UCKE
Abstract.
Motivated by results of Bagaria, Magidor and V¨a¨an¨anen, we study character-izations of large cardinal properties through reflection principles for classes of structures.More specifically, we aim to characterize notions from the lower end of the large cardinalhierarchy through the principle SR − introduced by Bagaria and V¨a¨an¨anen. Our resultsisolate a narrow interval in the large cardinal hierarchy that is bounded from below bytotal indescribability and from above by subtleness, and contains all large cardinals thatcan be characterized through the validity of the principle SR − for all classes of structuresdefined by formulas in a fixed level of the L´evy hierarchy. Moreover, it turns out thatno property that can be characterized through this principle can provably imply stronginaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals ,introduced by Rathjen in a proof-theoretic context, and embedding characterizations ofthese cardinals that resembles Magidor’s classical characterization of supercompactness.In addition, we show that several important weak large cardinal properties, like weakinaccessibility, weak Mahloness or weak Π n -indescribability, can be canonically character-ized through localized versions of the principle SR − . Finally, the techniques developed inthe proofs of these characterizations also allow us to show that Hamkin’s weakly compactembedding property is equivalent to L´evy’s notion of weak Π -indescribability. Introduction
The work presented in this paper is motivated by results of Bagaria, Magidor, V¨a¨an¨anenand others that establish deep connections between extensions of the
Downward L¨owenheim–Skolem Theorem , large cardinal axioms and set-theoretic reflection principles. Our resultswill focus on the characterization of large cardinal notions through reflection properties forclasses of structures. To motivate our work, we start by discussing results that provide suchcharacterizations for supercompact cardinals.First, recall that classical work of Magidor in [16] yields a characterization of super-compactness through model-theoretic reflection by showing that second-order logic has a L¨owenheim–Skolem–Tarski cardinal (see [5, Definition 6.1]) if and only if there exists a su-percompact cardinal. Moreover, Magidor’s results show that if these equivalent statementshold true, then the least L¨owenheim–Skolem–Tarski cardinal for second-order logic is equalto the least supercompact cardinal. Next, in order to connect supercompactness and reflec-tion principles for second-order logic to set-theoretic reflection properties, we make use ofthe following principle of structural reflection , formulated by Bagaria:
Definition 1.1 (Bagaria, [2]) . Given an infinite cardinal κ and a class C of structures ofthe same type, we let SR C ( κ ) denote the statement that for every structure A in C , thereexists an elementary embedding of a structure in C of cardinality less than κ into A . Mathematics Subject Classification.
Key words and phrases.
Structural reflection, large cardinals, Σ -definability, elementary embeddings,shrewd cardinals, weakly compact embedding property.The author would like to thank Joan Bagaria for several helpful discussions on the topic of this work andvarious comments on earlier versions of this paper. This project has received funding from the EuropeanUnion’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agree-ment No 842082 (Project SAIFIA: Strong Axioms of Infinity – Frameworks, Interactions and Applications ). Throughout this paper, we will use the term large cardinal notion to refer to properties of cardinalsthat imply weak inaccessibility. In the following, the term structure refers to structures for countable first-order languages. It should benoted that all results cited and proven below remain true if we restrict ourselves to finite languages.
In order to precisely formulate the relevant results from [2], [3] and [5], we first need todiscuss certain subclasses of Σ -formulas defined through standard refinements of the L´evyhierarchy of formulas. Given a first-order language L that extends the language L ∈ of settheory, an L -formula is a Σ -formula if it is contained in the smallest class of L -formulasthat contains all atomic L -formulas and is closed under negations, conjunctions and boundedexistential quantification. Moreover, given n < ω , an L -formula is a Π n -formula if it is thenegation of a Σ n -formula, and it is a Σ n +1 -formula if it is of the form ∃ x , . . . , x m − ϕ forsome Π n -formula ϕ . Finally, given a class R , n < ω and a set z , a class P is Σ n ( R ) -definable (respectively, Π n ( R ) -definable ) in the parameter z if there is a Σ n -formula (respectively, aΠ n -formula) ϕ ( v , v ) in the language L ˙ A that expands L ∈ by a unary relation symbol ˙ A such that P consists of all sets x with the property that ϕ ( x, z ) holds in h V , ∈ , R i . As usual,we call Σ n ( ∅ )-definable classes (i.e. classes defined by a Σ n -formula in L ∈ ) Σ n -definable andΠ n ( ∅ )-definable classes Π n -definable . The closure properties of the class of all L ∈ -formulasthat are ZFC-provably equivalent to Σ n +1 -formulas then ensure that if R is a class that isΠ n -definable in the parameter z , then every class that is Σ n ( R )-definable in the parameter z is also Σ n +1 -definable in this parameter.Using V¨a¨an¨anen’s notion of symbiosis between model- and set-theoretic reflection prin-ciples introduced in [19], Bagaria and V¨a¨an¨anen connected reflection principles for second-order logic with the principle SR by showing that a cardinal κ is a L¨owenheim–Skolem–Tarskicardinal for second-order logic if and only if SR C ( κ ) holds for every class C of structures thatis Σ ( P wSet )-definable without parameters (see [5, Theorem 5.5] and [5, Lemma 7.1]), where
P wSet denotes the Π -definable class of all pairs of the form h x, P ( x ) i . In combination withthe results from [16] discussed above, this equivalence provides a characterization of thefirst supercompact cardinal through the validity of the principle SR C ( κ ) for all Σ ( P wSet )-definable classes of structures. This connection between supercompactness and structuralreflection for Σ -definable classes was studied in depth by Bagaria and his collaboratorsin [2] and [3]. In the following, define V to be the class of all L ∈ -structures of the formV α for some ordinal α . It is easy to see that the class V is Σ ( P wSet )-definable withoutparameters. Using results from [2] and [3], the above characterization of supercompactnesscan now be phrased in the following way:
Theorem 1.2 ([2], [3]) . The following statements are equivalent for every cardinal κ :(i) κ is the least supercompact cardinal.(ii) κ is the least cardinal with the property that SR V ( κ ) holds.(iii) κ is the least cardinal with the property that SR C ( κ ) holds for every class of struc-tures of the same type that is definable by a Σ -formula with parameters in H( κ ) . It is natural to ask if other large cardinal notions can be characterized in similar ways, i.e.given some large cardinal property, is there a class of formulas such that the least cardinalwith the given property provably coincides with the least reflection point for all classes ofstructures defined by formulas from this class. Note that the validity of such characterizationcan be seen as a strong justification for the naturalness of large cardinal axioms (see [1]).Results contained in [2] and [3] already yield such characterizations for Σ n +2 -definable classesof structures and so-called C ( n ) -extendible cardinals (see [2, Definition 3.2]). These resultsalso provide a characterization of Vopˇenka’s Principle in terms of structural reflection. Inaddition, results in [1] provide a characterization of the existence of X for a set X ofordinals through structural reflection and results in [6] yield such characterization for largecardinal notions in the region between strong cardinals and “ Ord is Woodin ”. Finally, theresults of [4] give characterizations of remarkable cardinals and other virtual large cardinalsthrough principles of generic structural reflection .The work presented in this paper is devoted to the characterization of objects from thelower part of the large cardinal hierarchy through principles of structural reflection. The Throughout this paper, we will often identify a class M with the L ∈ -structure h M, ∈i to simplify ourformulations. It is easy to see that the parameters class H( κ ) is maximal in this setting. Fix z / ∈ H( κ ) and let L ∅ denote the trivial first-order language. Then the class C of all L -structures of cardinality | tc( z ) | is definableby a Σ -formula with parameter z , and the principle SR −C ( κ ) obviously fails. TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 3 starting point of this work is the following restriction of the principle SR, isolated by Bagariaand V¨a¨an¨anen:
Definition 1.3 (Bagaria–V¨a¨an¨anen, [5]) . Given an infinite cardinal κ and a class C ofstructures of the same type, we let SR −C ( κ ) denote the statement that for every structure A in C of cardinality κ , there exists an elementary embedding of a structure in C of cardinalityless than κ into A .Our results will show that there exists a narrow interval in the large cardinal hierarchythat contains all large cardinal notions for which there exists a natural number n > − for all Σ n -definable classes of structures. This interval is bounded from belowby total indescribability and from above by subtleness. Moreover, this analysis will showthat there is essentially only one large cardinal notion that can be characterized throughcanonical non-trivial Π -predicates R and the principle SR − for Σ ( R )-definable classes ofstructures. This unique large cardinal notion turns out to be closely related to the followingproperty of large cardinals, introduced by Rathjen in a proof-theoretic context: Definition 1.4 (Rathjen, [17]) . A cardinal κ is shrewd if for every L ∈ -formula Φ( v , v ),every ordinal α and every subset A of V κ such that Φ( A, κ ) holds in V κ + α , there existordinals ¯ κ and ¯ α below κ such that Φ( A ∩ V ¯ κ , ¯ κ ) holds in V ¯ κ +¯ α .The defining property of shrewd cardinals directly implies that all of these cardinals aretotally indescribable. Moreover, Rathjen showed that, given a subtle cardinal δ , the set ofcardinals κ < δ that are shrewd in V δ is stationary in δ (see [17, Lemma 2.7]). The followingresult now connects shrewdness to the consistency strength of principles the principle SR − for Σ -definable classes: Theorem 1.5.
The following statements are equiconsistent over the theory
ZFC :(i) There exists a shrewd cardinal.(ii) There exists a cardinal κ with the property that SR −C ( κ ) holds for every class C ofstructures of the same type that is definable by a Σ ( Cd ) -formula without parame-ters.(iii) There exists a cardinal κ with the property that SR −C ( κ ) holds for every class C ofstructures of the same type that is definable by a Σ -formula with parameters in H( κ ) . This result shows that for all large cardinal properties whose consistency strength isstrictly smaller than the existence of a shrewd cardinal, there is no characterization of thesenotions through canonical non-trivial Π -predicates R and the principle SR − for Σ ( R )-definable classes of structures. Moreover, it shows that the connection between the principleSR − and the strict L¨owenheim–Skolem–Tarski property (see [5, Definition 8.2]) as well asthe characterizations of weak inaccessibility, weak Mahloness and weak compactness statedin [5, Theorem 8.3] need to be reformulated, because, by the above theorem, all of thesestatements would imply that the consistency of the existence of a shrewd cardinal is strictlyweaker than the consistency strength of the existence of a total indescribable cardinal.The proof of Theorem 1.5 is based on the following weakening of Definition 1.4: Definition 1.6.
An infinite cardinal κ is weakly shrewd if for every L ∈ -formula Φ( v , v ),every cardinal θ > κ and every subset A of κ with the property that Φ( A, κ ) holds in H( θ ),there exist cardinals ¯ κ < ¯ θ with the property that ¯ κ < κ and Φ( A ∩ ¯ κ, ¯ κ ) holds in H(¯ θ ). Note that [2, Theorem 4.2] shows that SR C ( κ ) holds for every uncountable cardinal κ and every class C of structures that is definable by a Σ -formula with parameters in H( κ ). More precisely, through Π -predicates R with the property that the class Cd of all cardinals is Σ ( R )-definable. The class Cd , the class Rg of all regular cardinals, the class P wSet and the class obtained by auniversal Π -formula in L ∈ are all examples of such predicates. The problematic part in the adaption of the proof of [5, Theorem 5.5] to a proof of these statements isthe fact that the cardinality of all L ∈ -models witnessing that some structure of cardinality κ is contained ina Σ ( R )-definable class can be strictly greater than the given cardinal κ and therefore it is not possible toapply the strict L¨owenheim–Skolem–Tarski property at κ to these L ∈ -models. An example of such a classof structures is the Σ ( P wSet )-definable class W introduced below. PHILIPP L¨UCKE
The notion of weak shrewdness turns out to be closely connected to principles of structuralreflection. Our results will allow us to show that if κ is a weakly shrewd cardinal, thenSR −C ( κ ) holds for every class C of structures that is Σ -definable with parameters in H( κ )(see Lemma 4.1). Moreover, the next theorem shows that this large cardinal property canbe canonically characterized through the principle SR − for Σ ( P wSet )-definable classes ofstructures. In the following, let L ˙ c denote the first-order language that extends the language L ∈ of set theory by a constant symbol ˙ c and let W denote the class of all L ˙ c -structures h X, ∈ , κ i with the property that there exists a cardinal θ such that κ is an infinite cardinalsmaller than θ and X is an elementary submodel of H( θ ) of cardinality κ with κ + 1 ⊆ X .It is easy to see that the class W is definable by a Σ ( P wSet )-formula without parameters.
Theorem 1.7.
The following statements are equivalent for every cardinal κ :(i) κ is the least weakly shrewd cardinal.(ii) κ is the least cardinal with the property that SR −W ( κ ) holds.(iii) κ is the least cardinal with the property that SR −C ( κ ) holds for every class C ofstructures of the same type that is definable by a Σ -formula with parameters in H( κ ) . In combination with Theorem 1.5, this result directly yields the following equiconsistency:
Corollary 1.8.
The following statements are equiconsistent over the theory
ZFC :(i) There exists a shrewd cardinal.(ii) There exists a weakly shrewd cardinal. (cid:3)
In addition, the third statement listed in Theorem 1.7 shows that large cardinal propertiesof higher consistency strength than shrewdness cannot be characterized through the principleSR − for Σ -definable classes of structures. Together with our earlier observations, this showsthat weak shrewdness is basically the only large cardinal notions that can be characterizedwith the help of canonical Π -predicates R and the principle SR − for Σ ( R )-definable classesof structures.The above results directly motivate several follow-up questions. First, it is natural to askwhich large cardinal properties stronger than weak shrewdness can be characterized throughthe principle SR − for classes of structures defined by more complex formulas. Second, theseresults suggest to study the interactions between principles of structural reflection and thebehavior of the continuum function. In particular, it is interesting to ask whether anylarge cardinal property that entails strong inaccessibility can be characterized through theprinciple SR − . Finally, it is also natural to ask whether large cardinal notions weaker thanshrewdness can be characterized through further restrictions of the principle SR − .The answers to the first two questions turn out to be closely related to the existence ofweakly shrewd cardinals that are not shrewd. The following result positions the consistencystrength of the existence of weakly shrewd cardinals that are, for various reasons, not shrewdin the large cardinal hierarchy: Theorem 1.9. (i) If κ is a weakly shrewd cardinal that is not shrewd, then there existsan ordinal ε > κ with the property that ε is inaccessible in L and κ is a shrewdcardinal in L ε .(ii) The least subtle cardinal is a stationary limit of inaccessible weakly shrewd cardinalsthat are not shrewd.(iii) The following statements are equiconsistent over ZFC :(a) There exists an inaccessible weakly shrewd cardinal that is not shrewd.(b) There exists a weakly shrewd cardinal that is not inaccessible.(c) There exists a weakly shrewd cardinal smaller than ℵ . The techniques developed in the proofs of the above results will also allow us to showthat the existence of a weakly shrewd cardinal does not imply the existence of a cardinal κ with the property that SR −C ( κ ) holds for every class C of structures that is definable by a Σ -formula without parameters (see Corollary 4.4 below). In contrast, the following technicalresult shows that the existence of a weakly shrewd cardinal that is not shrewd directly impliesthe existence of reflection points for classes of structures of higher complexities. Moreover, TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 5 it will also allow us to show that the existence of reflection points for classes of structuresof arbitrary complexities has strictly weaker consistency strength than the existence of aweakly shrewd cardinal that is not shrewd.
Theorem 1.10.
Let κ be weakly shrewd cardinal that is not shrewd.(i) There is a cardinal δ > κ with the property that the set { δ } is definable by a Σ -formula with parameters in H( κ ) .(ii) Given < n < ω and α < κ , if δ > κ is a cardinal with the property that the set { δ } is definable by a Σ -formula with parameters in H( κ ) , then there exists a cardinal α < ρ < δ such that SR −C ( ρ ) holds for every class C of structures of the same typethat is definable by a Σ n -formula with parameters in H( ρ ) .(iii) Assume that does not exist and κ is inaccessible. If δ > κ is a cardinal withthe property that the set { δ } is definable by a Σ -formula with parameters in H( κ ) ,then there exists an inaccessible cardinal κ < ε < δ with the property that, in V ε ,the principle SR −C ( κ ) holds for every class C that is defined by a formula usingparameters from H( κ ) . Note that the set { ℵ } is always definable by a Σ -formula without parameters. Inparticular, the second part of the above theorem tells us that the existence of a weakly shrewdcardinal smaller than the cardinality of the continuum implies the existence of various localreflection points below 2 ℵ . By Theorem 1.9, the existence of such cardinals is consistentrelative to the existence of a subtle cardinal.Theorem 1.10 now allows us to show that ZFC is consistent with the existence of cardinalswith maximal local structural reflection properties. In the light of the results of [2, Section4], the existence of such cardinals can be seen as a localized version of Vopˇenka’s Principle .Our results show that the consistency strength of this local principle is surprisingly small.Moreover, they show that such reflection points can consistently exist below the cardinalityof the continuum. As above, we let L ˙ c denote the first-order language extending the language L ∈ by a constant symbol ˙ c . Given 0 < n < ω , we let SR − n denote the L ˙ c -sentence statingthat ˙ c is an infinite cardinal and SR −C (˙c) holds for every class C of structures of the sametype that is definable by a Σ n -formula in L ∈ with parameters in H( ˙ c ). Theorem 1.11. (i) The L ∈ -theory ZFC + “ There exists a weakly shrewd cardinal that is not shrewd ”proves the existence of a transitive model of the L ˙ c -theory ZFC + { SR − n | < n < ω } . (ii) The following theories are equiconsistent:(a) ZFC + “ There exists a weakly shrewd cardinal that is not shrewd ”.(b)
ZFC + { SR − n | < n < ω } + “ ˙ c < ℵ ”. This result answers the first two questions formulated above. The first part of the abovetheorem shows that no large cardinal notion with consistency strength greater than or equalto the existence of a weakly shrewd cardinal that is not shrewd can be characterized throughthe principle SR − . Moreover, the second part of the corollary shows that no large cardinalproperty that entails strong inaccessibility can be characterized through the principle SR − .In particular, this shows that the statement of [5, Theorem 3.5] needs extra assumptions. In order to answer the third of the above questions, we now turn to the characterizationsof large cardinal notions weaker than shrewdness through principles of structural reflection.In the light of the above results, we introduce further restricted forms of Σ -definability that Note that this conclusion is a statement about the structure h V ε , ∈ , κ i that holds in V and is formulatedwith the help of a formalized satisfaction relation (see, for example, [8, Section I.9]). In particular, thisstatement also applies to possible classes in V ε that are defined through formulas with non-standard G¨odelnumbers. More precisely, given some canonical formalization of this L ˙ c -theory, the above L ∈ -theory proves theexistence of a transitive set M such that for some ν ∈ M , every formalized axiom holds in the structure h M, ∈ , ν i with respect to some formalized satisfaction relation. For example, the argument presented in [5] works for all cardinals κ satisfying κ = κ <κ . PHILIPP L¨UCKE will enable us to characterize several classical weak large cardinal notions through principlesof structural reflection. To motivate the upcoming definition, first observe that for every0 < n < ω and every Σ n ( R )-definable class C of structures, the class of all isomorphic copiesof elements of C is again Σ n ( R )-definable from the same parameters. Next, note that Σ -absoluteness implies that a class Q is definable by a Σ -formula with parameter z if andonly if there is a Σ -formula ϕ ( v , v ) with the property that for every infinite cardinal δ with z ∈ H( δ + ), H( δ + ) ∩ Q = { x ∈ H( δ + ) | H( δ + ) | = ϕ ( x, z ) } holds. In contrast, let T denote the class of all triples h δ, x, a i with the property that δ is an infinite cardinal, x is an element of H( δ + ) and a is an element of the set Fml offormalized L ∈ -formulas with the property that Sat (H( δ + ) , x, a ) holds, where Sat denotesthe canonical formalized satisfaction relation for L ∈ -formulas. Then it is easy to see thatthe class T is definable by a Σ -formula without parameters and Tarski’s Undefinabilityof Truth Theorem implies that for every infinite cardinal δ , the intersection H( δ + ) ∩ T isnot definable in H( δ + ). These observations motivate the restricted form of Σ -definabilityintroduced in the definition below that provides us with a notion of complexity that liesstrictly in-between Σ - and Σ -definability (see Proposition 6.1 below). Definition 1.12.
Let R be a class and let n > z , a class S is uniformly locally Σ n ( R ) -definable in the parameter z ifthere is a Σ n ( R )-formula ϕ ( v , v ) with the property thatH( κ + ) ∩ S = { x ∈ H( κ + ) | h H( κ + ) , ∈ , R i | = ϕ ( x, z ) } holds for every infinite cardinal κ with z ∈ H( κ ).(ii) Given a class Z , a class C of structures of the same type is a local Σ n ( R ) -class over Z if the following statements hold:(a) C is closed under isomorphic copies.(b) C is uniformly locally Σ n ( R )-definable in a parameter contained in Z .It can easily be shown that no new large cardinal characterizations can be obtainedthrough canonical Π -classes R and the principle SR for local Σ ( R )-classes. First, notethat the class ¯ V of all L ∈ -structures that are isomorphic to an element of the class V defined above is a local Σ ( P wSet )-class over ∅ . This shows that a cardinal κ is the leastsupercompact cardinal if and only if it is the least cardinal with the property that SR C ( κ )holds for every local Σ ( P wSet )-class over ∅ . Moreover, if V = L holds, then the fact thatH( δ + ) = L δ + holds for every infinite cardinal δ implies that the class P wSet is Σ ( Cd )-definable and hence the class ¯ V is definable in the same way. This shows that no Π -class R with the property that the class Cd is Σ ( R )-definable can be used to characterize largecardinal notions compatible with the assumption V = L through the principle SR C ( κ ) forlocal Σ ( R )-classes.In contrast, the next result shows how weak inaccessibility, weak Mahloness and weakΠ n -indescribability, introduced by L´evy in [15], can all be characterized through the validityof the principle SR − for certain local Σ n ( R )-classes. Recall that, given natural numbers m and n , a cardinal κ is weakly Π mn -indescribable if for all relations A , . . . , A m − on the set κ and all Π mn -sentences Φ in L ∈ that hold in the structure h κ, ∈ , A , . . . , A m − i , there existsan ordinal λ < κ such that Φ holds in the corresponding substructure h λ, ∈ , A , . . . , A m − i with domain λ (see [15, Definition 1.(b)]). Note that a cardinal κ is Π mn -indescribable if andonly if it is weakly Π mn -indescribable and strongly inaccessible. Theorem 1.13. (i) The following statements are equivalent for every cardinal κ :(a) κ is the least weakly inaccessible cardinal.(b) κ is the least cardinal with the property that SR −C ( κ ) holds for every local Σ ( Cd ) -class C over ∅ . Note that the classes
Fml and
Sat are both defined by Σ -formulas. Moreover, by using codes fornegated formulas, it is easy to see that the complement of Sat is also definable by a Σ -formula. See [11, p. 295].
TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 7 (c) κ is the least cardinal with the property that SR −C ( κ ) holds for every local Σ ( Cd ) -class C over H( κ ) .(ii) The following statements are equivalent for every cardinal κ :(a) κ is the least weakly Mahlo cardinal.(b) κ is the least cardinal with the property that SR −C ( κ ) holds for every local Σ ( Rg ) -class C over ∅ .(c) κ is the least cardinal with the property that SR −C ( κ ) holds for every local Σ ( Rg ) -class C over H( κ ) .(iii) The following statements are equivalent for every cardinal κ and every < n < ω :(a) κ is the least weakly Π n -indescribable cardinal.(b) κ is the least cardinal with the property that SR −C ( κ ) holds for every local Σ n +1 -class over ∅ .(c) κ is the least cardinal with the property that SR −C ( κ ) holds for every local Σ n +1 -class over H( κ ) . The techniques developed in the proof of the above result will also allow us to show thata large cardinal property isolated by Hamkins is in fact equivalent to L´evy’s notion of weakΠ -indescribability. Hamkins defined a cardinal κ to have the weakly compact embeddingproperty if for every transitive set M of cardinality κ with κ ∈ M , there is a transitive set N and an elementary embedding j : M −→ N with crit ( j ) = κ (see [9]). He then showed thatthis property implies both weak Mahloness and the tree property. Moreover, he showedthat if κ is weakly compact and G is Add( ω, κ + )-generic over V, then κ has the weaklycompact embedding property in V[ G ]. In the proof of Theorem 1.13, we will show thatweak Π n -indescribability can be characterized through the existence of certain elementaryembedding and this equivalence also allows us to conclude that weak Π -indescribabilitycoincides with the weakly compact embedding property. These observations will also showthat the results of [10, Section 4] only work under the additional assumption that the givencardinal is strongly inaccessible.Finally, in unpublished work, Cody, Cox, Hamkins and Johnstone showed that variouscardinal invariants of the continuum do not possess the weakly compact embedding property(see [9]). We will extend these results by showing that various definable cardinals cannotbe reflection points of certain classes of structures. For examples, our methods will allowus to show that, although there can consistently exist weakly shrewd cardinals below the dominating number d , the cardinal d is neither weakly shrewd nor the successor of a weaklyshrewd cardinal. 2. Shrewd cardinals
In this section, we derive some consequences of shrewdness that will be used in the proof ofTheorem 1.5. The starting point of this analysis is the following embedding characterizationfor shrewd cardinals that resembles Magidor’s classical characterization of supercompactness(see [16] and also [14, Theorem 22.10]):
Lemma 2.1.
The following statements are equivalent for every cardinal κ :(i) κ is a shrewd cardinal.(ii) For all sufficiently large cardinals θ > κ , there exist cardinals ¯ κ < ¯ θ < κ , anelementary submodel X of H(¯ θ ) and an elementary embedding j : X −→ H( θ ) such that ¯ κ + 1 ⊆ X , j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ .(iii) For all cardinals θ > κ and all z ∈ H( θ ) , there exist cardinals ¯ κ < ¯ θ < κ , anelementary submodel X of H(¯ θ ) and an elementary embedding j : X −→ H( θ ) suchthat ¯ κ + 1 ⊆ X , j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ and z ∈ ran( j ) .Proof. First, assume that (i) holds. Fix an L ∈ -formula Φ( v , v ) with the property thatΦ( A, δ ) expresses, in a canonical way, that the conjunction of the following statements holdstrue:(a) There exist unboundedly many strong limit cardinals. Note that, in general, the elementary submodel X will not be transitive. PHILIPP L¨UCKE (b) δ is an inaccessible cardinal.(c) There is a cardinal θ > δ , a subset X of H( θ ) and a bijection b : δ −→ X such thatthe following statements hold: • δ + 1 ⊆ X , b (0) = δ and b ( ω · (1 + α )) = α for all α < δ . • The class H( θ ) is a set and, given α , . . . , α n − < δ and an element a of Fml that codes a formula with n free variables, we have h a, α , . . . , α n − i ∈ A ⇐⇒ Sat ( X, h b ( α ) , . . . , b ( α n − ) i , a ) ⇐⇒ Sat (H( θ ) , h b ( α ) , . . . , b ( α n − ) i , a ) . Fix a cardinal θ > κ , z ∈ H( θ ) and a strong limit cardinal λ > θ with the property thatV λ is sufficiently elementary in V. Pick an elementary submodel Y of H( θ ) of cardinality κ with κ ∪ { κ, z } ⊆ Y and a bijection b : κ −→ Y satisfying b (0) = κ , b (1) = z and b ( ω · (1 + α )) = α for all α < κ . Define A to be the set of all tuples h a, α , . . . , α n − i withthe property that α , . . . , α n − < κ , a ∈ Fml codes a formula with n free variables and Sat ( Y, h b ( α ) , . . . , b ( α n − ) i , a ) holds.Then κ + λ = λ and Φ( A, κ ) holds in V λ . In this situation, the shrewdness of κ yieldsordinals ¯ κ, ¯ λ < κ with the property that Φ( A ∩ V ¯ κ , ¯ κ ) holds in V ¯ κ +¯ λ . By the definition of theformula Φ, we know that ¯ κ is an inaccessible cardinal, ¯ λ is a strong limit cardinal and hence¯ κ + ¯ λ = ¯ λ < κ . Moreover, since statements of the form “ x = H( δ ) ” are absolute betweenV ¯ λ and V, and the formulas defining the classes Fml and
Sat are upwards absolute from V ¯ λ to V, there exists a cardinal ¯ κ < ¯ θ < ¯ λ , a subset X of H(¯ θ ) and a bijection ¯ b : ¯ κ −→ X suchthat the following statements hold: • ¯ κ + 1 ⊆ X , ¯ b (0) = ¯ κ and ¯ b ( ω · (1 + α )) = α for all α < ¯ κ . • Given an L ∈ -formula ϕ ( v , . . . , v n − ) and α , . . . , α n − < ¯ κ , we haveH( θ ) | = ϕ ( b ( α ) , . . . , b ( α n − )) ⇐⇒ Y | = ϕ ( b ( α ) , . . . , b ( α n − )) ⇐⇒ H(¯ θ ) | = ϕ (¯ b ( α ) , . . . , ¯ b ( α n − )) ⇐⇒ X | = ϕ (¯ b ( α ) , . . . , ¯ b ( α n − )) . This shows that X is an elementary submodel of H(¯ θ ) with ¯ κ + 1 ⊆ X and, if we define j = b ◦ ¯ b − : X −→ H( θ ) , then j is an elementary embedding with j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ and z ∈ ran( j ). This showsthat (iii) holds in this case.Now, assume that (ii) holds and assume, towards a contradiction, that there is an L ∈ -formula Φ( v , v ), an ordinal α and a subset A of V κ witnessesing that κ is not a shrewdcardinal. Pick a sufficiently large strong limit cardinal θ > κ + α with the property thatH( θ ) is sufficiently elementary in V. By our assumption, we can find cardinals ¯ κ < ¯ θ < κ and an elementary embedding j : X −→ H( θ ) such that ¯ κ + 1 ⊆ X ≺ H( θ ), j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ . Claim. κ is a strong limit cardinal.Proof of the Claim. Fix an ordinal µ < ¯ κ . Since we know that µ ∈ X , j ( µ ) = µ andH( θ ) | = “ P ( µ ) exists ”, elementarity yields an ordinal ν in X with X | = “ 2 µ = ν ”. But thenH(¯ θ ) | = “ 2 µ = ν ” and hence ν is a cardinal in V with 2 µ = ν < ¯ θ < κ . In this situation,elementarity implies that X | = “ 2 µ < ¯ κ ”. Hence, we know that ¯ κ is a strong limit cardinalin X and this allows us to conclude that κ is a strong limit cardinal. (cid:3) By elementarity, we can now find an ordinal α in X and a subset A of V ¯ κ in X with theproperty that the formula Φ( v , v ), the ordinal j ( α ) and the subset j ( A ) of V κ witness that κ is not a shrewd cardinal. Since the above claim shows that V ¯ κ ⊆ X and j ↾ V ¯ κ = id V ¯ κ ,we know that j ( A ) ∩ V ¯ κ = A . In particular, it follows that Φ( j ( A ) , κ ) holds in V κ + j ( α ) andΦ( A, ¯ κ ) does not hold in V ¯ κ + α . Since V ¯ κ + α is an element of X with j (V ¯ κ + α ) = V κ + j ( α ) ,we can now use elementarity to derive a contradiction. (cid:3) The above equivalence allows us to easily deduce several consequences of shrewdness.
Corollary 2.2.
Shrewd cardinals are totally indescribable stationary limits of totally inde-scribable cardinals.
TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 9
Proof.
By definition, all shrewd cardinals are totally indescribable. Now, let κ be a shrewdcardinal and let C be a closed unbounded subset of κ . Pick a cardinal θ > i ω ( κ ) and useLemma 2.1 to find cardinals ¯ κ < ¯ θ < κ and an elementary embedding j : X −→ H( θ ) with¯ κ + 1 ⊆ X ≺ H(¯ θ ), j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ and C ∈ ran( j ). Then ¯ κ ∈ C and elementarityimplies that ¯ θ > i ω (¯ κ ). But this setup ensures that the statement “ κ is totally indescribable ” is absolute between H( θ ) and V, and the statement “ ¯ κ is totally indescribable ” is absolutebetween H(¯ θ ) and V. In particular, we can use elementarity to conclude that ¯ κ ∈ C is totallyindescribable. (cid:3) The next consequence of Lemma 2.1 will be crucial for our characterization of weaklyshrewd cardinals that are not shrewd in the next section. This result should be comparedwith the corresponding statements for supercompact and remarkable cardinals (see [14,Proposition 22.3] and [20, Theorem 1.3]). Remember that, given a natural number n >
0, acardinal κ is Σ n -reflecting if it is inaccessible and V κ ≺ Σ n V holds.
Corollary 2.3.
Shrewd cardinals are Σ -reflecting.Proof. Pick a Σ -formula ϕ ( v , . . . , v m − ) and sets z , . . . , z m − ∈ V κ with the propertythat the statement ϕ ( z , . . . , z m − ) holds in V. By Σ -absoluteness, there exists a cardinal θ > κ with the property that ϕ ( z , . . . , z m − ) holds in H( θ ). An application of Lemma 2.1now yields cardinals ¯ κ < ¯ θ < κ and an elementary embedding j : X −→ H( θ ) such that¯ κ + 1 ⊆ X ≺ H(¯ θ ), j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ and z , . . . , z m − ∈ ran( j ). Since shrewd cardinalsare inaccessible, we have V ¯ κ ⊆ X and j ↾ V ¯ κ = id V ¯ κ . In particular, we know that z i ∈ V ¯ κ and j ( z i ) = z i holds for all i < m . But then ϕ ( z , . . . , z m − ) holds in H(¯ θ ) ⊆ V κ and henceΣ -absoluteness implies that this statement also holds in V κ . (cid:3) Weakly shrewd cardinals
This section contains an analysis of the basic properties of weakly shrewd cardinals.We start by slightly modifying the proof of Lemma 2.1 to obtain an analogous embeddingcharacterization for weakly shrewd cardinals.
Lemma 3.1.
The following statements are equivalent for every cardinal κ :(i) κ is a weakly shrewd cardinal.(ii) For all sufficiently large cardinals θ > κ , there exist cardinals ¯ κ < ¯ θ , an elementarysubmodel X of H(¯ θ ) and an elementary embedding j : X −→ H( θ ) with ¯ κ + 1 ⊆ X , j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ > ¯ κ .(iii) For all cardinals θ > κ and all z ∈ H( θ ) , there exist cardinals ¯ κ < ¯ θ , an elementarysubmodel X of H(¯ θ ) and an elementary embedding j : X −→ H( θ ) with ¯ κ + 1 ⊆ X , j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ > ¯ κ and z ∈ ran( j ) .Proof. Assume that (i) holds. Fix a recursive enumeration h a l | l < ω i of the class Fml . LetΦ( v , v ) be an L ∈ -formula such that Φ( A, δ ) expresses that the conjunction of the followingstatements holds true:(a) δ is an infinite cardinal.(b) There is a cardinal θ > δ , a subset X of H( θ ) and a bijection b : δ −→ X such thatthe following statements hold: • δ + 1 ⊆ X , b (0) = δ and b ( ω · (1 + α )) = α for all α < δ . • The class H( θ ) is a set and, given α , . . . , α n − < δ and l < ω with the propertythat a l codes a formula with n free variables, we have ≺ l, α , . . . , α n − ≻ ∈ A ⇐⇒ Sat ( X, h b ( α ) , . . . , b ( α n − ) i , a l ) ⇐⇒ Sat (H( θ ) , h b ( α ) , . . . , b ( α n − ) i , a l ) . Fix a cardinal θ > κ , z ∈ H( θ ) and a cardinal ϑ > θ with the property that H( ϑ ) issufficiently elementary in V. Pick an elementary submodel Y of H( θ ) of cardinality κ with κ ∪{ κ, z } ⊆ Y and a bijection b : κ −→ Y with b (0) = κ , b (1) = z and b ( ω · (1+ α )) = α for all α < κ . Define A to be the set of all ordinals of the form ≺ l, α , . . . , α n − ≻ such that l < ω , We let ≺· , . . . , ·≻ : Ord n +1 −→ Ord denote iterated G¨odel pairing . α , . . . , α n − < κ , a l codes a formula with n free variables and Sat ( Y, h b ( α ) , . . . , b ( α n − ) i , a l )holds. Then Φ( A, κ ) holds in H( ϑ ) and our assumption yields cardinals ¯ κ < ¯ ϑ such that¯ κ < κ and Φ( A ∩ ¯ κ, ¯ κ ) holds in H( ¯ ϑ ). Since the formula defining the predicate Sat is absolutebetween H( ϑ ), H( ¯ ϑ ) and V, the definition of Φ now yields a cardinal ¯ κ < ¯ θ < ¯ ϑ , a subset X of H(¯ θ ) and a bijection ¯ b : ¯ κ −→ X such that the following statements hold: • ¯ κ + 1 ⊆ X , ¯ b (0) = ¯ κ and ¯ b ( ω · (1 + α )) = α for all α < ¯ κ . • Given an L ∈ -formula ϕ ( v , . . . , v n − ) and α , . . . , α n − < ¯ κ , we haveH( θ ) | = ϕ ( b ( α ) , . . . , b ( α n − )) ⇐⇒ Y | = ϕ ( b ( α ) , . . . , b ( α n − )) ⇐⇒ H(¯ θ ) | = ϕ (¯ b ( α ) , . . . , ¯ b ( α n − )) ⇐⇒ X | = ϕ (¯ b ( α ) , . . . , ¯ b ( α n − )) . This shows that X is an elementary submodel of H(¯ θ ) with ¯ κ + 1 ⊆ X and the map j = b ◦ ¯ b − : X −→ H( θ ) is an elementary embedding with j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ and z ∈ ran( j ). These computations show that (iii) holds in this case.Now, assume that (ii) holds and (i) fails. Pick an L ∈ -formula Φ( v , v ) witnessing that κ is not a weakly shrewd cardinal, and a sufficiently large cardinal ϑ > κ with the propertythat H( ϑ ) is sufficiently elementary in V. Then there exists a cardinal κ < θ < ϑ withthe property that for some subset A of κ , the statement Φ( A, κ ) holds in H( θ ) and thereare no cardinals ¯ κ < ¯ θ such that ¯ κ < κ and Φ( A ∩ ¯ κ, ¯ κ ) holds in H(¯ θ ). Let θ be theminimal cardinal with this property. By our assumption, we can find cardinals ¯ κ < ¯ ϑ andan elementary embedding j : X −→ H( ϑ ) with ¯ κ + 1 ⊆ X ≺ H( ¯ ϑ ), j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ > ¯ κ . Since the cardinal θ is definable in H( ϑ ) by an L ∈ -formula with parameter κ , there is a cardinal ¯ θ in X with j (¯ θ ) = θ . Since H( ϑ ) is sufficiently elementary in V,elementarity now yields a subset A of ¯ κ in X with the property that Φ( j ( A ) , κ ) holds inH( θ ) and Φ( j ( A ) ∩ ¯ κ, ¯ κ ) does not hold in H(¯ θ ). Since j ( A ) ∩ ¯ κ = A , we can use elementarityonce more to derive a contradiction and conclude that (i) holds in this case. (cid:3) Corollary 3.2.
Shrewd cardinals are weakly shrewd. (cid:3)
Building upon the equivalence established in Lemma 3.1, we now focus on consequencesof weak shrewdness. The below results will allow us to precisely characterize the class ofstructural reflecting cardinals that are not shrewd. Moreover, they will allow us to showthat the existence of such cardinals below the continuum is consistent.We start by proving two basic observation about cardinal arithmetic properties of weaklyshrewd cardinals.
Proposition 3.3.
Weakly shrewd cardinals are weakly Mahlo.Proof.
Let κ be a weakly shrewd cardinal. Then we can find a cardinal ¯ κ < κ , an elementarysubmodel X of H(¯ κ + ) with ¯ κ + 1 ⊆ X and an elementary embedding j : X −→ H( κ + ) with j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ . First, assume that κ is singular. Then elementarity implies that¯ κ is singular and there is a cofinal function c : cof(¯ κ ) −→ ¯ κ that is an element of X . Since j (cof(¯ κ )) = cof(¯ κ ), we can use elementarity to conclude that j ( c )[cof(¯ κ )] = c [cof(¯ κ )] ⊆ ¯ κ is a cofinal subset of κ , a contradiction. Now, assume that κ is not weakly Mahlo. Byelementarity, there exists a closed unbounded subset C of ¯ κ in X that consists of singularordinals. But then ¯ κ ∈ j ( C ) implies that ¯ κ is singular and elementarity implies that ¯ κ issingular in X , contradicting the above computations. (cid:3) Proposition 3.4. If κ is a weakly shrewd cardinal with κ = κ <κ , then κ is a Mahlo cardinal.Proof. Pick a cardinal ¯ κ and an elementary embedding j : X −→ H( κ + ) with the propertythat ¯ κ + 1 ⊆ X ≺ H(¯ κ + ), j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ > ¯ κ . Assume, towards a contradiction,that κ is not Mahlo. By Proposition 3.3, this implies that κ is not a strong limit cardinaland hence our assumptions show that 2 α = κ holds for some α < κ . In this situation,elementarity yields an α < ¯ κ with X | = “ 2 α = ¯ κ ”. Another application of elementarity nowshows that ¯ κ = 2 α = κ holds, a contradiction. (cid:3) TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 11
We are now ready to provide the desired characterization of weakly shrewd cardinals thatare not shrewd. This results and its proof should be compared with [20, Theorem 1.3] thatprovides an analogous statement for weakly remarkable cardinals that are not remarkable.
Lemma 3.5.
The following statements are equivalent for every weakly shrewd cardinal κ :(i) κ is not a shrewd cardinal.(ii) κ is not a Σ -reflecting cardinal.(iii) There exists a cardinal δ > κ with the property that the set { δ } is definable by a Σ -formula with parameters in H( κ ) .Proof. First, the implication from (ii) to (i) is given by Corollary 2.3.Next, assume that (iii) holds and κ is an inaccessible cardinal. Fix an L ∈ -formula ϕ ( v , v ), a cardinal δ > κ and z ∈ H( κ ) with the property that ∀ x [ ϕ ( x, z ) ←→ x = δ ]holds. Then the statement ∃ x ϕ ( x, z ) holds in V and, since κ is inaccessible, Σ -absolutenessimplies that it fails in V κ . In particular, we know that (ii) holds in this situation.Now, assume that (ii) holds. If κ is not inaccessible, then Proposition 3.4 yields an α < κ with 2 α > κ and, since the set { α } is definable by a Σ -formula with parameter α , we canconclude that (iii) holds in this case. We may therefore assume that κ is an inaccessiblecardinal that is not Σ -reflecting. By standard arguments, this shows that there is an L ∈ -formula ϕ ( v ) and z ∈ V κ with the property that there is a cardinal δ such that z ∈ V δ and ϕ ( z ) holds in V δ , and there is no cardinal α ≤ κ such that z ∈ V α and ϕ ( z ) holds in V α .Let δ denote the least cardinal such that z ∈ V δ and ϕ ( z ) holds in V δ . Then δ > κ and theset { δ } is definable by a Σ -formula with parameter z . This shows that (iii) also holds inthis case.Finally, assume, towards a contradiction, that (i) holds and (ii) fails. By Lemma 2.1,there is a cardinal θ > κ with the property that for all cardinals µ < ν < κ , there is noelementary embedding j : X −→ H( θ ) with µ + 1 ⊆ X ≺ H( ν ), j ↾ µ = id µ and j ( µ ) = κ .Let θ be minimal with this property and pick a strong limit cardinal ϑ > θ with the propertythat H( ϑ ) is sufficiently elementary in V. Using the weak shrewdness of κ , we find cardinals¯ κ < ¯ θ < ¯ ϑ and an elementary embedding j : X −→ H( ϑ ) with ¯ κ ∪ { ¯ κ, ¯ θ } ⊆ X ≺ H( ¯ ϑ ), j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ > ¯ κ and j (¯ θ ) = θ . Since j ↾ ( X ∩ H(¯ θ )) : X ∩ H(¯ θ ) −→ H( θ ) isan elementary embedding with ¯ κ + 1 ⊆ X ∩ H(¯ θ ) ≺ H(¯ θ ), j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ , weknow that ¯ θ ≥ κ and hence we can conclude that ¯ ϑ > κ . Elementarity now shows that, inH( ¯ ϑ ), there is a cardinal ρ > ¯ κ with the property that, for all cardinals µ < ν < ¯ κ , thereis no elementary embedding k : Y −→ H( ρ ) with µ + 1 ⊆ Y ≺ H( ν ), k ↾ µ = id µ and k ( µ ) = ¯ κ . Since this statement can be expressed by a Σ -formula with parameter ¯ κ ∈ V κ ,Σ -absoluteness implies that it holds in V and hence the fact that κ is Σ -reflecting causesthe statement to also hold in V κ . Therefore, we can find ¯ κ < ρ < κ with the property that,in V κ , for all cardinals µ < ν < ¯ κ , there is no elementary embedding k : Y −→ H( ρ ) with µ + 1 ⊆ Y ≺ H( ν ), k ↾ µ = id µ and k ( µ ) = ¯ κ . Since this statement can be expressed inV κ by a Π -formula with parameters ¯ κ and ρ , Σ -absoluteness implies that it also holds inH( ¯ ϑ ). By the above computations, we have ¯ θ ∈ X ∩ [ κ, ¯ ϑ ) = ∅ and this allows us to define λ = min( X ∩ [ κ, ¯ ϑ )). We then know that, in H( ¯ ϑ ), there exists a cardinal ¯ κ < τ < λ suchthat for all cardinals µ < ν < ¯ κ , there is no elementary embedding k : Y −→ H( τ ) with µ + 1 ⊆ Y ≺ H( ν ), k ↾ µ = id µ and k ( µ ) = ¯ κ . By elementarity, we can find such a cardinal τ in X and, by the above setup, it follows that τ < κ . But then elementarity implies that,in H( ϑ ), the cardinal j ( τ ) has the property that for all cardinals µ < ν < κ , there is noelementary embedding k : Y −→ H( j ( τ )) with µ + 1 ⊆ Y ≺ H( ν ), k ↾ µ = id µ and k ( µ ) = κ .Since H( ϑ ) was chosen to be sufficiently elementary in V, this statement also holds in V.But this contradicts the minimality of θ , because τ < κ ≤ ¯ θ implies that j ( τ ) < θ . Thesecomputations show that (i) implies (ii). (cid:3) We now use the above characterization to show that, over the theory ZFC, the existenceof a shrewd cardinal is equiconsistent with the existence of an inaccessible weakly shrewdcardinal. The proof of this equiconsistency will make use of the following notion that willalso be central for the proofs of Theorems 1.9 and 1.10.
Definition 3.6.
Given infinite cardinals κ < δ , the cardinal κ is δ -hyper-shrewd if for allsufficiently large cardinals θ > δ and all z ∈ H( θ ), there exist cardinals ¯ κ < κ < δ < ¯ θ ,an elementary submodel X of H(¯ θ ) and an elementary embedding j : X −→ H( θ ) with¯ κ ∪ { ¯ κ, δ } ⊆ X , j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ , j ( δ ) = δ and z ∈ ran( j ).Note that the second item in Lemma 3.1 implies that all hyper-shrewd cardinals areweakly shrewd. The following lemma provides us with typical examples of hyper-shrewdcardinals: Lemma 3.7.
Let κ be a weakly shrewd cardinal that is not shrewd.(i) There exists δ > κ with the property that the set { δ } is definable by a Σ -formulawith parameters in H( κ ) .(ii) If δ > κ is a cardinal with the property that the set { δ } is definable by a Σ -formulawith parameters in H( κ ) , then κ is δ -hyper-shrewd.Proof. The first statement follows directly from Lemma 3.5. In the following, fix y ∈ H( κ ),a cardinal δ > κ and a Σ -formula ϕ ( v , v ) with the property that δ is the unique elementsatisfying ϕ ( δ, y ). Pick a cardinal θ > δ with the property that ϕ ( δ, y ) holds in H( θ ) and anelement z of H( θ ). Using Lemma 3.1, we find cardinals ¯ κ < ¯ θ and an elementary embedding j : X −→ H( θ ) with the property that ¯ κ + 1 ⊆ X ≺ H(¯ θ ), j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ > ¯ κ and y, z ∈ ran( j ). Then j ↾ (H(¯ κ ) ∩ X ) = id H(¯ κ ) ∩ X and this shows that y ∈ X and j ( y ) = y . In this situation, elementarity yields a cardinal ¯ δ in X such that j (¯ δ ) = δ and ϕ (¯ δ, y ) holds in H(¯ θ ). But then Σ -absoluteness implies that the statements ϕ ( δ, y ) and ϕ (¯ δ, y ) both hold in V. By our assumptions on the formula ϕ , this allows us to concludethat ¯ δ = δ = j (¯ δ ) = j ( δ ). (cid:3) We are now ready to show that shrewd and inaccessible weakly shrewd cardinals possessthe same consistency strength.
Lemma 3.8. If κ is an inaccessible cardinal that is δ -hyper-shrewd for some cardinal δ > κ ,then the interval ( κ, δ ) contains an inaccessible cardinal and, if ε is the least inaccessiblecardinal above κ , then κ is a shrewd cardinal in V ε .Proof. Pick a sufficiently large cardinal ϑ , cardinals ¯ κ < κ < δ < ¯ ϑ and an elementaryembedding j : X −→ H( ϑ ) with ¯ κ ∪ { ¯ κ, δ } ⊆ X ≺ H( ¯ ϑ ), j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ and j ( δ ) = δ . By elementarity, there is exists an inaccessible cardinal in the interval (¯ κ, δ ) thatis an element of X . This directly implies that there exists an inaccessible cardinal in theinterval ( κ, δ ).Now, let ε denote the least inaccessible cardinal above κ and assume, towards a con-tradiction, that κ is not a shrewd cardinal in V ε . By Lemma 2.1, there exists a cardinal κ < θ < ε such that for all cardinals µ < ν < κ , there is no elementary embedding k : Y −→ H( θ ) in V ε such that ¯ κ + 1 ⊆ Y ≺ H( ν ), k ↾ µ = id µ and k ( µ ) = κ . Let θ denote the least cardinal with this property. Since ε and θ are both definable in H( ϑ ) by L ∈ -formulas that only use the parameter κ , we can find cardinals ¯ ε and ¯ θ in X with j (¯ ε ) = ε and j (¯ θ ) = θ . Then ¯ ε is the least inaccessible cardinal above ¯ κ and, since Proposition 3.3shows that κ is a Mahlo cardinal, it follows that ¯ θ < ¯ ε < κ . But this yields a contradic-tion, because the map j ↾ (H(¯ θ ) ∩ X ) : H(¯ θ ) ∩ X −→ H( θ ) is an elementary embedding with¯ κ + 1 ⊆ H(¯ θ ) ∩ X ≺ H(¯ θ ), j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ > ¯ κ , and the inaccessibility of ε impliesthat this map is an element of V ε . (cid:3) We now use a small variation of a standard argument, commonly attributed to Kunen,to show that both weak shrewdness and hyper-shrewdness are downwards absolute to L.
Proposition 3.9.
Let M be an inner model, let κ < ϑ and ¯ κ < ¯ ϑ be M -cardinals, and let j : Y −→ H( ϑ ) M be an elementary embedding with ¯ κ + 1 ⊆ Y ≺ H( ¯ ϑ ) M , j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ > ¯ κ . If ¯ κ < ¯ θ < ¯ ϑ is an M -cardinal in Y with H(¯ θ ) M ∈ H( ¯ ϑ ) M , then there existsan elementary submodel X ∈ Y of H(¯ θ ) M with ¯ κ + 1 ⊆ X ⊆ Y and the property that theembedding j ↾ X is an element of M . TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 13
Proof.
Our assumptions ensure that Y contains an elementary submodel X of H(¯ θ ) M ofcardinality ¯ κ with ¯ κ + 1 ⊆ X . By elementarity, there exists a bijection b : ¯ κ −→ X in Y .Since we have ¯ κ ∪ { b } ⊆ Y ≺ H( ¯ ϑ ) M , we know that X is a subset of Y . Given x ∈ X ,we have j ( b − ( x )) = b − ( x ) and hence we know that j ( x ) = ( j ( b ) ◦ b − )( x ) holds. Since b ∈ H( ¯ ϑ ) M and j ( b ) ∈ H( ϑ ) M , this shows that j ↾ X is an element of M . (cid:3) Corollary 3.10. (i) Every weakly shrewd cardinal is a weakly shrewd cardinal in L .(ii) Given a cardinal δ , every δ -hyper-shrewd cardinal is a δ -hyper-shrewd cardinal in L .Proof. (i) Assume that κ is a weakly shrewd cardinal. Fix an L-cardinal θ > κ and a regularcardinal ϑ > θ . By Lemma 3.1, there exists cardinals ¯ κ < ¯ ϑ and an elementary embedding k : Y −→ H( ϑ ) with ¯ κ + 1 ⊆ Y ≺ H( ϑ ), k ↾ ¯ κ = id ¯ κ , k (¯ κ ) = κ > ¯ κ and θ ∈ ran( k ). Pick¯ θ ∈ Y with k (¯ θ ) = θ . Since ¯ θ ∈ L ¯ ϑ ∩ Y ≺ L ¯ ϑ , we can apply Proposition 3.9 to find anelementary submodel X ∈ L ¯ ϑ ∩ Y of L ¯ θ with | X | L = ¯ κ , ¯ κ + 1 ⊆ X and j = k ↾ X ∈ L. Butthen j : X −→ L θ is an elementary embedding in L with j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ > ¯ κ .These computations show that κ is weakly shrewd in L.(ii) Now, assume that κ is δ -hyper-shrewd for some cardinal δ > κ . Fix an L-cardinal θ > δ , z ∈ L θ and a sufficiently large regular cardinal ϑ > θ . Pick cardinals ¯ κ < ¯ ϑ andan elementary embedding k : Y −→ H( ϑ ) with ¯ κ ∪ { ¯ κ, δ } ⊆ Y ≺ H( ϑ ), k ↾ ¯ κ = id ¯ κ , k (¯ κ ) = κ > ¯ κ , k ( δ ) = δ and θ, z ∈ ran( k ). In addition, fix ¯ θ, ¯ z ∈ Y with j (¯ θ ) = θ and j (¯ z ) = z . Now, use Proposition 3.9 to find an elementary submodel X ∈ L ¯ ϑ ∩ Y of L ¯ θ with | X | L = ¯ κ , ¯ κ ∪ { ¯ κ, δ, ¯ z } ⊆ X and j = k ↾ X ∈ L. Then j witnesses that κ is δ -hyper-shrewdwith respect to θ and z in L. In particular, we have shown that κ is a δ -hyper-shrewdcardinal in L. (cid:3) Corollary 3.11.
The following statements are equiconsistent over
ZFC :(i) There exists a shrewd cardinal.(ii) There exists an inaccessible weakly shrewd cardinal.(iii) There exists a weakly shrewd cardinal.Proof.
Let κ be a weakly shrewd cardinal. Then Corollary 3.10 shows that κ is a weaklyshrewd cardinal in L and Proposition 3.4 implies that κ is inaccessible in L. If κ is not ashrewd cardinal in L, then a combination of Lemma 3.7 with Lemma 3.8 yields an ordinal ε > κ that is inaccessible in L and has the property that κ is a shrewd cardinal in L ε . Thisshows that, over ZFC, the consistency of (iii) implies the consistency of (i). Since all shrewdcardinals are inaccessible weakly shrewd cardinals, this implication yields the statement ofthe corollary. (cid:3) We now show that the existence of weakly shrewd cardinals that are not shrewd is con-sistent by proving that subtle cardinals provide a proper upper bound for the consistencystrength of the existence of hyper-shrewd cardinals. Remember that a cardinal δ is subtle if for every sequence h d α | α < δ i with d α ⊆ α for all α < δ and every closed unboundedsubset C of δ , there exist α, β ∈ C with α < β and d α = d β ∩ α (see [13]). Lemma 3.12. If δ is a subtle cardinal, then the set of all inaccessible δ -hyper-shrewd car-dinals is stationary in δ .Proof. Let δ be a subtle cardinal and assume, towards a contradiction, that the set ofinaccessible δ -hyper-shrewd cardinals is not stationary in δ . Since δ is an inaccessible limitof inaccessible cardinals, there is a closed unbounded subset C of δ consisting of cardinalsthat are not δ -hyper-shrew and are limits of inaccessible cardinals. Given an element κ of C that is inaccessible, our assumptions yield a cardinal θ κ > δ and an element z κ of H( θ κ )with the property that for all cardinals ¯ κ < κ < δ < ¯ θ there is no elementary embedding j : X −→ H( θ κ ) satisfying ¯ κ ∪ { ¯ κ, δ } ⊆ X ≺ H(¯ θ ), j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ , j ( δ ) = δ and z κ ∈ ran( j ). For each inaccessible cardinal κ in C , fix an elementary submodel X κ of H( θ κ )of cardinality κ with κ ∪ { κ, δ, z κ } ⊆ X κ , and a bijection b κ : κ −→ X κ with b (0) = κ , b (1) = δ , b (2) = z κ and b ( ω · (1 + α )) = α for all α < κ . Next, if λ is an element of C that is not inaccessible, then λ is a singular cardinal and we fix a strictly increasing cofinal function c λ : cof( λ ) −→ λ with c λ (0) = 0.Fix an enumeration h a l | l < ω i of Fml and pick a sequence h d α | α < δ i such that thefollowing statements hold for all α < κ : • d α is a subset of α . • If α is an inaccessible cardinal in C , then the set d α consists of all ordinals of theform ≺ l, α , . . . , α n − ≻ such that l < ω , α , . . . , α n − < α , a l codes a formula with n free variables and Sat ( X α , h b ( α ) , . . . , b ( α n − ) i , a l ) holds. • If α is a singular cardinal in C , then the set d α consists of all ordinals of the form ≺ cof( α ) , c α ( ξ ) ≻ with ξ < cof( α ).The subtlety of δ now yields α, β ∈ C with α < β and d α = d β ∩ α . Claim.
The ordinals α and β are both inaccessible.Proof of the Claim. Assume, towards a contradcition, that either α or β is not inaccessible.If α is not inaccessible, then ≺ cof( α ) , ≻ ∈ d α ⊆ d β and hence cof( α ) = cof( β ) < α < β .In the other case, if β is not inaccessible, then the fact that d β ∩ α = d α = ∅ yields a ξ < cof( β ) with ≺ cof( β ) , c β ( ξ ) ≻ ∈ d α and this shows that cof( α ) = cof( β ) < α < β alsoholds in this case. Let ζ be the minimal element of cof( α ) with c β ( ζ ) ≥ α . Then thereexists ζ < ζ < cof( α ) such that c β ( ξ ) < c α ( ζ ) holds for all ξ < ζ . Since our setup ensuresthat ≺ cof( α ) , c α ( ζ ) ≻ ∈ d α ⊆ d β , there exists ξ < cof( α ) with c α ( ζ ) = c β ( ξ ). Moreover,since c β is strictly increasing, the fact that c β ( ξ ) = c α ( ζ ) < α ≤ c β ( ζ ) implies that ξ < ζ and hence we can conclude that c β ( ξ ) < c α ( ζ ) = c β ( ξ ), a contradiction. (cid:3) By the definition of the sequence h d α | α < δ i , we now know that X α | = ϕ ( b α ( α ) , . . . , b α ( α n − )) ⇐⇒ H( θ β ) | = ϕ ( b β ( α ) , . . . , b β ( α n − ))holds for every L ∈ -formula ϕ ( v , . . . , v n − ) and all α , . . . , α n − < α . In particular, if wedefine j = b β ◦ b − α : X α −→ H( θ β ) , then j is an elementary embedding with α ∪ { α, δ } ⊆ X α ≺ H( θ α ), j ↾ α = id α , j ( α ) = β , j ( δ ) = δ and z β ∈ ran( j ), contradicting the above assumptions. (cid:3) We continue by showing that weakly shrewd cardinals can exist below the cardinality ofthe continuum:
Lemma 3.13. If κ is a cardinal that is δ -hyper-shrewd for some cardinal δ > κ and G is Add( ω, δ ) -generic over V , then κ is δ -hyper-shrewd in V[ G ] .Proof. Work in V and pick a sufficiently large cardinal θ > δ and an Add( ω, δ )-name ˙ z inH( θ ). By our assumptions, there are cardinals ¯ κ < κ < δ < ¯ θ and an elementary embedding k : Y −→ H( θ ) with ¯ κ ∪ { ¯ κ, δ } ⊆ Y ≺ H(¯ θ ), k ↾ ¯ κ = id ¯ κ , k (¯ κ ) = κ , k ( δ ) = δ and˙ z ∈ ran( k ). Let X be an elementary submodel of Y of cardinality ¯ κ with the property that¯ κ ∪ { ¯ κ, δ } ⊆ X and ˙ z ∈ k [ X ]. Then the map j = k ↾ X : X −→ H( θ ) is an elementaryembedding with j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ , j ( δ ) = δ and ˙ z ∈ ran( j ). Since | X | = ¯ κ < δ ,there exists a permutation σ of δ that extends the injection j ↾ ( X ∩ δ ). Let τ denotethe automorphism of the partial order Add( ω, δ ) that is induced by the action of σ on thesupports of conditions, i.e. given a condition p in Add( ω, δ ), we have supp( τ ( p )) = σ [supp( p )]and τ ( p )( σ ( α )) = p ( α ) for all α ∈ supp( p ). Then it is easy to see that τ ( p ) = j ( p ) holds forevery condition in Add( ω, δ ) that is an element of X . Moreover, since elementarity impliesthat 2 δ < ¯ θ , we also know that the automorphism τ is an element of H(¯ θ ).Now, work in V[ G ] and set ¯ G = τ − [ G ]. Then j [ ¯ G ∩ X ] ⊆ G and we can thereforeconstruct a canonical lifting j ∗ : X [ ¯ G ∩ X ] −→ H( θ ) V [ G ]; ˙ x ¯ G ∩ X j ( ˙ x ) G of j . Then H( θ ) V [ G ] = H( θ ) V[ G ] and, since τ ∈ H(¯ θ ) V , we also haveH(¯ θ ) V [ G ] = H(¯ θ ) V [ ¯ G ] = H(¯ θ ) V[ G ] . TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 15
In particular, we know that the set X [ ¯ G ∩ X ] is an elementary submodels of H(¯ θ ) V[ G ] andwe can conclude that the embedding j ∗ witnesses the δ -hyper-shrewdness of κ with respectto θ and ˙ z G in V[ G ]. (cid:3) The next lemma will allow us to show that the existence of a hyper-shrewd cardinal isequiconsistent to the existence of a weakly shrewd cardinal that is not shrewd.
Lemma 3.14.
Assume that
V = L . If κ is a cardinal that is δ -hyper-shrewd for somecardinal δ > κ and G is Add( δ + , -generic over V , then, in V[ G ] , the set { δ } is definableby a Σ -formula without parameters and the cardinal κ is inaccessible, weakly shrewd andnot shrewd.Proof. Work in V and fix an L ∈ -formula ϕ ( v , v ), a cardinal θ > κ and a subset A of κ withthe property that the statement ϕ ( A, κ ) holds in H( θ ) V[ G ] . Pick a sufficiently large cardinal ϑ > max { δ + , θ } , cardinals ¯ κ < κ < δ + < ¯ ϑ , and an elementary embedding j : X −→ H( ϑ + )with the property that ¯ κ ∪ { ¯ κ, δ, ¯ ϑ } ⊆ X ≺ H( ¯ ϑ + ), j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ , j ( δ ) = δ and A ∈ ran( j ). Since Proposition 3.4 implies that κ is an inaccessible cardinal, we know that¯ κ is also inaccessible, V ¯ κ is a subset of X and j ↾ V ¯ κ = id V ¯ κ . In particular, the fact that A is an element of ran( j ) implies that A ∩ ¯ κ ∈ X and j ( A ∩ ¯ κ ) = A . Using the weakhomogeneity of Add( δ + ,
1) and the fact that H( ϑ + ) V [ G ] = H( ϑ + ) V[ G ] , our assumptions nowimply that, in H( ϑ + ), every condition in the partial order Add( δ + ,
1) forces the statement ϕ ( A, κ ) to hold in the H( θ ) of the generic extension of H( ϑ + ). Pick a cardinal ¯ θ in X with j (¯ θ ) = θ . Since δ is a fixed point of j , elementarity implies that, in H( ¯ ϑ + ), every conditionin Add( δ + ,
1) forces the statement ϕ ( A ∩ ¯ κ, ¯ κ ) to hold in the H(¯ θ ) of the generic extensionof H( ¯ ϑ + ).Next, observe that we have H( ¯ ϑ + ) V [ G ] = H( ¯ ϑ + ) V[ G ] and therefore the fact that ¯ θ + < ¯ ϑ + holds in V implies that H(¯ θ ) H( ¯ ϑ + ) V[ G ] = H(¯ θ ) V[ G ] . This allows us to conclude that ϕ ( A ∩ ¯ κ, ¯ κ ) holds in H(¯ θ ) V[ G ] . Using the fact that, in V,the partial order Add( δ + ,
1) is <δ + -closed and satisfies the δ ++ -chain condition, the abovecomputations allow us conclude that κ is weakly shrewd in V[ G ].Now, work in V[ G ]. Then δ + is the least ordinal containing a non-constructible subsetand hence the set { δ } is definable by a Σ -formula without parameters. By Lemma 3.5, thisallows us to conclude that κ is an inaccessible weakly shrewd cardinal that is not a shrewdcardinal. (cid:3) We are now ready to determine the position of accessible weakly shrewd cardinals in thelarge cardinal hierarchy.
Proof of Theorem 1.9. (i) Let κ be a weakly shrewd cardinal that is not shrewd. ThenLemma 3.7 shows that κ is δ -hyper-shrewd for some cardinal δ > κ and Corollary 3.10implies that κ is a δ -hyper-shrewd cardinal in L. Since Proposition 3.4 ensures that κ isan inaccessible cardinal in L, Lemma 3.8 now allows us to find an ordinal ε > κ that isinaccessible in L and has the property that κ is a shrewd cardinal in L ε .(ii) Let δ be the least subtle cardinal and let C be a closed unbounded subset of δ . ByLemma 3.12, there is an inaccessible weakly shrewd cardinal κ in C . Since the statement “ µ is subtle ” is absolute between H( µ + ) and V for every infinite cardinal µ , the minimality of δ implies that the set { δ } is definable by a Σ -formula without parameters. An applicationof Lemma 3.5 now allows us to conclude that κ is not a shrewd cardinal.(iii) First, assume that κ is a weakly shrewd cardinal that is not shrewd. Using Lemma3.7, we find a cardinal δ > κ such that κ is δ -hyper-shrewd. Let G be Add( ω, δ )-genericover V. Then Lemma 3.13 implies that κ is a weakly shrewd cardinal smaller than 2 ℵ inV[ G ]. These arguments show that, over the theory ZFC, the consistency of the statement(a) listed in the theorem implies the consistency of the statement (c) listed in the theorem.Now, assume that κ is a weakly shrewd cardinal that is not inaccessible. Then κ is nota shrewd cardinal and hence Lemma 3.7 shows that κ is δ -hyper-shrewd for some cardinal δ > κ . In this situation, Corollary 3.10 implies that κ is a δ -hyper-shrewd cardinal in L.Let G be Add( δ + , L -generic over L. Then Lemma 3.14 shows that, in L[ G ], the cardinal κ inaccessible, weakly shrewd and not shrewd. These arguments show that the consistency ofthe statement (b) listed in the theorem implies the consistency of the statement (a) listedin the theorem. (cid:3) We end this section by showing that weak shrewdness is also a direct consequence of largecardinal notions introduced by Schindler in [18] and Wilson in [20]. Note that the resultsof [4] show that the below definition of remarkability is equivalent to Schindler’s originaldefinition.
Definition 3.15. (i) (Schindler) A cardinal κ is remarkable if for every ordinal α > κ ,there is an ordinal β < κ and a generic elementary embedding j : V β −→ V α with j (crit ( j )) = κ .(ii) (Wilson) A cardinal κ is weakly remarkable if for every ordinal α > κ , there is anordinal β and a generic elementary embedding j : V β −→ V α with j (crit ( j )) = κ .Results of Wilson in [20] now allow us to find additional natural examples of hyper-shrewdcardinals. Lemma 3.16.
Every weakly remarkable cardinal is weakly shrewd.Proof.
Let κ be a weakly remarkable cardinal. Assume, towards a contradiction, that κ isnot weakly shrewd and let θ > κ denote the least cardinal with the property that for allcardinals ¯ κ < ¯ θ , there is no elementary embedding j : X −→ H( θ ) with ¯ κ + 1 ⊆ X ≺ H( θ ), j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ > ¯ κ . Pick a strong limit cardinal ϑ > θ with the propertythat H( ϑ ) = V ϑ is sufficiently elementary in V. We then know that κ is definable in V ϑ by a formula with parameter κ . By our assumptions, we can now find ordinals ¯ κ < ¯ ϑ and a generic elementary embedding j : V ¯ ϑ −→ V ϑ with crit ( j ) = ¯ κ and j (¯ κ ) = κ . Thenelementarity implies that ¯ κ is a cardinal in V and ¯ ϑ is a strong limit cardinal in V. Moreover,the definability of θ yields a V-cardinal ¯ θ with j (¯ θ ) = θ . An application of Proposition 3.9in the given generic extension of V now yields an elementary submodel X of H(¯ θ ) V in Vwith ¯ κ + 1 ⊆ X and the property that the embedding j ↾ X : X −→ H( θ ) is an element ofV. Since j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ > ¯ κ , the existence of such an embedding contradicts ourassumptions on θ . (cid:3) Corollary 3.17.
A weakly remarkable cardinal is remarkable if and only if it is shrewd.Proof.
By [20, Theorem 1.3], a weakly remarkable cardinal is remarkable if and only if it isΣ -reflecting. In combination with Lemmas 3.5 and 3.16, this result directly provides thedesired equivalence. (cid:3) Note that [20, Theorem 1.4] shows that every ω -Erd˝os cardinal is a limit of weaklyremarkable cardinals that are not remarkable. By the above results, all of these cardinalsare weakly shrewd and not shrewd.4. Structural reflection
We now connect weak shrewdness with principles of structural reflection.
Lemma 4.1. If κ is weakly shrewd and C is a class of structures of the same type that isdefinable by a Σ -formula with parameters in H( κ ) , then SR − κ ( C ) holds.Proof. Fix a Σ -formula ϕ ( v , v ) and z in H( κ ) with C = { A | ϕ ( A, z ) } . Pick a structure B in C of cardinality κ . Then there exists a cardinal θ > κ with the property that B ∈ H( θ )and ϕ ( B, z ) holds in H( θ ). Using Lemma 3.1, we find cardinals ¯ κ < ¯ θ and an elementaryembedding j : X −→ H( θ ) satisfying ¯ κ + 1 ⊆ X ≺ H(¯ θ ), j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ > ¯ κ and B, z ∈ ran( j ). Then we have j ↾ (H(¯ κ ) ∩ X ) = id H(¯ κ ) ∩ X , and hence we know that z ∈ H(¯ κ )and j ( z ) = z . Pick A ∈ X with j ( A ) = B . Then elementarity and Σ -absoluteness impliesthat ϕ ( A, z ) holds and hence A is a structure in C . Since the structure B has cardinality κ in H( θ ), we know that the structure A has cardinality ¯ κ , and the fact that ¯ κ is a subset of X allows us to conclude that j induces an elementary embedding of A into B . (cid:3) TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 17
By combining the above lemma with the results of the previous sections, we can now provethe desired characterization of weak shrewdness through the principle SR − for Σ -definableclasses of structures. Proof of Theorem 1.7.
Fix an infinite cardinal κ .First, assume that κ is the least cardinal with the property that SR −W ( κ ) holds. Fix acardinal ϑ > κ with the property that H( ϑ ) is sufficiently elementary in V and an elementarysubmodel Y of H( ϑ ) of cardinality κ with κ + 1 ⊆ Y . Then ϑ witnesses that the structure h Y, ∈ , κ i is an element of the class W . By our assumption, we can find an elementaryembedding j of a structure h X, ∈ , ¯ κ i of cardinality smaller than κ in W into h Y, ∈ , κ i . Fixa cardinal ¯ ϑ witnessing that h X, ∈ , ¯ κ i is an element of W . Then we know that ¯ κ < ¯ ϑ are cardinals, X is an elementary submodel of H( ¯ ϑ ) and j : X −→ H( ϑ ) is an elementaryembedding with ¯ κ + 1 ⊆ X and j (¯ κ ) = κ > ¯ κ . Claim. j ↾ ¯ κ = id ¯ κ .Proof of the Claim. Let µ ≤ ¯ κ be the minimal ordinal with j ( µ ) > µ . Assume, towards acontradiction, that µ < ¯ κ . Since µ +1 ⊆ ¯ κ +1 ⊆ X , elementarity implies that µ is a cardinal.Moreover, since H( ϑ ) was chosen to be sufficiently elementary in V, the minimality of κ andthe fact that j ( µ ) < κ allow us to use elementarity to find a cardinal µ < θ < ¯ ϑ in X and anelementary submodel Z of H( θ ) in X with the property that the cardinal j ( θ ) witnesses thatthe structure h j ( Z ) , ∈ , j ( µ ) i is an element of W and there is no elementary embedding froman element of W of cardinality less than j ( µ ) into h j ( Z ) , ∈ , j ( µ ) i . In this situation, we knowthat the structure h Z, ∈ , µ i is an element of W of cardinality µ and, since µ ⊆ X impliesthat Z ⊆ X , the map j induces an elementary embedding from h Z, ∈ , θ i into h j ( Z ) , ∈ , j ( µ ) i ,a contradiction. (cid:3) By Lemma 3.1, the above claim shows that κ is weakly shrewd in this case. By Lemma4.1, the minimality of κ implies that there are no weakly shrewd cardinals smaller than κ .In particular, we know that (ii) implies (i).Now, assume that κ is the least cardinal with the property that SR −C ( κ ) holds for everyclass C of structures of the same type that is definable by a Σ -formula with parametersin H( κ ). Then SR −W ( κ ) holds and we can define µ ≤ κ to be the least cardinal such thatSR −W ( µ ) holds. By the above computations, we know that µ is a weakly shrewd cardinaland therefore Lemma 4.1 implies that SR −C ( µ ) holds for every class C of structures of thesame type that is definable by a Σ -formula with parameters in H( µ ). The minimality of κ then implies that κ = µ and hence κ is a weakly shrewd cardinal. Moreover, anotherapplication of Lemma 4.1 shows that the minimality of κ implies that there are no weaklyshrewd cardinals below κ . These arguments show that (iii) also implies (i).Now, assume that κ is the least weakly shrewd cardinal. Then Lemma 4.1 and the abovecomputations show that SR −C ( κ ) holds for every class C of structures of the same type that isdefinable by a Σ -formula with parameters in H( κ ), and that SR −W ( µ ) fails for all cardinals µ smaller than κ . In combination, this shows that κ is both the least cardinal with theproperty that SR −W ( κ ) holds and the least cardinal with the property that SR −C ( κ ) holds forevery class C of structures of the same type that is definable by a Σ -formula with parametersin H( κ ). This shows that (i) implies that both (ii) and (iii) hold. (cid:3) Next, we determine the consistency strength of structural reflection for Σ -definableclasses of structures. Proof of Theorem 1.5.
By Corollary 3.2 and Lemma 4.1, it suffices to show that, over thetheory ZFC, the consistency of statement (ii) implies the consistency of the statement (i).Hence, assume that there exists a cardinal κ with the property that SR −C ( κ ) holds for everyclass C of structures of the same type that is definable by a Σ ( Cd )-formula without param-eters. Moreover, we may assume that 0 does not exist, because otherwise a combination of[18, Lemma 1.3] with Lemma 3.16 and Corollary 3.17 ensures the existence of many shrewdcardinals in L. In the following, we let L denote the first-order language extending of L ∈ bytwo constant symbols and let K denote the class of all L -structures h X, ∈ , δ, θ i such that δ is an infinite cardinal, θ is an ordinal greater than δ and there exists a cardinal ϑ > θ with the property that X is an elementary submodel of L ϑ with δ ∪ { δ, θ } ⊆ X . It is easy to seethat the class K is definable by Σ ( Cd )-formula without parameters.Now, pick an L-cardinal θ > κ , a cardinal ϑ > θ and an elementary submodel Y of L ϑ of cardinality κ with κ ∪ { κ, θ } ⊆ Y . Then the L -structure h Y, ∈ , κ, θ i is an element of K and therefore our assumptions yield an elementary embedding j of a structure h X, ∈ , ¯ κ, ¯ θ i of cardinality less than κ in K into h Y, ∈ , κ, θ i . Pick a cardinal ¯ ϑ witnessing that h X, ∈ , ¯ κ, ¯ θ i is contained in K . Then we know that ¯ κ is a cardinal smaller than κ with j (¯ κ ) = κ , ¯ θ is anL-cardinal with j (¯ θ ) = θ and X is an elementary submodel of L ¯ ϑ with ¯ κ ∪ { ¯ κ, ¯ θ } ⊆ X . Claim. j ↾ ¯ κ = id ¯ κ .Proof of the Claim. Let µ ≤ ¯ κ be the minimal ordinal in X with j ( µ ) > µ . Assume, towardsa contradiction that µ < ¯ κ holds. Note that, since j (¯ κ ) = κ holds and ¯ κ + 1 ⊆ X impliesL ¯ κ ∪ { L ¯ κ } ⊆ X , we know that the map j ↾ L ¯ κ : L ¯ κ −→ L κ is an elementary embeddingbetween transitive structures. In this situation, our assumption implies that this embeddinghas a critical point. But then the fact that ¯ κ is a cardinal implies that | crit ( j ↾ L ¯ κ ) | ≤ µ < ¯ κ and hence [11, Theorem 18.27] shows that 0 exists, a contradiction. (cid:3) Since the above claim shows that j : Y −→ L ϑ is an elementary embedding with ¯ κ ∪{ ¯ κ, ¯ θ } ⊆ Y ≺ L ¯ ϑ , j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ , we can apply Proposition 3.9 to find anelementary submodel X of L θ in L with X ⊆ Y and the property that the map j ↾ X is alsoan element of L. These computations now allow us to conclude that κ is a weakly shrewdcardinal in L. By Corollary 3.11, this argument shows that, over ZFC, the consistency ofthe second statement listed in the theorem implies the consistency of the first statementlisted there. (cid:3) Using ideas from the above proofs, we can show that a failure of the converse implicationof Lemma 4.1 has non-trivial consistency strength. This argument again makes use of theclass W defined in Section 1. Lemma 4.2. If SR −W ( κ ) holds for some infinite cardinal κ that is not a weakly shrewdcardinal, then exists.Proof. By Lemma 3.1, our assumptions yield a cardinal θ > κ with the property that for allcardinals ¯ κ < ¯ θ , there is no elementary embedding j : X −→ H( θ ) with ¯ κ + 1 ⊆ X ≺ H(¯ θ ), j ↾ ¯ κ = id ¯ κ and j (¯ κ ) = κ > ¯ κ . As in the proof of Theorem 1.7, we can use these assumptionto find cardinals ¯ κ < ¯ θ and an elementary embedding j : X −→ H( θ ) with ¯ κ + 1 ⊆ X ≺ H(¯ θ )and j (¯ κ ) = κ . By our assumptions, we know that j ↾ ¯ κ = id ¯ κ . Since L ¯ κ ∪ { L ¯ κ } ⊆ X , wenow know that j ↾ L ¯ κ : L ¯ κ −→ L κ is an elementary embedding between transitive structuresthat has a critical point. As in the proof of Theorem 1.5, the fact that ¯ κ is a cardinal and | crit ( j ↾ L ¯ κ ) | < ¯ κ now allows us to apply [11, Theorem 18.27] to conclude that 0 exists. (cid:3) By combining the above observation with Lemma 4.1, we can now conclude that, in theconstructible universe L, weak shrewdness is equivalent to the validity of the principle SR − for Σ -definable classes. Corollary 4.3. If V = L holds, then the following statements are equivalent for everyinfinite cardinal κ :(i) κ is a weakly shrewd cardinal.(ii) SR −W ( κ ) holds.(iii) SR −C ( κ ) holds for every class C of structures of the same type that is definable by a Σ -formula with parameters in H( κ ) . (cid:3) In addition, we can also use the above lemma to motivate the statement of Theorem 1.10by showing that the existence of a weakly shrewd cardinal does not imply the existenceof a reflection point for classes of structures defined by Σ -formulas. Note that the aboveresults show that, over ZFC, the consistency of the existence of a shrewd cardinal impliesthe consistency of the assumptions of the following corollary. Corollary 4.4. If V = L holds and there exists a single weakly shrewd cardinal, then thereis no cardinal ρ with the property that SR −C ( ρ ) holds for every class C of structures of thesame type that is definable by a Σ -formula without parameters. TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 19
Proof.
Let κ denote the unique weakly shrewd cardinal. Then it is easy to see that the set { κ } is definable by a Σ -formula without parameters. Therefore, there is a non-empty class C of structures of the same type that is definable by a Σ -formula without parameters andconsists of structures of cardinality κ . In particular, we know that SR −C ( κ ) fails.Now, assume, towards a contradiction, that there is a cardinal ρ with the property thatSR −C ( ρ ) holds for every class C of structures of the same type that is definable by a Σ -formula without parameters. An application of Corollary 4.3 then directly shows that ρ isa weakly shrewd cardinal and hence we can conclude that κ = ρ holds. Since SR −C ( κ ) fails,this yields a contradiction. (cid:3) In the remainder of this section, we show that hyper-shrewd cardinals imply the existenceof cardinals with strong structural reflection properties.
Proof of Theorem 1.10.
Let κ be a weakly shrewd cardinal that is not shrewd. Then Lemma3.5 already shows that there is a cardinal δ > κ with the property that the set { δ } is definableby a Σ -formula with parameters in H( κ ). For the remainder of this proof, fix such a cardinal δ . (ii) Given a natural number n > α < κ , assume, towards a contradiction,that for every cardinal α < ρ < δ , there exists a class C of structures of the same type thatis definable by a Σ n -formula with parameters in H( ρ ) such that SR −C ( ρ ) fails. Now, let ε denote the least strong limit cardinal above δ with the property that, in H( ε ), for everycardinal α < ρ < δ , there exists a class C of structures of the same type that is definableby a Σ n -formula with parameters in H( ρ ) such that SR −C ( ρ ) fails. Then the definability of δ ensures that the set { ε } is also definable by a Σ -formula with parameters in H( κ ). Inthis situation, in H( ε ), there is a class C of structures of the same type and a structure B of cardinality κ in C such that C is definable by a Σ n -formula ϕ ( v , v ) with parameter z ∈ H( κ ) and there is no elementary embedding of a structure of cardinality less than κ in C into B . Since Lemma 3.7 shows that κ is ε -hyper-shrewd, there are cardinals ¯ κ < κ < ε < ¯ θ and an elementary embedding j : X −→ H( θ ) with ¯ κ ∪ { ¯ κ, ε } ⊆ X ≺ H(¯ θ ), j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ , j ( ε ) = ε and B, z ∈ ran( j ). Moreover, since j ↾ (H(¯ κ ) ∩ X ) = id H(¯ κ ) ∩ X , we knowthat z ∈ H(¯ κ ) ∩ X and j ( z ) = z . Pick A ∈ H(¯ θ ) ∩ X with j ( A ) = B . Then A ∈ H( ε ) and,since both ε and z are fixed points of j , we know that ϕ ( A, z ) holds in H( ε ). In particular,we can conclude that A is a structure of cardinality ¯ κ in C and, since ¯ κ is a subset of X ,the embedding j induces an elementary embedding of A into B . But now, the fact A and B are both contained in H( ε ) implies that this embedding is also an element of H( ε ), acontradiction.(iii) Assume that κ is inaccessible and 0 does not exist. Using Lemma 3.7, we find acardinal θ > δ , cardinals ¯ κ < κ < δ < ¯ θ and an elementary embedding j : X −→ H( θ )with ¯ κ ∪ { ¯ κ, δ } ⊆ X ≺ H(¯ θ ), j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ and j ( δ ) = δ . Let ε denote the minimalelement in X ∩ [¯ κ, δ ] satisfying j ( ε ) = ε . Claim. ε is an inaccessible cardinal.Proof of the Claim. First, assume, towards a contradiction, that ε is singular. Since wehave | ε | = | ε | H( θ ) = | ε | H(¯ θ ) ∈ X , it follows that j ( | ε | ) = | ε | ≥ ¯ κ and hence the minimalityof ε implies that ε is a cardinal. Then the non-existence of 0 implies that ε is a singularcardinal in L. Set λ = cof( ε ) L < ε ≤ δ and let c : λ −→ ε be the < L -least cofinal map from λ to ε in L. Since L δ ⊆ H( θ ) ∩ H(¯ θ ), we now know that λ, c ∈ X , j ( λ ) = λ and j ( c ) = c .But this implies that λ < ¯ κ and hence λ ⊆ X . Pick ξ < λ with c ( ξ ) > ¯ κ . Then¯ κ < κ = j (¯ κ ) < j ( c ( ξ )) = j ( c )( j ( ξ )) = c ( ξ ) < ε, contradicting the minimality of ε . This shows that ε is a regular cardinal.Now, assume, towards a contradiction, that ε is not a strong limit cardinal and let µ < ε be the least cardinal satisfying 2 µ ≥ ε . Then the inaccessibility of κ implies that µ ≥ κ .Since elementarity implies that 2 δ ∈ H( θ ) ∩ H(¯ θ ), we know that µ is the least cardinal with Note that, by using an universal Σ n -formula, this statement can be expressed by a single L ∈ -formulawith parameters α and δ . µ ≥ ε in both H( θ ) and H(¯ θ ). But this implies that µ ∈ X ∩ [ κ, ε ) with j ( µ ) = µ , acontradiction. (cid:3) Now, assume, towards a contradiction, that there exists
C ⊆ V ε , a ∈ Fml and z ∈ H( κ )such that C = { A ∈ V ε | Sat (V ε , h A, z i , a ) } and, in V ε , the class C consists of structuresof the same type and SR −C ( κ ) fails. By elementarity, we find C ∈ V ε +1 ∩ X , a ∈ Fml ⊆ X and z ∈ H(¯ κ ) ∩ X such that the elements j ( C ), j ( a ) and j ( z ) witness the above statement.Since we have j ( a ) = a and j ( z ) = z , we also know that j ( C ) = C . Another applicationof elementarity now yields B ∈ C ∩ X such that Sat (V ε , h j ( B ) , z i , a ) holds and, in V ε , thestructure j ( B ) witnesses the failure of SR −C ( κ ). We then know that Sat (V ε , h B, z i , a ) holdsand hence B is also an element of C . Since B is a structure of cardinality ¯ κ in V ε , it followsthat the domain of B is a subset of X and, by the inaccessibility of ε , the restriction of j tothis domain is an element of V ε . But this restricted map is an elementary embedding from B into j ( B ) in V ε , a contradiction. The above arguments show that, in V ε , the principleSR −C ( κ ) holds for every class C that is defined by a formula using parameters from H( κ ). (cid:3) We now use Theorems 1.9 and 1.10 to show that the consistency strength of the existenceof cardinals with maximal local structural reflection properties is strictly smaller than theexistence of a weakly shrewd cardinal that is not shrewd. Moreover, by combining Theorem1.10 with the compactness theorem, we can prove that the principle SR − cannot be used tocharacterize large cardinal properties that imply strong inaccessibility. Proof of Theorem 1.11. (i) Let κ be a weakly shrewd cardinal that is not shrewd. ThenLemma 3.7 shows that κ is δ -hyper-shrewd for some cardinal δ > κ and we can applyCorollary 3.10 to conclude that κ is δ -hyper-shrewd in L. Let G be Add( δ + , L -generic overL. Then Lemma 3.14 shows that, in L[ G ], the set { δ } is definable by a Σ -formula withoutparameters and κ is an inaccessible weakly shrewd cardinal that is not shrewd. Since 0 does not exist in L[ G ] and the partial order Add( δ + , L is <δ -closed in L, we can applythe last part of Theorem 1.10 in L[ G ] to find κ < ε < δ with the property that h L ε , ∈ , κ i is a transitive model of the formalized L ˙ c -theory ZFC + { SR − n | < n < ω } with respect tosome canonical formalized satisfaction predicate. Since such a satisfaction predicate can bedefined by a ∆ ZFC − -formula, the model h L ε , ∈ , κ i also has these properties in both L andV. These computations prove the first part of the theorem.(ii) Assume that the existence of a weakly shrewd cardinal that is not shrewd is consistentwith the axioms of ZFC. By Theorem 1.9, this assumption implies that the existence of aweakly shrewd cardinal smaller than 2 ℵ is consistent with ZFC. Since the set { ℵ } is alwaysdefinable by a Σ -formula without parameters, we can now apply Theorem 1.10 to show thatfor all 0 < n < ω , the L ˙ c -theory ZFC + SR − n + “ ˙ κ < ℵ ” is consistent. By the CompactnessTheorem , this allows us to conclude that our assumption implies the consistency of the L ˙ c -theory ZFC + { SR − n | < n < ω } + “ ˙ κ < ℵ ” . Now, assume that the above L ˙ c -theory is consistent. By Theorem 1.7, this implies thatZFC is consistent with the existence of a weakly shrewd cardinal smaller than 2 ℵ andtherefore ZFC is consistent with the existence of a weakly shrewd cardinal that is notshrewd. (cid:3) Fragments of weak shrewdness
Motivated by Rathjen’s definition of A - η -shrewd cardinals (see [17, Definition 2.2]), wenow study restrictions of weak shrewdness and derive embedding characterizations for theresulting large cardinal notions. Together with the concept of local Σ n ( R )-classes, thisanalysis will allow us to characterize several classical notions from the lower part of the largecardinal hierarchy through the principle SR − in the next section. Moreover, our results willallow us to show that Hamkins’ weakly compact embedding property is equivalent to L´evy’snotion of weak Π -indescribability . This statement about the structure h V ε , ∈i is again stated using the formalized satisfaction relation Sat . TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 21
Remember that L ˙ A denotes the first-order language that extends L ∈ by a unary predicatesymbol ˙ A . Definition 5.1.
Given a class R , a natural number n > θ , an infinitecardinal κ < θ is weakly (Σ n , R, θ ) -shrewd if for every Σ n -formula Φ( v , v ) in L ˙ A and every A ⊆ κ with the property that Φ( A, κ ) holds in h H( θ ) , ∈ , R i , there exist cardinals ¯ κ < ¯ θ suchthat ¯ κ < κ and Φ( A ∩ ¯ κ, ¯ κ ) holds in h H(¯ θ ) , ∈ , R i .It is easy to see that a cardinal κ is weakly shrewd if and only if it is weakly (Σ n , R, θ )-shrewd for all cardinals θ > κ , all 0 < n < ω and all classes R that are definable byΠ -formulas with parameters in H( κ ). The following variation of Lemma 3.1 provides acharacterization of restricted weak shrewdness in terms of elementary embeddings betweenset-sized structures: Lemma 5.2.
The following statements are equivalent for all classes R , all natural numbers n > and all infinite cardinals κ < θ :(i) κ is weakly (Σ n , R, θ ) -shrewd.(ii) For every A ⊆ κ , there exist cardinals ¯ κ < ¯ θ and a Σ n -elementary embedding j : h X, ∈ , R i −→ h H( θ ) , ∈ , R i satisfying h X, ∈ , R i ≺ Σ n − h H(¯ θ ) , ∈ , R i , ¯ κ + 1 ⊆ X , j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ > ¯ κ and A ∈ ran( j ) .(iii) For every z ∈ H( θ ) , there exist cardinals ¯ κ < ¯ θ and an elementary embedding j : h X, ∈ , R i −→ h H( θ ) , ∈ , R i satisfying h X, ∈ , R i ≺ Σ n − h H(¯ θ ) , ∈ , R i , ¯ κ + 1 ⊆ X , j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ > ¯ κ and z ∈ ran( j ) .Proof. First, assume that (ii) holds. Fix a Σ n -formula Φ( v , v ) in L ˙ A and a subset A of κ such that Φ( A, κ ) holds in h H( θ ) , ∈ , R i . Pick cardinals ¯ κ < ¯ θ and a Σ n -elementaryembedding j : h X, ∈ , R i −→ h H( θ ) , ∈ , R i such that h X, ∈ , R i ≺ Σ n − h H(¯ θ ) , ∈ , R i , ¯ κ + 1 ⊆ X , j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ > ¯ κ and A ∈ ran( j ). Then A ∩ ¯ κ ∈ X and j ( A ∩ ¯ κ ) = A . Thereforeelementarity implies that the Σ n -statement Φ( A ∩ ¯ κ, ¯ κ ) holds in h X, ∈ , R i and, since ourassumptions ensure that Σ n -statements are upwards-absolute from h X, ∈ , R i to h H(¯ θ ) , ∈ , R i ,we can conclude that this statement also holds in h H(¯ θ ) , ∈ , R i . These computations showthat (i) holds in this case.Now, assume that (i) holds and fix an element z of H( θ ). Pick an elementary submodel h Y, ∈ , R i of h H( θ ) , ∈ , R i of cardinality κ with κ ∪ { κ, z } ⊆ Y , and a bijection b : κ −→ Y with b (0) = κ , b (1) = z and b ( ω · (1 + α )) = α for all α < κ . Let Fml ∗ denote the class ofall formalized L ˙ A -formulas and let Sat ∗ denote the formalized satisfaction relation for L ˙ A -structures. Then the classes Fml ∗ and Sat ∗ are both defined by Σ -formulas in L ∈ . Fix arecursive enumeration h a l | l < ω i of the class Fml ∗ and let A be the subset of κ consisting ofall ordinals of the form ≺ l, α , . . . , α m − ≻ with the property that l < ω , α , . . . , α m − < κ , a l codes a formula with m free variables and Sat ∗ ( Y, R ∩ Y, h b ( α ) , . . . , b ( α m − ) i , a l ) holds.Let ψ ( v , v ) be a universal Π n -formula in L ˙ A (as constructed in [12, Section 1]). Now, picka Σ n -formula Φ( v , v ) in L ˙ A such that Φ( B, δ ) holds in structures of the form h H( ϑ ) , ∈ , R i if and only if δ < ϑ is a limit ordinal and, in H( ϑ ), there exists a set X and a bijection b : δ −→ X such that the following statements hold:(i) If x ∈ X and k < ω such that ψ ( k, x ) holds in h H( ϑ ) , ∈ , R i , then ψ ( k, x ) holds in h X, ∈ , R i . In the setting of this lemma, the notions of Σ n -elementary embeddings and submodels are definedthrough the absoluteness of Σ n -formulas in the extended language L ˙ A . Note that for every infinite cardinal ϑ and every class R , the structure h H( ϑ ) , ∈ , R i is closed underfunctions which are rudimentary in R . This shows that for every Π n -formula ϕ ( v ) in L ˙ A , we can find k < ω such that h X, ∈ , R i | = ∀ x [ ϕ ( x ) ←→ ψ ( k, x )]holds for every class R , every infinite cardinal ϑ and every elementary submodel h X, ∈ , R i of h H( ϑ ) , ∈ , R i . Note that, given 0 < n < ω , a standard induction shows that the class of all L ˙ A -formulas ϕ ( v , . . . , v m − ) with the property that there exists a Σ n -formula ψ ( v , . . . , v m − ) in L ˙ A such that ZFCproves that h H( ϑ ) , ∈ , R i | = ∀ x , . . . , x m − [ ϕ ( x , . . . , x m − ) ←→ ψ ( x , . . . , x m − )]holds for every class R and every infinite cardinal ϑ is closed under conjunctions, disjunctions, boundeduniversal quantification and unbounded existential quantification. (ii) δ + 1 ⊆ X , b (0) = δ and b ( ω · (1 + α )) = α for all α < δ .(iii) Given α , . . . , α m − < δ and l < ω such that a l codes a formula with m freevariables, we have ≺ l, α , . . . , α m − ≻ ∈ B ⇐⇒ Sat ∗ ( X, R ∩ X, h b ( α ) , . . . , b ( α m − ) i , a l ) . Then Φ(
A, κ ) holds in h H( θ ) , ∈ , R i and hence we can find cardinals ¯ κ < ¯ θ such that ¯ κ < κ and Φ( A ∩ ¯ κ, ¯ κ ) holds in h H(¯ θ ) , ∈ , R i . Pick a set X ∈ H(¯ θ ) and a bijection ¯ b : ¯ κ −→ X witnessing this. Then the definition of the formula Φ ensures that all Π n -formulas in L ˙ A aredownwards-absolute from h H(¯ θ ) , ∈ , R i to h X, ∈ , R i and this directly allows us to concludethat h X, ∈ , R i ≺ Σ n − h H(¯ θ ) , ∈ , R i . Define j = b ◦ ¯ b : X −→ H( θ ) . Then we know that j is an elementary embedding of h X, ∈ , R i into h H( θ ) , ∈ , R i satisfying j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ and z ∈ ran( j ). This shows that (iii) holds in this case. (cid:3) We now derive some easy consequences of the equivalences established above.
Corollary 5.3.
Given infinite cardinals κ < θ , if κ is weakly (Σ , ∅ , θ ) -shrewd, then κ isweakly inaccessible.Proof. Assume, towards a contradiction, that κ is not weakly inaccessible. Since Lemma 5.2directly implies that all weakly (Σ , ∅ , θ )-shrewd cardinals are uncountable limit cardinals,this assumption allows us to find a cofinal subset A of κ of order-type λ < κ . By Lemma5.2, there exist cardinals ¯ κ < ¯ θ and an elementary embedding j : X −→ H( θ ) satisfying X ≺ Σ H(¯ θ ), ¯ κ + 1 ⊆ X , j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ > ¯ κ and A, λ ∈ ran( j ). But then we have λ ∪ { λ, A ∩ ¯ κ } ⊆ X , j ↾ ( λ + 1) = id λ +1 and j ( A ∩ ¯ κ ) = A . By elementarity, this impliesthat otp ( A ) = λ = otp ( A ∩ ¯ κ ) and hence A ⊆ ¯ κ , a contradiction. (cid:3) Next, we observe that, in the case θ = κ + , the Lemma 5.2 yields non-trivial elementaryembeddings between transitive structures. Note that the same argument as for Σ -formulasin L ∈ shows that h M, ∈ , R i ≺ Σ h N, ∈ , R i holds for every class R and all transitive classes M and N with M ⊆ N . Corollary 5.4.
Let R be a class, let n > be a natural number, let κ be a weakly (Σ n , R, κ + ) -shrewd cardinal and let z ∈ H( κ + ) . Then there exists a transitive set N and a non-trivialelementary embedding j : h N, ∈ , R i −→ h H( κ + ) , ∈ , R i with the property that crit ( j ) is acardinal, h N, ∈ , R i ≺ Σ n − h H(crit ( j ) + ) , ∈ , R i , j (crit ( j )) = κ and z ∈ ran( j ) .Proof. With the help of Lemma 5.2, we can find cardinals ¯ κ < ¯ θ and an elementary em-bedding j : h N, ∈ , R i −→ h H( κ + ) , ∈ , R i with h N, ∈ , R i ≺ Σ n − h H(¯ θ ) , ∈ , R i , ¯ κ + 1 ⊆ N , j ↾ ¯ κ = id ¯ κ , j (¯ κ ) = κ > ¯ κ and z ∈ ran( j ). In this situation, elementarity implies that N = H(¯ κ + ) N = H(¯ κ + ) ∩ N ⊆ H(¯ κ + ) and, since ¯ κ ⊆ N , this shows that N is transitive. Inparticular, we know that h N, ∈ , R i ≺ Σ h H(¯ κ + ) , ∈ , R i and this completes the proof in thecase n = 1. Next, if n >
2, then elementarity directly implies that ¯ θ = ¯ κ + and this showsthat N also possesses the desired properties in this case. Finally, if n = 2, then we have h N, ∈ , R i ≺ Σ h H(¯ θ ) , ∈ , R i , transitivity implies that all Σ -formulas are upwards-absolutefrom h N, ∈ , R i to h H(¯ κ + ) , ∈ , R i and from h H(¯ κ + ) , ∈ , R i to h H(¯ θ ) , ∈ , R i , and therefore wecan conclude that h N, ∈ , R i ≺ Σ h H(¯ κ + ) , ∈ , R i holds in this case. (cid:3) The next lemma will allow us to show that both weak inaccessibility and weak Mahlonessare equivalent to certain restrictions of weak shrewdness.
Lemma 5.5.
Given a class R of cardinals, the following statements are equivalent for everycardinal κ in R :(i) κ is regular and the set R ∩ κ is stationary in κ .(ii) κ is weakly (Σ , R, κ + ) -shrewd.Proof. First, assume that (i) holds and fix z ∈ H( κ + ). With the help of an elementary chainof submodels, we can use the stationarity of R ∩ κ in κ to find a cardinal ¯ κ ∈ R ∩ κ andan elementary substructure h X, ∈ , R i of h H( κ + ) , ∈ , R i of cardinality ¯ κ such that X ∩ κ = ¯ κ and tc( { z } ) ⊆ X . Let π : X −→ N denote the corresponding transitive collapse. TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 23
Claim. π [ R ∩ X ] = N ∩ R .Proof of the Claim. In one direction, if δ ∈ R ∩ X ⊆ κ + , then the fact that R consists ofcardinals implies that either δ = κ and π ( δ ) = ¯ κ ∈ R , or δ ∈ X ∩ κ = ¯ κ and π ( δ ) = δ ∈ R .In the other direction, if α ∈ N with π − ( α ) / ∈ R , then either α < ¯ κ and α = π − ( α ) / ∈ R ,or α > ¯ κ , α has cardinality ¯ κ in N and we can again make use of the fact that R consists ofcardinals to conclude that α / ∈ R . (cid:3) The above claim now shows that π : h N, ∈ , R i −→ h X, ∈ , R i is an isomorphism and thisdirectly implies that π − : h N, ∈ , R i −→ h H( κ + ) , ∈ , R i is an elementary embedding withcrit (cid:0) π − (cid:1) = ¯ κ , π − (¯ κ ) = κ and z ∈ ran( π − ). Moreover, we know that N is a transitive setof cardinality ¯ κ and hence h N, ∈ , R i ≺ Σ h H(¯ κ + ) , ∈ , R i . By Lemma 5.2, this shows that (ii)holds in this case.Now, assume that (ii) holds. Then Corollary 5.3 shows that κ is regular. Fix a closedunbounded subset C of κ and use Corollary 5.4 to find a transitive set N and a non-trivial elementary embedding j : h N, ∈ , R i −→ h H( κ + ) , ∈ , R i such that j (crit ( j )) = κ and C ∈ ran( j ). Then C ∩ crit ( j ) ∈ N , j ( C ∩ crit ( j )) = C and hence crit ( j ) ∈ Lim( C ) ⊆ C .Since elementarity implies that crit ( j ) ∈ R , we can now conclude that C ∩ R = ∅ . Thisshows that (i) holds in this case. (cid:3) Corollary 5.6. (i) A cardinal κ is weakly inaccessible if and only if it is weakly (Σ , Cd, κ + ) -shrewd.(ii) A cardinal κ is weakly Mahlo if and only if it is weakly (Σ , Rg, κ + ) -shrewd.Proof. Note that a regular cardinal is weakly inaccessible if and only if it is a stationarylimit of cardinals. Moreover, a regular cardinal is defined to be weakly Mahlo if and onlyit is a stationary limit of regular cardinals. Therefore both statement follow directly fromLemma 5.5. (cid:3)
Next, we show that weak Π n -indescribability also corresponds to a certain canonicalrestrictions of weak shrewdness. Lemma 5.7.
The following statements are equivalent for every cardinal κ and every naturalnumber n > :(i) κ is weakly Π n -indescribable.(ii) κ is weakly (Σ n +1 , ∅ , κ + ) -shrewd.Proof. First, assume that (ii) holds, A , . . . , A m − are relations on κ and Φ is a Π n -sentencethat holds in h κ, ∈ , A , . . . , A m − i . Using Corollary 5.4, we find a transitive set N anda non-trivial elementary embedding j : N −→ H( κ + ) with the property that crit ( j ) is acardinal, N ≺ Σ n H(crit ( j ) + ), j (crit ( j )) = κ and A , . . . , A m − ∈ ran( j ). Given i < m , if A i is a k i -ary relation on κ , then have A ∩ crit ( j ) k i ∈ N with j ( A ∩ crit ( j ) k i ) = A . Therefore,elementarity implies that, in N , the sentence Φ holds in the structure h crit ( j ) , ∈ , A ∩ crit ( j ) k , . . . , A m − ∩ crit ( j ) k m − i . Since this statement can be expressed by a Π n -formula with parameters in N , our as-sumptions imply that it also holds in H(crit ( j ) + ) and therefore it holds in V too. Thesecomputations show that (i) holds in this case.Now, assume that (i) holds. In order to derive the desired conclusion, we introduce acanonical translation of Π n -statements over H( κ + ) into Π n -statements over some expansionsof the structure h κ, ∈i by finitely-many relation symbols. First, we set P = {h α, β, ≺ α, β ≻i | α, β ∈ Ord } . For each infinite cardinal δ , we now define C ( δ ) to consists of all subsets B of δ with theproperty that E B = {h α, β i | α, β < δ, ≺ α, β ≻ ∈ B } is an extensional and well-founded relation on δ . Note that with the help of rank functions,it is easy to see that there is a Σ -formula that uniformly defines C ( δ ) in h δ, ∈ , P ∩ δ i . Given B ∈ C ( δ ), we let π B : h δ, E B i −→ h z B , ∈i denote the corresponding transitive collapse and set x B = π B (0). Then x B , z B ∈ H( δ + ) for all B ∈ C ( δ ) and it is also easy to see that everyelement of H( δ + ) is of the form x B for some B ∈ C ( δ ). Now, given an infinite cardinal δ and subsets A, B , . . . , B m − of δ with B , . . . , B m − ∈ C ( δ ), we say that B ∈ C ( δ ) codes amodel containing A, x B , . . . , x B m − if the following statements hold: • x B = κ , π B (1) = A and π − B ( α ) = ω · (1 + α ) for all α < κ . • For every i < m , the transposition τ i = (0 , i + 2) is an isomorphism of h κ, E B i and h κ, E B i i .Note that, in the above situation, an easy induction shows that for all i < m , we have π B = π B i ◦ τ i , z B = z B i and x B i = π B ( i + 2). Moreover, it is easy to see that for every m < ω , there is a Σ -formula Ψ( v , . . . , v m +1 ) with second-order variables v , . . . , v m +1 andthe property that for every infinite cardinal δ and all A, B, B , . . . , B m − ⊆ δ , the statementΨ( A, B, B , . . . , B m − ) holds in h δ, ∈ , P ∩ δ i if and only if B, B , . . . , B m − ∈ C ( δ ) and B codes a model containing A, x B , . . . , x B m − . Given some Σ -formula ϕ ( v , . . . , v m ) in L ∈ ,we can now use the fact that a Σ -formula in L ∈ holds true if and only if it holds truein a transitive set containing all of its parameters to find a Σ -formula Φ( v , . . . , v m ) withsecond-order variables v . . . , v m and the property that for every infinite cardinal δ , all A ⊆ δ and all B , . . . , B m − ∈ C ( δ ), we haveH( δ + ) | = ϕ ( A, x B , . . . , x B m − ) ⇐⇒ h δ, ∈ , P ∩ δ i | = Φ( A, B , . . . , B m − ) . Now, fix a Π n -formula ϕ ( v , v , v ) and a subset A of κ with the property that thestatement ∃ x ϕ ( x, A, κ ) holds in H( κ + ). With the help of the above constructions, we cannow find a Π n -sentence Φ with the property thatH( δ + ) | = ∃ x ϕ ( x, B, δ ) ⇐⇒ ∃ X ⊆ δ h δ, P ∩ δ , B, X i | = Φholds for every infinite cardinal δ and every subset B of δ . Then there exists X ⊆ κ with h κ, P ∩ κ , A, X i | = Φ and hence we can apply [15, Theorem 6] to find a cardinal ¯ κ < κ suchthat Φ holds in h ¯ κ, P ∩ ¯ κ , A ∩ ¯ κ, X i . But this shows that ∃ x ϕ ( x, A ∩ ¯ κ, ¯ κ ) holds in H(¯ κ + ).These computations allow us to conclude that (ii) holds in this case. (cid:3) We end this section by using the embedding characterization for weakly Π -indescribablecardinals provided by Corollary 5.4 and Lemma 5.7 to show that this large cardinal propertyis equivalent to Hamkins’ weakly compact embedding property . Remember that a cardinal κ has the weakly compact embedding property if for every transitive set M of cardinality κ with κ ∈ M , there is a transitive set N and an elementary embedding j : M −→ N withcrit ( j ) = κ . Corollary 5.8.
The following statements are equivalent for every infinite cardinal κ :(i) κ has the weakly compact embedding property.(ii) κ is weakly Π -indescribable.(iii) There is a transitive set N and a non-trivial elementary embedding j : N −→ H( κ + ) with the property that crit ( j ) is a cardinal, j (crit ( j )) = κ and N ≺ Σ H(crit ( j ) + ) .(iv) For every cardinal θ > κ and all z ∈ H( θ ) , there is a transitive set N and a non-trivial elementary embedding j : N −→ H( θ ) with the property that crit ( j ) is acardinal, j (crit ( j )) = κ , H(crit ( j ) + ) N ≺ Σ H(crit ( j ) + ) and z ∈ ran( j ) .Proof. First, assume that (i) holds. Fix a cardinal θ > κ and z ∈ H( θ ). Pick an elementarysubmodel X of H( θ ) of cardinality κ with κ ∪ { κ, z } ⊆ X and let π : h X, ∈i −→ h B, ∈i denote the corresponding transitive collapse. Note that elementarity directly implies that π − ↾ H( κ + ) B = id H( κ + ) B . Pick an elementary submodel M of H( κ + ) of cardinality κ with κ ∪ { B } ⊆ M and fix a bijection b : κ −→ B in M . Since M is transitive, we can use theweakly compact embedding property to find a transitive set N and an elementary embedding j : M −→ N with crit ( j ) = κ . Define D = {≺ α, β ≻ | b ( α ) ∈ b ( β ) } ∈ M and let E ∈ M be the binary relation on κ coded by D . Then h κ, E i is well-foundedand extensional, and b : h κ, E i −→ h B, ∈i is the corresponding transitive collapse. Now, I.e. τ i is the unique permutation of κ with supp( τ i ) = { , } , τ i (0) = i + 2 and τ i ( i + 2) = 0. TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 25 since D = j ( D ) ∩ κ ∈ N implies that E is contained in N and N is a transitive modelof ZFC − , the above observations show that B and b are also elements of N . Moreover,since j ↾ κ = id κ , we have j ( x ) = ( j ( b ) ◦ b − )( x ) for all x ∈ B and therefore we knowthat the elementary embedding j ↾ B : B −→ j ( B ) is an element of N . In addition, sincewe have H( κ + ) B ⊆ H( κ + ) N ⊆ H( κ + ) and our setup ensures that H( κ + ) B = H( κ + ) X is anelementary submodel of H( κ + ), we know that H( κ + ) B ≺ Σ H( κ + ) N . Using elementarity,these computations yield a transitive set A of cardinality less than κ and an elementaryembedding k : A −→ B with the property that crit ( k ) is a cardinal with k (crit ( k )) = κ , π ( z ) ∈ ran( k ) and H(crit ( k ) + ) A ≺ Σ H(crit ( k ) + ). Finally, using the fact that π − ( κ ) = κ ,we can conclude that the elementary embedding π − ◦ k : A −→ H( θ ) possesses all of theproperties listed in (iv). Therefore, these arguments show that (iv) holds in this case.Next, a combination of Lemma 5.2 with Lemma 5.7 directly shows that (iv) implies (ii)and (ii) implies (iii).Finally, assume, towards a contradiction, that (iii) holds and (i) fails. By elementarity, weknow that, in N , there is a transitive set M of cardinality crit ( j ) with crit ( j ) ∈ M and theproperty that for every transitive set B , there is no elementary embedding k : M −→ B withcrit ( k ) = crit ( j ). Since this statement can be formalized by a Π -formula with parameters M and crit ( j ), our assumptions imply that it holds in H(crit ( j ) + ) and, by Σ -absoluteness,it also holds in V. But this yields a contradiction, because the map j ↾ M : M −→ j ( M ) isan elementary embedding with these properties. (cid:3) Note that the above result also shows that one has to add inaccessibility to the assump-tions of the statements in [10, Section 4] to obtain correct results.6.
Characterizations of small large cardinals
In this section, we will complete the proof of Theorem 1.13. We start by showing that alllocal Σ n ( R )-classes considered in this section are in fact Σ -definable. Proposition 6.1.
Let R be a class, let n > be a natural number and let S be a class thatis uniformly locally Σ n ( R ) -definable in some parameter z .(i) If n = 1 , then the class S is Σ ( R ) -definable in the parameter z .(ii) If the class P wSet is Σ ( R ) -definable in the parameter z , then the class S is defin-able in the same way.(iii) If the class R is definable by a Π -formula with parameter z , then the class S isdefinable by a Σ -formula with parameter z .Proof. Let ϕ ( v , v ) be a Σ n -formula in L ˙ A witnessing that S is uniformly locally Σ n ( R )-definable.(i) If n = 1, then the absoluteness of Σ -formulas between transitive structures impliesthat S consists of all sets x with the property that there exists a transitive set N suchthat x, z ∈ N and ϕ ( x, z ) holds in h N, ∈ , R i . This equality then directly provides a Σ ( R )-definition of S in the parameter z .(ii) If the class P wSet is Σ ( R )-definable in parameter z , then the class Cd of all cardinalsand the function that sends a cardinal δ to the set H( δ + ) are definable in the same way.It now follows that S consists of all sets x such that there exists a cardinal δ such that x, z ∈ H( δ + ) and ϕ ( x, z ) holds in h H( δ + ) , ∈ , R i . This again provides a Σ ( R )-definition of S in the parameter z .(iii) Assume that there is a Π -formula ψ ( v , v ) in L ∈ that witnesses that the class R isΠ -definable in the parameter z . By Σ -absoluteness, the class S consists of all sets x withthe property that there exists a cardinal δ and a set Q such that x, z ∈ H( δ + ), ϕ ( x, z ) holdsin h H( δ + ) , ∈ , Q i and Q = { y ∈ H( δ + ) | H( δ + ) | = ψ ( y, z ) } . Since the function that sends a cardinal δ to the set H( δ + ) is definable by a Σ -formulawithout parameters, we can conclude that S is Σ -definable in the parameter z . (cid:3) We now show how the restricted forms of weak shrewdness introduced above are connectedto principles of structural reflection for local Σ n ( R )-classes. Lemma 6.2.
Let R be a class, let n > be a natural number and let ¯ κ < κ be infinitecardinals. If there exists a transitive set N and an elementary embedding j : h N, ∈ , R i −→h H( κ + ) , ∈ , R i with ¯ κ ∈ N , h N, ∈ , R i ≺ Σ n − h H(¯ κ + ) , ∈ , R i and j (¯ κ ) = κ , then j (crit ( j )) is a regular cardinal and SR −C (j(crit (j))) holds for every local Σ n ( R ) -class over the set { x ∈ N | j ( x ) = x } .Proof. Set µ = j (crit ( j )). First, assume, towards a contradiction, that µ is singular. Thencrit ( j ) is singular in N , and, since N is a transitive model of ZFC − , this shows that N contains a cofinal function c : λ −→ crit ( j ) for some ordinal λ < crit ( j ). Then elementarityimplies that the range of the function j ( c ) : λ −→ µ is cofinal in µ and this yields acontradiction, because elementarity also implies thatran( j ( c )) = ran( c ) ⊆ crit ( j ) < µ. Now, let C be a local Σ n ( R )-class over the set F = { x ∈ N | j ( x ) = x } . Pick a Σ n ( R )-formula ϕ ( v , v ) and z ∈ F witnessing that the class C is uniformly locally Σ n ( R )-definable.Assume, towards a contradiction, that C contains a structure A of cardinality µ with theproperty that for every structure B in C of cardinality less than µ , there is no elementaryembedding from B into A . Since C is closed under isomorphic copies, this implies that, in h H( κ + ) , ∈ , R i , there is a structure A of the given type of cardinality µ such that ϕ ( A, z )holds and for all structures B of the given type of cardinality less than µ , if ϕ ( B, z ) holds,then there is no elementary embedding of B into A . By our assumptions, elementarity allowsus to find a structure A with these properties that is contained in ran( j ). Pick B ∈ N with j ( B ) = A . Then elementarity ensures that ϕ ( B, z ) holds in h N, ∈ , R i . Since Σ n -formulas in L ˙ A are upwards-absolute from h N, ∈ , R i to h H(¯ κ + ) , ∈ , R i , this shows that ϕ ( B, z ) holds in h H(¯ κ + ) , ∈ , R i and therefore B is contained in C . Next, since N is transitive, the embedding j induces an elementary embedding of B into A . Finally, since B has cardinality less than µ ,we can pick an isomorphic copy C of B that is an element of H( κ + ). Then the structure C iscontained in C and therefore ϕ ( C, z ) holds in h H( κ + ) , ∈ , R i . But then the above argumentsyield an elementary embedding from C into A and this map is also contained in H( κ + ),contradicting the properties of A . (cid:3) Corollary 6.3.
Let κ be an infinite cardinal, let R be a class and let n > be a naturalnumber. If κ is weakly (Σ n , R, κ + ) -shrewd, then SR −C ( κ ) holds for every local Σ n ( R ) -class C over H( κ ) .Proof. Let C be a local Σ n ( R )-class over H( κ ). Pick z ∈ H( κ ) witnessing that the class C isuniformly locally Σ n ( R )-definable. Using Corollary 5.4, our assumptions allow us to find atransitive set N and a non-trivial elementary embedding j : h N, ∈ , R i −→ h H( κ + ) , ∈ , R i withthe property that crit ( j ) is a cardinal, h N, ∈ , R i ≺ Σ n − h H(crit ( j ) + ) , ∈ , R i , j (crit ( j )) = κ and z ∈ ran( j ). Since we now have z ∈ H(crit ( j )) ∩ N and j ( z ) = z , Lemma 6.2 directlyimplies that SR −C ( κ ) holds. (cid:3) The following partial converse of Lemma 6.2 is the last ingredient needed for our charac-terization of small large cardinals through principles of structural reflection:
Lemma 6.4.
Let n > be a natural number, let R be a class and let z be a set such thatthe class Cd of all cardinals is uniformly locally Σ n ( R ) -definable in the parameter z . Thenthere is a local Σ n ( R ) -class C over { z } with the property that whenever SR −C ( κ ) holds forsome infinite cardinal κ with z ∈ H( κ ) , then for every y ∈ H( κ + ) , there exists a cardinal ¯ κ < κ , a transitive set N and an elementary embedding j : h N, ∈ , R i −→ h H( κ + ) , ∈ , R i suchthat ¯ κ, z ∈ N , h N, ∈ , R i ≺ Σ n − h H(¯ κ + ) , ∈ , R i , y ∈ ran( j ) , j (¯ κ ) = κ and j ( z ) = z .Proof. Let L denote the first-order language that extends L ˙ A by three constant symbols.Define C to be the class of all L -structures that are isomorphic to an L -structure of theform h N, ∈ , Q, δ, y, z i with the property that δ is an infinite cardinal, N is a transitive setof cardinality δ , Q = N ∩ R and h N, ∈ , R i ≺ Σ n − h H( δ + ) , ∈ , R i . Claim. C is a local Σ n ( R ) -class over { z } . TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 27
Proof.
By definition, the class C is closed under isomorphic copies. Note that, given aninfinite cardinal ν with z ∈ H( ν ), an L -structure A in H( ν + ) is contained in C if and onlyif there exists a cardinal µ ≤ ν , a transitive set N of cardinality µ and y ∈ N such that z ∈ N , h N, ∈ , R i ≺ Σ n − h H( µ + ) , ∈ , R i and H( ν + ) contains an isomorphism between A andthe resulting L -structure h N, ∈ , N ∩ R, µ, y, z i . In the case n = 1, our assumptions on R together with the fact that h N, ∈ , R i ≺ Σ h H( | N | + ) , ∈ , R i holds for every infinite transitiveset N directly yield a Σ ( R )-formula witnessing that C is uniformly locally Σ ( R )-definablein the parameter z . Now, assume that n >
1. Since the classes H( µ + ) are uniformly Σ -definable in the parameter µ and the relativization of a Σ n -formula in L ˙ A to a Σ -class againyields a Σ n -formula in L ˙ A , we can use a universal Σ n − -formula in L ˙ A to find a Σ n -formula ψ ( v , v ) in L ˙ A with the property that for all infinite cardinals µ ≤ ν and all N ∈ H( ν + ),the statement ψ ( µ, N ) holds in h H( ν + ) , ∈ , R i if and only if N ∈ H( µ + ), N is transitive and h N, ∈ , R i ≺ Σ n − h H( µ + ) , ∈ , R i . Together with the above observations and our assumptionson R , we can again conclude that there is a Σ n -formula in L ˙ A that witnesses that the class C is uniformly locally Σ n ( R )-definable in the parameter z . (cid:3) Now, let κ be a cardinal with the property that z ∈ H( κ ) and SR −C ( κ ) holds. Fix y ∈ H( κ ) and pick an elementary submodel h X, ∈ , R i of h H( κ + ) , ∈ , R i of cardinality κ suchthat κ ∪{ κ, y, z } ⊆ X . Then the resulting L -structure h X, ∈ , R ∩ X, κ, y, z i is an element of C of cardinality κ . By the definition of C and our assumptions on κ , there exists an elementaryembedding j of a structure h N, ∈ , Q, ¯ κ, x, z i in C of cardinality less than κ with the propertythat ¯ κ is a cardinal, N is a transitive set, Q = N ∩ R and h N, ∈ , R i ≺ Σ n − h H(¯ κ + ) , ∈ , R i . Inparticular, we can conclude that j : h N, ∈ , R i −→ h H( κ + ) , ∈ , R i is an elementary embeddingwith y ∈ ran( j ), j (¯ κ ) = κ and j ( z ) = z . (cid:3) Corollary 6.5.
Let n > be a natural number and let R be a class with the property that theclass Cd is uniformly locally Σ n ( R ) -definable without parameters. If κ is the least cardinalwith the property that SR −C ( κ ) holds for all local Σ n ( R ) -classes over ∅ , then κ is weakly (Σ n , R, κ + ) -shrewd.Proof. Fix z ∈ H( κ + ). By Lemma 6.4, there exists a cardinal ¯ κ < κ , a transitive set N andan elementary embedding j : h N, ∈ , R i −→ h H( κ + ) , ∈ , R i with the property that ¯ κ ∈ N , h N, ∈ , R i ≺ Σ n − h H(¯ κ + ) , ∈ , R i , j (¯ κ ) = κ and z ∈ ran( j ). An application of Lemma 6.2 nowshows that the minimality of κ implies that crit ( j ) = ¯ κ . By Lemma 5.2, these computationsshow that κ is weakly (Σ n , R, κ + )-shrewd. (cid:3) Corollary 6.6.
Let n > be a natural number and let R be a class with the property thatthe class Cd is uniformly locally Σ n ( R ) -definable without parameters. Then the followingstatements are equivalent for every cardinal κ :(i) κ is the least regular cardinal with the property that κ is weakly (Σ n , R, κ + ) -shrewd.(ii) κ is the least cardinal with the property that SR −C ( κ ) holds for every local Σ n ( R ) -class over ∅ .(iii) κ is the least cardinal with the property that SR −C ( κ ) holds for every local Σ n ( R ) -class over H( κ ) .Proof. First, assume that (i) holds. Then we can use Corollary 6.3 to conclude that SR −C ( κ )holds for every local Σ n ( R )-class C over H( κ ). Moreover, the minimality of κ allows us toapply Corollary 6.5 to show that κ is the least cardinal with the property that SR −C ( κ ) holdsfor all local Σ n ( R )-classes over ∅ . But this also shows that κ is the least cardinal with theproperty that SR −C ( κ ) holds for all local Σ n ( R )-classes over H( κ ).Now, assume that (ii) holds. An application of Corollary 6.5 then shows that κ is weakly(Σ n , R, κ + )-shrewd. Moreover, we can use Corollary 6.3 to conclude that κ is the leastcardinal with this property.Finally, assume that (iii) holds and let ν ≤ κ be the least cardinal with the property thatSR −C ( ν ) holds for every local Σ n ( R )-class over ∅ . Then the above computations show that ν is weakly (Σ n , R, ν + )-shrewd and Corollary 6.3 implies that SR −C ( ν ) holds for every localΣ n ( R )-class over H( ν ). This allows us to conclude that ν = κ . (cid:3) Proof of Theorem 1.13. (i) The desired equivalence is a direct consequence of a combinationof Corollary 5.6 with Corollary 6.6.(ii) Note that the class Cd is uniformly locally Σ ( Rg )-definable without parameters,because a limit ordinal is a cardinal if and only if it is either regular or a limit of regularcardinals. Therefore, we can again combine Corollary 5.6 with Corollary 6.6 to derive thedesired equivalence.(iii) Since the class Cd is uniformly locally Σ -definable without parameters, the desiredequivalence for weakly Π n -indescribable cardinals is a direct consequence of Lemma 5.7 andCorollary 6.6. (cid:3) We end this paper by studying restrictions on the reflection properties of various cardinals.These results are motivated by unpublished work of Brent Cody, Sean Cox, Joel Hamkinsand Thomas Johnstone that shows that various cardinal invariants of the continuum do notpossess the weakly compact embedding property (see [9]). The following results place thisimplication into the general framework developed above.
Proposition 6.7.
Given a class I of infinite cardinals, there exists a class C of structuresof the same type such that SR −C (min(I)) fails and the following statements hold for everynatural number n > , every class R and every set z :(i) If I is Σ n ( R ) -definable in the parameter z , then C is definable in the same way.(ii) If I is uniformly locally Σ n ( R ) -definable in the parameter z , then C is a local Σ n ( R ) -class over { z } .Proof. Let L denote the trivial first-order language and define C to be the class of L -structures whose cardinality is an element of I . Then SR −C (min(I)) fails and both definabilitystatements are immediate. (cid:3) The next proposition provides examples of cardinal invariants of the continuum (thecardinality 2 ℵ of the continuum, bounding number b and the dominating number d ) that canbe represented as minima of universally locally Σ -definable classes of cardinals. It shouldbe noted that a combination of Lemma 5.6, Corollary 6.3 and Proposition 6.7 shows that, bystarting with a Mahlo cardinal κ and forcing Martin’s Axiom together with 2 ℵ = κ to hold ina generic extension, it is possible to obtain a model in which the set { ℵ } = [ b , ℵ ] = [ d , ℵ ]is not uniformly locally Σ ( Rg )-definable with parameters in H( κ ). Proposition 6.8.
The sets { ℵ } , [ b , ℵ ] and [ d , ℵ ] are all uniformly locally Σ -definablewithout parameters.Proof. First, let ϕ ( v ) be the canonical Σ -formula stating that v is a cardinal and there existsa bijection between v and P ( ω ). Fix an infinite cardinal ν . In one direction, if 2 ℵ ≤ ν , thenH( ν + ) contains a bijection between 2 ℵ and the reals, and hence ϕ (2 ℵ ) holds in H( ν + ).In the other direction, if ϕ ( µ ) holds in H( ν + ), then the fact that H( ν + ) contains all realsimplies that 2 ℵ = µ . This shows that ϕ ( v ) witnesses that the set { ℵ } is uniformly locallyΣ -definable without parameters.Next, let ϕ ( v ) be the canonical Σ -formula stating that v is a cardinal and there exists anunbounded family of cardinality v in h ω ω, < ∗ i . Fix an infinite cardinal ν . Given a cardinal b ≤ µ ≤ min( ν, ℵ ), the set H( ν + ) contains a bijection between µ and an unbounded familyin h ω ω, < ∗ i , and hence we know that ϕ ( µ ) holds in H( ν + ). In the other direction, if ϕ ( µ )holds in H( ν + ), then there exists an unbounded family in h ω ω, < ∗ i of cardinality µ andhence b ≤ µ ≤ ℵ , because ω ω is a subset of H( ν + ). Therefore the formula ϕ ( v ) witnessesthe desired definability of the set [ b , ℵ ].Finally, if ϕ ( v ) denotes the canonical Σ -formula stating that v is a cardinal and thereexists a dominating family of cardinality v in h ω ω, < ∗ i , then we may argue as above to showthat ϕ ( v ) witnesses the desired definability of the set [ d , ℵ ]. (cid:3) In combination with Lemma 5.7, Corollary 5.8 and Corollary 6.3, the above propositionsprovide a general reason for the fact that the cardinals 2 ℵ , b and d do not have the weaklycompact embedding property. In contrast, as discussed in the introduction, Hamkins ob-served that a cardinal with the weakly compact embedding property can be the predecessor TRUCTURAL REFLECTION, SHREWD CARDINALS AND THE CONTINUUM 29 of 2 ℵ . Now, note that if I is a class of cardinals that is definable by a Σ -formula with pa-rameter z and β is an ordinal, then the class of all cardinals of the form ℵ α with the propertythat the cardinal ℵ α + β is an element of I is definable by a Σ -formula with parameters β and z . In particular, the above propositions show that if κ is a weakly shrewd cardinal and α < κ , then κ + α / ∈ { ℵ , b , d } . Finally, note that a combination of Theorem 1.10, Lemma3.12 and Lemma 3.13 shows that, if δ is a subtle cardinal and G is Add( ω, δ )-generic overV, then, in V[ G ], the interval ( b , d ) contains unboundedly many weakly shrewd cardinals. References
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