aa r X i v : . [ m a t h . L O ] S e p CLOSURE PROPERTIES OF MEASURABLE ULTRAPOWERS
PHILIPP L ¨UCKE AND SANDRA M ¨ULLER
Abstract.
We study closure properties of measurable ultrapowers with re-spect to Hamkin’s notion of freshness and show that the extent of these prop-erties highly depends on the combinatorial properties of the underlying modelof set theory. In one direction, a result of Sakai shows that, by collapsing astrongly compact cardinal to become the double successor of a measurable car-dinal, it is possible to obtain a model of set theory in which such ultrapowerspossess the strongest possible closure properties. In the other direction, weuse various square principles to show that measurable ultrapowers of canon-ical inner models only possess the minimal amount of closure properties. Inaddition, the techniques developed in the proofs of these results also allow usto derive statements about the consistency strength of the existence of mea-surable ultrapowers with non-minimal closure properties. Introduction
The present paper studies the structural properties of ultrapowers of models ofset theory constructed with the help of normal ultrafilters on measurable cardi-nals. Two of the most fundamental properties of these ultrapowers are that thesemodels do not contain the ultrafilter utilized in their construction and that theyare closed under sequences of length equal to the relevant measurable cardinal. Inthe following, we want to further analyze the closure and non-closure properties ofmeasurable ultrapowers through the following notion introduced by Hamkins in [8].
Definition 1.1 (Hamkins) . Given a class M , a set A of ordinals is fresh over M if A / ∈ M and A ∩ α ∈ M for all α < lub( A ). Given a normal ultrafilter U on a measurable cardinal, we let Ult(V , U ) denotethe (transitive collapse of the) induced ultrapower and we let j U : V −→ Ult(V , U )denote the corresponding elementary embedding. For notational simplicity, we con-fuse Ult(V , U ) and its elements with their transitive collapses. In this paper, for agiven normal ultrafilter U , we aim to determine the class of limit ordinals containing Mathematics Subject Classification.
Key words and phrases.
Measurable Cardinals, Ultrapowers, Fresh Subsets, Square Sequences,Canonical Inner Models.The authors would like to thank Peter Koepke for a discussion that motivated the work pre-sented in this paper. This project has received funding from the European Union’s Horizon 2020research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 842082of the first author (Project
SAIFIA: Strong Axioms of Infinity – Frameworks, Interactions andApplications ). During the preparation of this paper, the first author was partially supported bythe Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Ex-cellence Strategy – EXC-2047/1 – 390685813. The second author gratefully acknowledges fundingfrom L’OR´EAL Austria, in collaboration with the Austrian UNESCO Commission and in coop-eration with the Austrian Academy of Sciences - Fellowship
Determinacy and Large Cardinals . Here lub( A ) denotes the least upper bound of A . an unbounded subset that is fresh over the ultrapower Ult(V , U ). For the imagesof regular cardinals under the embedding j U , this question was already studied byShani in [26]. Moreover, Sakai investigated closure properties of measurable ultra-powers that imply the non-existence of unbounded fresh subsets at many ordinalsin [21].The following proposition lists the obvious closure properties of measurable ul-trapowers with respect to the non-existence of fresh subsets. Note that the secondpart of the second statement also follows directly from [21, Corollary 3.3]. Theproof of this proposition and the next one will be given in Section 2. Proposition 1.2.
Let U be a normal ultrafilter on a measurable cardinal δ and let λ be a limit ordinal.(i) If the cardinal cof( λ ) is either smaller than δ + or weakly compact, then nounbounded subset of λ is fresh over Ult(V , U ) .(ii) If cof( λ ) > δ and there exists a < (2 δ ) + -closed ultrafilter on cof( λ ) thatcontains all cobounded subsets of cof( λ ) , then no unbounded subset of λ is fresh over Ult(V , U ) . In particular, if there exists a strongly compactcardinal κ with δ < κ ≤ cof( λ ) , then no unbounded subset of λ is freshover Ult(V , U ) . In the other direction, the fact that normal ultrafilters are not contained in thecorresponding ultrapowers directly yields the following non-closure properties ofthese ultrapowers.
Proposition 1.3.
Let U be a normal ultrafilter on a measurable cardinal δ .(i) If κ > δ is the minimal cardinal with P ( κ ) * Ult(V , U ) , then there is anunbounded subset of κ that is fresh over Ult(V , U ) .(ii) If λ is a limit ordinal with cof( λ ) Ult(V ,U ) = j U ( δ + ) , then there is an un-bounded subset of λ that is fresh over Ult(V , U ) .(iii) If δ = δ + holds and λ is a limit ordinal with cof( λ ) = δ + , then there isan unbounded subset of λ that is fresh over Ult(V , U ) . In the following, we will present results that show that the above propositionsalready cover all provable closure and non-closure properties of measurable ultra-powers, in the sense that there are models of set theory in which fresh subsets existat all limit ordinals that are not ruled out by Proposition 1.2 and models in whichfresh subsets only exist at limit ordinals, where their existence is guaranteed byProposition 1.3.Results of Sakai on the approximation properties of measurable ultrapowers in[21] can directly be used to prove the following result that shows that models withminimal non-closure properties can be constructed by collapsing a strongly compactcardinal to become the double successor of a measurable cardinal. Note that wephrase the following result in a non-standard way to clearly distinguish betweenthe ground model of the forcing extension and the model used in the correspondingultrapower construction.
Theorem 1.4.
Let δ be a measurable cardinal and let W be an inner model suchthat the GCH holds in W , V is a Col(( δ + ) V , < ( δ ++ ) V ) W -generic extension of W and ( δ ++ ) V is strongly compact in W . Given a normal ultrafilter U on δ , thefollowing statements are equivalent for every limit ordinal λ :(i) There is an unbounded subset of λ that is fresh over Ult(V , U ) . LOSURE PROPERTIES OF MEASURABLE ULTRAPOWERS 3 (ii) cof( λ ) = δ + .Proof. For one direction, our assumptions directly imply that the GCH holds andtherefore Proposition 1.3 shows that for every limit ordinal λ with cof( λ ) = δ + ,there exists an unbounded subset of λ that is fresh over Ult(V , U ). For the otherdirection, assume, towards a contradiction, that λ is a limit ordinal with cof( λ ) = δ + and the property that some unbounded subset A of λ is fresh over Ult(V , U ). ThenProposition 1.2 implies that cof( λ ) > δ + . By our assumption, [21, Corollary 3.3]directly implies that Ult(V , U ) has the δ ++ -approximation property , i.e., we have X ∈ Ult(V , U ) whenever a set X of ordinals has the property that X ∩ a ∈ Ult(V , U )holds for all a ∈ Ult(V , U ) with | a | Ult(V ,U ) < δ ++ . In particular, there exists some a ∈ Ult(V , U ) with | a | Ult(V ,U ) < δ ++ and A ∩ a / ∈ Ult(V , U ). But then the fact thatcof( λ ) ≥ δ ++ > | a | Ult(V ,U ) ≥ | A ∩ a | implies that A ∩ a is bounded in λ and hencethe assumption that A is fresh over Ult(V , U ) allows us to conclude that A ∩ a isan element of Ult(V , U ), a contradiction. (cid:3) For the other direction, we will prove results that show that canonical innermodels provide measurable ultrapowers with closure properties that are minimal inthe above sense. These arguments make use of the validity of various combinatorialprinciples in these models. In particular, they heavily rely on the existence ofsuitable square sequences . For specific types of cardinals, similar constructionshave already been done in [21, Section 3.3] and [26, Section 3].
Definition 1.5. (i) Given an uncountable regular cardinal κ , a sequence h C γ | γ ∈ Lim ∩ κ i is a (cid:3) ( κ ) -sequence if the following statements hold:(a) C γ is a closed unbounded subset of γ for all γ ∈ Lim ∩ κ .(b) If γ ∈ Lim ∩ κ and β ∈ Lim( C γ ), then C β = C γ ∩ β .(c) There is no closed unbounded subset C of κ with C ∩ γ = C γ for all γ ∈ Lim( C ).(ii) Given an infinite cardinal κ , a (cid:3) ( κ + )-sequence h C γ | γ ∈ Lim ∩ κ + i is a (cid:3) κ -sequence if otp ( C γ ) ≤ κ holds for all γ ∈ Lim ∩ κ + .The next result shows that, in certain models of set theory, fresh subsets formeasurable ultrapowers exist at all limit ordinals that are not ruled out by theconclusions of Proposition 1.2. Theorem 1.6.
Let U be a normal ultrafilter on a measurable cardinal δ . Assumethat the following statements hold:(a) The GCH holds at all cardinals greater than or equal to δ .(b) If κ > δ + is a regular cardinal that is not weakly compact, then there existsa (cid:3) ( κ ) -sequence.(c) If κ > δ is a singular cardinal, then there exists a (cid:3) κ -sequence.Then the following statements are equivalent for every limit ordinal λ :(i) There is an unbounded subset of λ that is fresh over Ult(V , U ) .(ii) The cardinal cof( λ ) is greater than δ and not weakly compact. With the help of results of Schimmerling and Zeman in [25] and [29], the aboveresult will allow us to show that measurable ultrapowers of a large class of canonicalinner models, so called
Jensen-style extender models , possess the minimal amount ofclosure properties with respect to freshness. These inner models go back to Jensenin [12], following a suggestion of S. Friedman, and can have various large cardinals
PHILIPP L¨UCKE AND SANDRA M¨ULLER below supercompact cardinals. As for example in [29], we demand that they satisfyclassical consequences of iterability such as solidity and condensation. The theorembelow holds for Mitchell-Steel extender models with the same properties constructedas in [20] as well. But it turned out that Jensen-style constructions are more naturalin the proof of (cid:3) -principles in canonical inner models, so this is what Schimmerlingand Zeman use in [25] and [29], and we decided to follow their notation.
Theorem 1.7.
Assume that V is a Jensen-style extender model that does not havea subcompact cardinal. Then the statements (i) and (ii) listed in Theorem 1.6 areequivalent for every normal ultrafilter U on a measurable cardinal δ and every limitordinal λ . We restrict ourselves to inner models without subcompact cardinals in the state-ment of Theorem 1.7, as the non-existence of (cid:3) κ -sequences in Jensen-style extendermodels is equivalent to κ being subcompact (see [25]). Results of Kypriotakis andZeman in [15] show that (cid:3) ( κ + )-sequences can exists even if κ is subcompact, butwe decided to not discuss this further here.The techniques developed in the proof of Theorem 1.6 also allows us to derivelarge lower bounds for the consistency strength of the conclusion of Theorem 1.4. Theorem 1.8.
Let U be a normal ultrafilter on a measurable cardinal δ .(i) If there exists a regular cardinal κ > δ + and a cardinal θ ∈ { κ, κ + } withthe property that there exists a (cid:3) ( θ ) -sequence, then there exists a limitordinal λ of cofinality θ and an unbounded subset of λ that is fresh over Ult(V , U ) .(ii) If there exists a singular strong limit cardinal κ with the property that cof( κ ) = δ , the SCH holds at κ and there exists a (cid:3) κ -sequence, then thereexists a limit ordinal λ with cof( λ ) = κ + and an unbounded subset of λ that is fresh over Ult(V , U ) . If U is a normal ultrafilter on a measurable cardinal δ with the property that thestatements (i) and (ii) listed in 1.4 are equivalent for every limit ordinal λ , then thefirst part of the above theorem shows that κ = δ ++ is a countably closed regularcardinal that is greater than max { ℵ , ℵ } and has the property that there are no (cid:3) ( κ )- and no (cid:3) κ -sequences. By [23, Theorem 5.6], the existence of such a cardinalimplies Projective Determinacy . In addition, [13, Theorem 0.1] derives the existenceof a sharp for a proper class model with a proper class of strong cardinals and aproper class of Woodin cardinals from the existence of such a cardinal. Moreover,note that the results of [27] show that the existence of a singular strong limitcardinal κ with the property that there are no (cid:3) κ -sequences implies that AD holdsin L( R ), and even stronger consequences of this assumptions are given by the resultsof [22]. Finally, note that work of Gitik and Mitchell (see [7] and [19]) shows that afailure of SCH at a singular strong limit cardinal κ of uncountable cofinality impliesthat κ is a measurable cardinal of high Mitchell order in a canonical inner model.Finally, our techniques also allow us to determine the exact consistency strengthof the existence of a measurable ultrapower that has the property that no un-bounded subsets of the double successor of the corresponding measurable cardinal Remember that the SCH states that κ cof( κ ) = κ + holds for every singular cardinal κ with2 cof( κ ) < κ . Remember that a cardinal κ is countably closed if µ ω < κ holds for all cardinals µ < κ . LOSURE PROPERTIES OF MEASURABLE ULTRAPOWERS 5 are fresh over it. This result is motivated by results of Cummings in [1] that deter-mine the exact consistency strength of the existence of a measurable ultrapower thatcontains the power set of the successor of the corresponding measurable cardinal.In our setting, Cummings’ results can be rephrased in the following way:
Theorem 1.9.
The following statements are equiconsistent over the theory
ZFC :(i) There exists a ( δ + 2) -strong cardinal δ .(ii) There exists a normal ultrafilter U on a measurable cardinal δ with theproperty that no unbounded subset of δ + is fresh over Ult(V , U ) .Proof. In one direction, [1, Theorem 1] shows that, starting with a model of ZFCcontaining a ( δ + 2)-strong cardinal δ , it is possible to construct a model in whichthere exists a normal ultrafilter U on δ satisfying P ( δ + ) ⊆ Ult(V , U ). In particular,no subset of δ + is fresh over Ult(V , U ) in this model. In the other direction, if U is anormal ultrafilter on a measurable cardinal δ with the property that no unboundedsubset of δ + is fresh over Ult(V , U ), then the closure of Ult(V , U ) under κ -sequencesimplies that P ( δ + ) ⊆ Ult(V , U ) holds and hence [1, Theorem 2] yields an innermodel with a ( δ + 2)-strong cardinal δ . (cid:3) The following theorem determines the exact consistency of the correspondingstatement for double successors of measurable cardinals.
Theorem 1.10.
The following statements are equiconsistent over the theory
ZFC :(i) There exists a weakly compact cardinal above a measurable cardinal.(ii) There exists a normal ultrafilter U on a measurable cardinal δ with theproperty that no unbounded subset of δ ++ is fresh over Ult(V , U ) . Simple closure and non-closure properties
In this section, we prove the two propositions stated in the introduction.
Proof of Proposition 1.2. (i) If cof( λ ) ≤ δ , then the desired statement follows di-rectly from the closure of Ult(V , U ) under δ -sequences. Hence, we may assume that κ = cof( λ ) is a weakly compact cardinal greater than δ . Pick a cofinal sequence h γ α | α < κ i in λ and fix an unbounded subset A of λ such that A ∩ γ ∈ Ult(V , U )for all γ < λ . Given α < κ , fix functions f α and g α with domain δ such that[ f α ] U = γ α and [ g α ] U = A ∩ γ α (recall that we are identifying Ult(V , U ) with itstransitive collapse). Let c : [ κ ] −→ U denote the unique function with the propertythat c ( { α, β } ) = { ξ < δ | f α ( ξ ) < f β ( ξ ) , g α ( ξ ) = f α ( ξ ) ∩ g β ( ξ ) } holds for all α < β < κ . In this situation, since κ > δ is weakly compact, weknow that | U | = 2 δ < κ and hence the weak compactness of κ yields an unboundedsubset H of κ and an element X of U with the property that c [ H ] = { X } . Picka function g with domain δ and the property that g ( ξ ) = S { g α ( ξ ) | α ∈ H } holdsfor all ξ ∈ X . This construction ensures that [ g ] U ∩ γ α = [ g α ] U holds for every α ∈ H and we can conclude that [ g ] U = A . In particular, the set A is not freshover Ult(V , U ).(ii) Fix a < (2 δ ) + -closed ultrafilter F on cof( λ ) that contains all cobounded sub-sets of cof( λ ) and assume, towards a contradiction, that A is an unbounded subsetof λ that is fresh over Ult(V , U ). Pick a strictly increasing sequence h γ η | η < cof( λ ) i that is cofinal in λ . Given η < cof( λ ), fix functions f η and g η with domain δ that PHILIPP L¨UCKE AND SANDRA M¨ULLER satisfy [ f η ] U = γ η and [ g η ] U = A ∩ γ η . Moreover, given η < cof( λ ) and X ∈ U , weset A η,X = { ζ ∈ ( η, cof( λ )) | X = { ξ < δ | f η ( ξ ) < f ζ ( ξ ) , g η ( ξ ) = g ζ ( ξ ) ∩ f η ( ξ ) }} . Then [ { A η,X | X ∈ U } = ( η, cof( λ ))holds for every η < cof( λ ). By our assumptions on F , there exists a sequence h X η | η < cof( λ ) i of elements of U with the property that A η,X η ∈ F holds for all η < cof( λ ). Furthermore, as cof( λ ) > δ , there is a set X ∗ ∈ U and an unboundedsubset E of cof( λ ) with X η = X ∗ for all η ∈ E . By construction, we now have f η ( ξ ) < f ζ ( ξ ) and g η ( ξ ) = g ζ ( ξ ) ∩ f η ( ξ ) for all ξ ∈ X ∗ and all η, ζ ∈ E with η < ζ .Pick a function g with domain δ and g ( ξ ) = [ { g η ( ξ ) | η ∈ E } for all ξ ∈ X ∗ . Then [ g ] U ∩ γ η = [ g η ] U = A ∩ γ η for all η ∈ E and hence wecan conclude that [ g ] U = A ∈ Ult(V , U ), a contradiction. The second part of thestatement follows directly from the first part and the filter extension property ofstrongly compact cardinals (see [14, Proposition 4.1]). (cid:3)
Proof of Proposition 1.3. (i) If κ > δ is the minimal cardinal with P ( κ ) * Ult(V , U ),then every element of P ( κ ) \ Ult(V , U ) is unbounded in κ and fresh over Ult(V , U ).(ii) Fix a limit ordinal λ with cof( λ ) Ult(V ,U ) = j U ( δ + ) and pick a strictly in-creasing, cofinal function c : j U ( δ + ) −→ λ in Ult(V , U ). Since j U [ δ + ] is a cofinalsubset of j U ( δ + ) of order-type δ + , we know that ( c ◦ j U )[ δ + ] is a cofinal subset of λ of order-type δ + . In particular, the closure of Ult(V , U ) under δ -sequences impliesthat every proper initial segment of ( c ◦ j U )[ δ + ] is an element of Ult(V , U ). Finally,since [14, Proposition 22.4] shows that j U [ δ + ] / ∈ Ult(V , U ), we can conclude thatthe set ( c ◦ j U )[ δ + ] is fresh over Ult(V , U ).(iii) First, assume that cof( λ ) Ult(V ,U ) = δ + . Since 2 δ = δ + and U / ∈ Ult(V , U ),we have P ( δ + ) * Ult(V , U ), and we can use (i) to find an unbounded subset A of δ + that is fresh over Ult(V , U ). Let h γ α | α < δ + i be the monotone enumerationof an unbounded subset of λ of order-type δ + in Ult(V , U ). Set B = { γ α | α ∈ A } .Then B is unbounded in λ and it is easy to see that B is fresh over Ult(V , U ).Now, assume that cof( λ ) Ult(V ,U ) > δ + and fix an unbounded subset A of λ oforder-type δ + . Then the closure of Ult(V , U ) under δ -sequences implies that A isfresh over Ult(V , U ). (cid:3) Note that, in the situation of Proposition 1.3, we have cof( λ ) = δ + for everylimit ordinal λ with cof( λ ) Ult(V ,U ) = j U ( δ + ). In particular, if κ is a strong limitcardinal of cofinality δ + , then the fact that j U ( κ ) = κ holds allows us to use thesecond part of the above proposition to conclude that there is an unbounded subsetof κ that is fresh over Ult(V , U ). Moreover, the results of Cummings in [1] discussedin the first section show that the cardinal arithmetic assumption in the third partof the proposition can, in general, not be omitted.3. Fresh subsets at image points of ultrapower embeddings
In this section, we will use a modified square principle introduced in [16] toshow that the existence of a (cid:3) ( κ )-sequence allows us to construct a fresh subset LOSURE PROPERTIES OF MEASURABLE ULTRAPOWERS 7 of j U ( κ ). The principle defined in the next definition is a variation of the indexedsquare principles studied in [4] and [5]. Definition 3.1 (Lambie-Hanson) . Let δ < κ be infinite regular cardinals. A (cid:3) ind ( κ, δ )-sequence is a matrix h C γ,ξ | γ < κ, i ( γ ) ≤ ξ < δ i satisfying the following statements:(i) If γ ∈ Lim ∩ κ , then i ( γ ) < δ .(ii) If γ ∈ Lim ∩ κ and i ( γ ) ≤ ξ < δ , then C γ,ξ is a closed unbounded subsetof γ .(iii) If γ ∈ Lim ∩ κ and i ( γ ) ≤ ξ < ξ < δ , then C γ,ξ ⊆ C γ,ξ .(iv) If β, γ ∈ Lim ∩ κ and i ( γ ) ≤ ξ < δ with β ∈ Lim( C γ,ξ ), then ξ ≥ i ( β ) and C β,ξ = C γ,ξ ∩ β .(v) If β, γ ∈ Lim ∩ κ with β < γ , then there is an i ( γ ) ≤ ξ < δ such that β ∈ Lim( C γ,ξ ).(vi) There is no closed unbounded subset C of κ with the property that, forall γ ∈ Lim( C ), there is ξ < δ such that C γ,ξ = C ∩ γ holds.The main result of [17] now shows that for all infinite regular cardinals δ < κ ,the existence of a (cid:3) ( κ )-sequence implies the existence of a (cid:3) ind ( κ, δ )-sequence. Theproof of the following result is based on this implication. Theorem 3.2.
Let U be a normal ultrafilter on a measurable cardinal δ and let κ > δ be a regular cardinal. If there exists a (cid:3) ( κ ) -sequence, then there is a closedunbounded subset of j U ( κ ) that is fresh over Ult(V , U ) .Proof. By [17, Theorem 3.4], our assumptions allow us to fix a (cid:3) ind ( κ, δ )-sequence h C γ,ξ | γ < κ, i ( γ ) ≤ ξ < δ i . Given γ ∈ Lim ∩ κ , let f γ : δ −→ P ( γ ) denote the unique function with f γ ( ξ ) = ∅ for all ξ < i ( γ ) and f γ ( ξ ) = C γ,ξ for all i ( γ ) ≤ ξ < δ . In this situation, Los’Theorem directly implies that for all β, γ ∈ Lim ∩ κ with β ≤ γ , the set [ f γ ] U isclosed unbounded in j U ( γ ) and [ f β ] U = [ f γ ] U ∩ j U ( β ) holds. Define A = [ { [ f γ ] U | γ ∈ Lim ∩ κ } . By our assumptions on κ , we know that j U ( κ ) = sup( j U [ κ ]) and therefore A is aclosed unbounded subset of j U ( κ ) with A ∩ j U ( γ ) = [ f γ ] U for all γ ∈ Lim ∩ κ .Assume, towards a contradiction, that A is an element of Ult(V , U ). Then thereis f : δ −→ P ( κ ) with [ f ] U = A and the property that f ( ξ ) is a closed unboundedsubset of κ for all ξ < δ . Since we have { ξ < δ | f γ ( ξ ) = f ( γ ) ∩ γ = ∅} ∈ U for all γ ∈ Lim ∩ κ , we can find ξ < δ with the property that ξ ≥ i ( γ ) and f ( ξ ) ∩ γ = C γ,ξ holds for unboundedly many γ below κ . Pick β ∈ Lim( f ( ξ )) and β < γ < κ with ξ ≥ i ( γ ) and f ( ξ ) ∩ γ = C γ,ξ . Then β ∈ Lim( C γ,ξ ) and this implies that ξ ≥ i ( β )and C β,ξ = C γ,ξ ∩ β = f ( ξ ) ∩ β . These computations show that there is a closedunbounded subset C of κ and ξ < δ such that ξ ≥ i ( β ) and C ∩ β = C β,ξ holds forall β ∈ Lim( C ). But this contradicts (vi) in Definition 3.1. (cid:3) PHILIPP L¨UCKE AND SANDRA M¨ULLER Fresh subsets of successors of singular cardinals
We now aim to construct fresh subsets of cardinals that are not contained inthe image of the corresponding ultrapower embedding, e.g. successors of singularcardinals whose cofinality is equal to the relevant measurable cardinal. Our argu-ments will rely on two standard observations about measurable ultrapowers and (cid:3) κ -sequences that we present first. A proof of the following lemma is contained inthe proof of [18, Lemma 1.3]. Lemma 4.1.
Let U be a normal ultrafilter on a measurable cardinal δ . If ν > δ is a cardinal with cof( ν ) = δ and λ δ < ν for all λ < ν , then j U ( ν ) = ν and j U ( ν + ) = ν + . The next lemma contains a well-known construction (see [2, Section 4]) thatshows that, in the situations relevant for our proofs, the existence of some (cid:3) κ -sequence already implies the existence of such a sequence with certain additionalstructural properties. Lemma 4.2.
Let κ be a singular cardinal and let S be a stationary subset of κ + .If there exists a (cid:3) κ -sequence, then there exists a (cid:3) κ -sequence h C γ | γ ∈ Lim ∩ κ + i and a stationary subset E of S such that otp ( C γ ) < κ and Lim( C γ ) ∩ E = ∅ forall γ ∈ Lim ∩ κ + .Proof. Fix a (cid:3) κ -sequence h A γ | γ ∈ Lim ∩ κ + i and a closed unbounded subset C of κ of order-type cof( κ ). Given γ ∈ Lim ∩ κ + , let λ γ = otp ( A γ ) ≤ κ and let h β γα | α < λ γ i denote the monotone enumeration of A γ . Given γ ∈ Lim ∩ κ + with λ γ ∈ Lim( C ) ∪ { κ } , let B γ = { β ∈ A γ | otp ( A γ ∩ β ) ∈ C } . Next, if γ ∈ Lim ∩ κ + with λ γ / ∈ Lim( C ) ∪ { κ } and Lim( C ) ∩ λ γ = ∅ , then we define B γ = A γ . Finally,if γ ∈ Lim ∩ κ + with λ γ / ∈ Lim( C ) ∪ { κ } and Lim( C ) ∩ λ γ = ∅ , then we set α = max(Lim( C ) ∩ λ γ ) < λ γ and we define B γ = B β γα ∪ ( A γ \ B β γα ). Claim.
The sequence h B γ | γ ∈ Lim ∩ κ + i is a (cid:3) κ -sequence with otp ( B γ ) < κ forall γ ∈ Lim ∩ κ + . (cid:3) With the help of Fodor’s Lemma, we can now find a stationary subset E of S and λ < κ with otp ( B γ ) = λ for all γ ∈ E . Then we have | Lim( B γ ) ∩ E | ≤ γ ∈ Lim ∩ κ + . Given γ ∈ Lim ∩ κ + , define C γ = B γ if otp ( B γ ) ≤ λ and let C γ = { β ∈ B γ | otp ( B γ ∩ β ) > λ } if otp ( B γ ) > λ . Claim.
The sequence h C γ | γ ∈ Lim ∩ κ + i is a (cid:3) κ -sequence with otp ( C γ ) < κ and Lim( C γ ) ∩ E = ∅ for all γ ∈ Lim ∩ κ + . (cid:3) This completes the proof of the lemma. (cid:3)
We are now ready to prove the main result of this section that will allow usto handle successors of singular cardinals of measurable cofinality in the proof ofTheorem 1.6.
Theorem 4.3.
Let U be a normal ultrafilter on a measurable cardinal δ and let κ be a singular cardinal of cofinality δ with κ = κ + and the property that λ δ < κ holds for all λ < κ . If there exists a (cid:3) κ -sequence, then there is a closed unboundedsubset of κ + that is fresh over Ult(V , U ) .Proof. By our assumptions, we can apply Lemma 4.2 to obtain a (cid:3) κ -sequence h C γ | γ ∈ Lim ∩ κ + i and a stationary subset E of S κ + δ such that otp ( C γ ) < κ and LOSURE PROPERTIES OF MEASURABLE ULTRAPOWERS 9
Lim( C γ ) ∩ E = ∅ for all γ ∈ Lim ∩ κ + . Next, note that Lemma 4.1 implies that j U (( ν δ ) + ) = ( ν δ ) + < κ holds for all cardinals ν < κ . This allows us to fix themonotone enumeration h κ ξ | ξ < δ i of a closed unbounded subset of κ of order-type δ with the property that j U ( κ ξ ) = κ ξ holds for all ξ < δ . In this situation,the normality of U implies that [ ξ κ ξ ] U = κ and [ ξ κ + ξ ] U ≤ κ + . Given γ ∈ Lim ∩ κ + , let ξ γ denote the minimal element ξ of δ with κ + ξ > otp ( C γ ). Notethat ξ γ ≥ ξ β holds for all γ ∈ Lim ∩ κ + and β ∈ Lim( C γ ).In the following, we inductively construct a sequence h f γ ∈ Y ξ<δ κ + ξ | γ < κ + i . The idea behind this construction is that these functions represent a cofinal subsetof κ + and thereby in particular witness that [ ξ κ + ξ ] U = κ + . We identify each f γ ∈ Q ξ<δ κ + ξ with a function with domain δ in the obvious way and define: • f ( ξ ) = 0 for all ξ < δ . • f γ +1 ( ξ ) = f γ ( ξ ) + 1 for all γ < κ + and ξ < δ . • If γ ∈ Lim ∩ κ + with Lim( C γ ) bounded in γ and ξ < δ , then f γ ( ξ ) = min { λ ∈ Lim | λ > f β ( ξ ) for all β ∈ C γ \ max(Lim( C γ )) } . • If γ ∈ Lim ∩ κ + with Lim( C γ ) unbounded in γ and ξ < ξ γ , then f γ ( ξ ) = ω . • If γ ∈ Lim ∩ κ + with Lim( C γ ) unbounded in γ and ξ γ ≤ ξ < δ , then f γ ( ξ ) = sup { f β ( ξ ) | β ∈ Lim( C γ ) } . Claim. (i) If β < γ < κ + , then f β ( ξ ) < f γ ( ξ ) for coboundedly many ξ < δ .(ii) If γ ∈ Lim ∩ κ + , β ∈ Lim( C γ ) and ξ γ ≤ ξ < δ , then f β ( ξ ) < f γ ( ξ ) .(iii) If γ ∈ Lim ∩ κ + , then f γ ( ξ ) ∈ Lim for all ξ < δ .Proof of the Claim. (i) We prove the statement by induction on 0 < γ < κ + , wherethe successor step follows trivially from our induction hypothesis. Now, assumethat γ ∈ Lim ∩ κ + with Lim( C γ ) bounded in γ . Since δ is an uncountable regularcardinal, our induction hypothesis allows us to find ζ < δ with the property that f β ( ξ ) < f β ( ξ ) holds for all β , β ∈ C γ \ max(Lim( C γ )) with β < β and all ζ ≤ ξ < δ . By definition, we now have f γ ( ξ ) = sup { f β ( ξ ) | β ∈ C γ \ max(Lim( C γ )) } for all ζ ≤ ξ < δ . Since C γ \ max(Lim( C γ )) is a cofinal subset of γ , the desiredstatement for γ now follows directly from our induction hypothesis. Finally, if γ ∈ Lim ∩ κ + with Lim( C γ ) unbounded in γ , then the desired statement for γ follows directly from the definition of f γ and our induction hypothesis.(ii) We prove the claim by induction on γ ∈ Lim ∩ κ + . First, if γ ∈ Lim ∩ κ + with Lim( C γ ) bounded in γ and β = max(Lim( C γ )), then our definition ensuresthat f β ( ξ ) < f γ ( ξ ) holds for all ξ < δ and hence the desired statement followsdirectly from our induction hypothesis. Next, if γ ∈ Lim ∩ κ + with Lim( C γ )unbounded in γ , then our induction hypothesis implies that f β ( ξ ) < f β ( ξ ) holdsfor all β , β ∈ Lim( C γ ) with β < β and all ξ β ≤ ξ < δ . Since ξ γ ≥ ξ β holds forall β ∈ Lim( C γ ), this fact together with our definition yields the desired statementfor γ .(iii) This statement is a direct consequence of the definition of the sequence h f γ | γ < κ + i and statement (ii). (cid:3) Note that the first part of the above claim in particular shows that we have[ f β ] U < [ f γ ] U < [ ξ κ + ξ ] U for all β < γ < κ + . Since we already observed that[ ξ κ + ξ ] U = ( κ + ) Ult(V ,U ) ≤ κ + holds, we can conclude that [ ξ κ + ξ ] U = κ + .Next, notice that the fact that 2 κ = κ + holds allows us to fix an enumeration h h α | α < κ + i of Q ξ<δ P ( κ + ξ ) of order-type κ + . In addition, let h γ α | α < κ + i denote the monotone enumeration of E . We now inductively define a sequence h c γ | γ ∈ Lim ∩ κ + i of functions with domain δ satisfying the following statements for all γ ∈ Lim ∩ κ + :(a) c γ ( ξ ) is a closed unbounded subset of f γ ( ξ ) for all ξ < δ .(b) If β ∈ Lim ∩ γ , then f β ( ξ ) < f γ ( ξ ) and c β ( ξ ) = c γ ( ξ ) ∩ f β ( ξ ) for cobound-edly many ξ < δ .(c) If γ / ∈ E , then c β ( ξ ) = c γ ( ξ ) ∩ f β ( ξ ) for all β ∈ Lim( C γ ) and ξ γ ≤ ξ < δ .(d) If γ ∈ E and α < κ + with γ = γ α , then c γ ( ξ ) = h α ( ξ ) ∩ f γ ( ξ ) for all ξ γ ≤ ξ < δ .The idea behind this definition is to use the fact that the sequence h [ f γ ] U | γ < κ + i is not continuous at ordinals of cofinality δ to diagonalize against the sequence h [ h α ] U | α < κ + i of subsets of κ + in Ult(V , U ) in (d). The inductive definition ofthis sequence is straightforward, but we decided to give the details to convince thereader that it works. We distinguish between the following cases: Case . γ ∈ Lim ∩ κ + with Lim ∩ γ bounded in γ .First, we set β = 0 if Lim( C γ ) = ∅ and β = max(Lim( C γ )) otherwise. Next, weset β = 0 if γ = ω and β = max(Lim ∩ γ ) otherwise. We than have β ≤ β < γ and f β ( ξ ) < f γ ( ξ ) for all ξ < δ . Using our induction hypothesis, we can find ξ γ ≤ ζ < δ with the property that f β ( ξ ) ≤ f β ( ξ ) < f γ ( ξ ) holds for all ζ ≤ ξ < δ and, if β >
0, then c β ( ξ ) = c β ( ξ ) ∩ f β ( ξ ) for all ζ ≤ ξ < δ . Note that ourassumptions imply that Lim( C γ ) is bounded in γ and hence the definition of f γ ( ξ )ensures that cof( f γ ( ξ )) = ω holds for every ξ < δ . Therefore, we can fix a sequenceof strictly increasing functions h k ξ : ω −→ f γ ( ξ ) | ξ < δ i with the property that k ξ is cofinal in f γ ( ξ ) for all ξ < δ , k ξ (0) = f β ( ξ ) for all ξ < ζ , and k ξ (0) = f β ( ξ ) forall ζ ≤ ξ < δ . Define c γ ( ξ ) = { k ξ ( n ) | n < ω } , for all ξ < ζ , if β = 0 . c β ( ξ ) ∪ { k ξ ( n ) | n < ω } , for all ξ < ζ , if β > . { k ξ ( n ) | n < ω } , for all ζ ≤ ξ < δ , if β = 0 . c β ( ξ ) ∪ { k ξ ( n ) | n < ω } , for all ζ ≤ ξ < δ , if β > . These definitions ensure that c γ ( ξ ) is a closed unbounded subset of f γ ( ξ ) forall ξ < δ . Moreover, if β >
0, then c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) holds for all ξ < δ .This inductively implies that c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) holds for all β ∈ Lim( C γ )and all ξ γ ≤ ξ < δ . Next, if β > ζ ≤ ξ < δ , then f β ( ξ ) < f γ ( ξ ) and c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ ). This allows us to conclude that for all β ∈ Lim ∩ γ , wehave f β ( ξ ) < f γ ( ξ ) and c β ( ξ ) = c γ ( ξ ) ∩ f β ( ξ ) for coboundedly many ξ < δ . Case . γ ∈ Lim ∩ κ + with Lim ∩ γ unbounded in γ and Lim( C γ ) bounded in γ .Since our assumptions imply that cof( γ ) = ω , there is a strictly increasing se-quence h β n | n < ω i cofinal in γ such that β n ∈ Lim ∩ γ for all 0 < n < ω , β = 0 LOSURE PROPERTIES OF MEASURABLE ULTRAPOWERS 11 in case Lim( C γ ) = ∅ , and β = max(Lim( C γ )) in case Lim( C γ ) = ∅ . By the reg-ularity of δ , we can find ξ γ ≤ ζ < δ such that f β n ( ξ ) < f β n +1 ( ξ ) < f γ ( ξ ) and c β n +2 ( ξ ) ∩ f β n +1 ( ξ ) = c β n +1 ( ξ ) for all ζ ≤ ξ < δ and all n < ω and, if β > c β ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) for all ζ ≤ ξ < δ . By the definition of f γ , we thenhave f γ ( ξ ) = sup { f β n ( ξ ) | n < ω } for all ζ ≤ ξ < δ . Since the definition of f γ alsoimplies that cof( f γ ( ξ )) = ω and f β ( ξ ) < f γ ( ξ ) for all ξ < δ , we can fix a sequenceof strictly increasing functions h k ξ : ω −→ f γ ( ξ ) | ξ < ζ i with the property that k ξ is cofinal in f γ ( ξ ) for all ξ < ζ and k ξ (0) = f β ( ξ ) for all ξ < ζ . Define c γ ( ξ ) = { k ξ ( n ) | n < ω } , for all ξ < ζ , if β = 0 . c β ( ξ ) ∪ { k ξ ( n ) | n < ω } , for all ξ < ζ , if β > . S { c β n ( ξ ) | < n < ω } , for all ζ ≤ ξ < δ , if β = 0 . S { c β n ( ξ ) | n < ω } , for all ζ ≤ ξ < δ , if β > . Then the set c γ ( ξ ) is closed and unbounded in f γ ( ξ ) for all ξ < δ . In addition,if β >
0, then c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) for all ξ < δ . In particular, we have c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) for all β ∈ Lim( C γ ) and all ξ γ ≤ ξ < δ . Next, if 0 < n < ω and ζ ≤ ξ < δ , then f β n ( ξ ) < f γ ( ξ ) and c γ ( ξ ) ∩ f β n ( ξ ) = c β n ( ξ ). This directlyimplies that for all β ∈ Lim ∩ γ , we have f β ( ξ ) < f γ ( ξ ) and c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ )for coboundedly many ξ < δ . Case . γ ∈ Lim ∩ κ + with γ / ∈ E and Lim( C γ ) unbounded in γ .Let c γ ( ξ ) = ( ω, for all ξ < ξ γ . S { c β ( ξ ) | β ∈ Lim( C γ ) } , for all ξ γ ≤ ξ < δ . Fix β , β ∈ Lim( C γ ) with β < β . Then β ∈ Lim( C β ) and β / ∈ E by thechoice of the (cid:3) κ -sequence and the stationary set E , because we have β ∈ Lim( C γ ).Moreover, if ξ γ ≤ ξ < δ , then the set c β ( ξ ) is unbounded in f β ( ξ ), ξ ≥ ξ β , f β ( ξ ) < f β ( ξ ) and c β ( ξ ) = c β ( ξ ) ∩ f β ( ξ ). This shows that c γ ( ξ ) is a closedunbounded subset of f γ ( ξ ) for all ξ < δ , and c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) holds for all β ∈ Lim( C γ ) and all ξ γ ≤ ξ < δ . Moreover, if β ∈ Lim ∩ γ and β ∈ Lim( C γ ) with β < β , then there is ξ γ ≤ ζ < δ with f β ( ξ ) < f β ( ξ ) and c β ( ξ ) = c β ( ξ ) ∩ f β ( ξ )for all ζ ≤ ξ < δ and hence we have f β ( ξ ) < f γ ( ξ ) and c β ( ξ ) = c γ ( ξ ) ∩ f β ( ξ ) forall ζ ≤ ξ < δ . Case . γ ∈ E .Fix α < κ + with γ = γ α . Let h β ξ | ξ < δ i be the monotone enumeration of asubset of Lim( C γ ) of order-type δ that is closed unbounded in γ . Given ξ γ ≤ ξ < δ ,we have f β ξ ( ξ ) < f γ ( ξ ) and we can therefore pick a closed unbounded subset C γξ of f γ ( ξ ) with min( C γξ ) = f β ξ ( ξ ) and C γξ = h α ( ξ ) ∩ [ f β ξ ( ξ ) , f γ ( ξ )). Now, define c γ ( ξ ) = ( ω, for all ξ < ξ γ . c β ξ ( ξ ) ∪ C γξ , for all ξ γ ≤ ξ < δ . Then c γ ( ξ ) is a closed unbounded subset of f γ ( ξ ) for all ξ < δ and, if ξ γ ≤ ξ < δ ,then c γ ( ξ ) = h α ( ξ ) ∩ f γ ( ξ ). Moreover, if ξ γ ≤ ζ < ξ < δ , then β ζ ∈ Lim( C β ξ ), β ξ / ∈ E , ξ > ξ β ξ , f β ζ ( ξ ) < f β ξ ( ξ ) < f γ ( ξ ) and c β ζ ( ξ ) = c β ξ ( ξ ) ∩ f β ζ ( ξ ) = f γ ( ξ ) ∩ f β ζ ( ξ ) . In particular, this shows that for all β ∈ Lim ∩ γ , we have f β ( ξ ) < f γ ( ξ ) and c β ( ξ ) = c γ ( ξ ) ∩ f β ( ξ ) for coboundedly many ξ < δ .The above construction ensures that [ c γ ] U is a closed unbounded subset of [ f γ ] U for all γ ∈ Lim ∩ κ + . Moreover, we have [ c β ] U = [ c γ ] U ∩ [ f β ] U for all β, γ ∈ Lim ∩ κ + with β < γ . In particular, there is a closed unbounded subset C of κ + with C ∩ [ f γ ] U = [ c γ ] U for all γ ∈ Lim ∩ κ + . Claim.
The set C is fresh over Ult(V , U ) .Proof of the Claim. First, if β < κ + , then there is γ ∈ Lim ∩ κ + with [ f γ ] U > β and C ∩ β = ( C ∩ [ f γ ] U ) ∩ β = [ c γ ] U ∩ β ∈ Ult(V , U ) . Next, assume, towards a contradiction, that C is an element of Ult(V , U ). Thenthere is an α < κ + with C = [ h α ] U . Since we have[ h α ] U ∩ [ f γ α ] U = C ∩ [ f γ α ] U = [ c γ α ] U , we know that the set { ξ < δ | h α ( ξ ) ∩ f γ α ( ξ ) = c γ α ( ξ ) } is an element of U . Inparticular, we can find ξ γ α ≤ ξ < δ with h α ( ξ ) ∩ f γ α ( ξ ) = c γ α ( ξ ), contradicting thedefinition of c γ α . (cid:3) This completes the proof of the theorem. (cid:3) Regular cardinals in
Ult(V , U )We now turn to the construction of fresh subsets of limit ordinals that are notcardinals in V. We first observe that we can restrict ourselves to ordinals that areregular cardinals in the corresponding ultrapower.
Proposition 5.1.
Let U be a normal ultrafilter on a measurable cardinal δ and let λ be a limit cardinal. If there is an unbounded subset of cof( λ ) Ult(V ,U ) that is freshover Ult(V , U ) , then there is an unbounded subset of λ that is fresh over Ult(V , U ) .Proof. Set λ = cof( λ ) Ult(V ,U ) . Let A be an unbounded subset of λ that is freshover Ult(V , U ) and let h γ η | η < λ i be a strictly increasing sequence that is cofinal in λ and an element of Ult(V , U ). In this situation, the set { γ η | η ∈ A } is unboundedin λ and fresh over Ult(V , U ). (cid:3) In the proof of the following result, we modify techniques from the proof ofTheorem 4.3 to cover the non-cardinal case in Theorem 1.6.
Theorem 5.2.
Let U be a normal ultrafilter on a measurable cardinal δ , let κ bea singular cardinal of cofinality δ with the property that µ δ < κ holds for all µ < κ and let κ + < λ < j U ( κ ) be a limit ordinal of cofinality κ + that is a regular cardinalin Ult(V , U ) . If there is a (cid:3) κ -sequence, then there is an unbounded subset of λ thatis fresh over Ult(V , U ) .Proof. As in the proof of Theorem 4.3, we can apply Lemma 4.1 to find the mono-tone enumeration h κ ξ | ξ < δ i of a closed unbounded subset of κ of order-type δ with the property that j U ( κ ξ ) = κ ξ holds for all ξ < δ . Then normality impliesthat [ ξ κ ξ ] U = κ and we can repeat arguments from the first part of the proof ofTheorem 4.3 to see that [ ξ κ + ξ ] U = κ + . By our assumptions, there is a function h with domain δ , [ h ] U = λ and the property that h ( ξ ) is a regular cardinal in the LOSURE PROPERTIES OF MEASURABLE ULTRAPOWERS 13 interval ( κ + ξ , κ ) for all ξ < δ . Fix a sequence h h γ ∈ Q ξ<δ h ( ξ ) | γ < κ + i such thatthe sequence h [ h γ ] U | γ < κ + i is strictly increasing and cofinal in λ .Pick a (cid:3) κ -sequence h C γ | γ ∈ Lim ∩ κ + i with otp ( C γ ) < κ for all γ ∈ Lim ∩ κ + .Given γ ∈ Lim ∩ κ + , we let ξ γ denote the minimal element ξ of δ with κ + ξ > otp ( C γ ).We now inductively construct a sequence h f γ ∈ Y ξ<δ h ( ξ ) | γ < κ + i by setting: • f ( ξ ) = 0 for all ξ < δ . • f γ +1 ( ξ ) = max( f γ ( ξ ) , h γ ( ξ )) + 1 for all γ < κ + and ξ < δ . • If γ ∈ Lim ∩ κ + with Lim( C γ ) bounded in γ and ξ < δ , then f γ ( ξ ) = min { λ ∈ Lim | λ > f β ( ξ ) for all β ∈ C γ \ max(Lim( C γ )) } . • If γ ∈ Lim ∩ κ + with Lim( C γ ) unbounded in γ and ξ < ξ γ , then f γ ( ξ ) = ω . • If γ ∈ Lim ∩ κ + with Lim( C γ ) unbounded in γ and ξ γ ≤ ξ < δ , then f γ ( ξ ) = sup { f β ( ξ ) | β ∈ Lim( C γ ) } . As in the proof of Theorem 4.3, we have the following claim.
Claim. (i) If β < γ < κ + , then f β ( ξ ) < f γ ( ξ ) for coboundedly many ξ < δ .(ii) If γ ∈ Lim ∩ κ + , then f γ ( ξ ) ∈ Lim for all ξ < δ .(iii) If γ ∈ Lim ∩ κ + , β ∈ Lim( C γ ) and ξ γ ≤ ξ < δ , then f β ( ξ ) < f γ ( ξ ) . (cid:3) In particular, this shows that the sequence h [ f γ ] U | γ < κ + i is strictly increasing.Since the above definition ensures that [ h γ ] U < [ f γ +1 ] U holds for all γ < κ + , wealso know that this sequence is cofinal in λ .Next, we inductively define a sequence h c γ | γ ∈ Lim ∩ κ + i of functions withdomain δ such that the following statements hold for all γ ∈ Lim ∩ κ + :(a) c γ ( ξ ) is a closed unbounded subset of f γ ( ξ ) with otp ( c γ ( ξ )) < κ + ξ for all ξ < δ .(b) If β ∈ Lim ∩ γ , then f β ( ξ ) < f γ ( ξ ) and c β ( ξ ) = c γ ( ξ ) ∩ f β ( ξ ) for cobound-edly many ξ < δ .(c) If β ∈ Lim( C γ ) and ξ γ ≤ ξ < δ , then c β ( ξ ) = c γ ( ξ ) ∩ f β ( ξ ).Our inductive construction distinguishes between the following cases: Case . γ ∈ Lim ∩ κ + with Lim ∩ γ bounded in γ .We set β = 0 if Lim( C γ ) = ∅ and β = max(Lim( C γ )) otherwise. Moreover,we set β = 0 if γ = ω and β = max(Lim ∩ γ ) otherwise. This definition ensuresthat β ≤ β < γ and f β ( ξ ) < f γ ( ξ ) for all ξ < δ . We can now find ξ γ ≤ ζ < δ with the property that f β ( ξ ) ≤ f β ( ξ ) < f γ ( ξ ) for all ζ ≤ ξ < δ and, if β > c β ( ξ ) = c β ( ξ ) ∩ f β ( ξ ) for all ζ ≤ ξ < δ . Since the definition of f γ impliesthat cof( f γ ( ξ )) = ω holds for all ξ < δ , we can pick a sequence of strictly increasingfunctions h k ξ : ω −→ f γ ( ξ ) | ξ < δ i with the property that k ξ is cofinal in f γ ( ξ ) forall ξ < δ , k ξ (0) = f β ( ξ ) for all ξ < ζ , and k ξ (0) = f β ( ξ ) for all ζ ≤ ξ < δ . Let c γ ( ξ ) = { k ξ ( n ) | n < ω } , for all ξ < ζ , if β = 0 . c β ( ξ ) ∪ { k ξ ( n ) | n < ω } , for all ξ < ζ , if β > . { k ξ ( n ) | n < ω } , for all ζ ≤ ξ < δ , if β = 0 . c β ( ξ ) ∪ { k ξ ( n ) | n < ω } , for all ζ ≤ ξ < δ , if β > . Then c γ ( ξ ) is a closed unbounded subset of f γ ( ξ ) of order-type less than κ + ξ forall ξ < δ and, if β >
0, then c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) for all ξ < δ . In particular, weknow that c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) for all β ∈ Lim( C γ ) and all ξ γ ≤ ξ < δ . Finally,notice that β > f β ( ξ ) < f γ ( ξ ) and c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) hold forall ζ ≤ ξ < δ . This shows that for all β ∈ Lim ∩ γ , we have f β ( ξ ) < f γ ( ξ ) and c β ( ξ ) = c γ ( ξ ) ∩ f β ( ξ ) for coboundedly many ξ < δ . Case . γ ∈ Lim ∩ κ + with Lim ∩ γ unbounded in γ and Lim( C γ ) bounded in γ .Since the limit points of C γ are bounded in γ , we have cof( γ ) = ω and we canpick a strictly increasing sequence h β n | n < ω i cofinal in γ such that β n ∈ Lim ∩ γ for all 0 < n < ω , β = 0 in case Lim( C γ ) = ∅ , and β = max(Lim( C γ )) incase Lim( C γ ) = ∅ . Fix ξ γ ≤ ζ < δ such that f β n ( ξ ) < f β n +1 ( ξ ) < f γ ( ξ ) and c β n +2 ( ξ ) ∩ f β n +1 ( ξ ) = c β n +1 ( ξ ) for all ζ ≤ ξ < δ and all n < ω , and, if β > c β ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) for all ζ ≤ ξ < δ . Then the definition of f γ ensures thatcof( f γ ( ξ )) = ω and f β ( ξ ) < f γ ( ξ ) for all ξ < δ . Moreover, it also directly impliesthat f γ ( ξ ) = sup { f β n ( ξ ) | n < ω } holds for all ζ ≤ ξ < δ . Fix a sequence of strictlyincreasing functions h k ξ : ω −→ f γ ( ξ ) | ξ < ζ i such that k ξ (0) = f β ( ξ ) and k ξ iscofinal in f γ ( ξ ) for all ξ < ζ . Define c γ ( ξ ) = { k ξ ( n ) | n < ω } , for all ξ < ζ , if β = 0 . c β ( ξ ) ∪ { k ξ ( n ) | n < ω } , for all ξ < ζ , if β > . S { c β n ( ξ ) | < n < ω } , for all ζ ≤ ξ < δ , if β = 0 . S { c β n ( ξ ) | n < ω } , for all ζ ≤ ξ < δ , if β > . Given ξ < δ , the set c γ ( ξ ) is closed and unbounded in f γ ( ξ ) and the regularity of κ + ξ implies that otp ( c γ ( ξ )) < κ + ξ . Next, β > c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ )for all ξ < δ , and therefore c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) for all β ∈ Lim( C γ ) and all ξ γ ≤ ξ < δ . Finally, we have f β n ( ξ ) < f γ ( ξ ) and c γ ( ξ ) ∩ f β n ( ξ ) = c β n ( ξ ) for all0 < n < ω and ζ ≤ ξ < δ , and hence for all β ∈ Lim ∩ γ , we have f β ( ξ ) < f γ ( ξ )and c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) for coboundedly many ξ < δ . Case . γ ∈ Lim ∩ κ + with Lim( C γ ) unbounded in γ .Let c γ ( ξ ) = ( ω, for all ξ < ξ γ . S { c β ( ξ ) | β ∈ Lim( C γ ) } , for all ξ γ ≤ ξ < δ . Given β , β ∈ Lim( C γ ) with β < β and ξ γ ≤ ξ < δ , the above definitionensures that f β ( ξ ) < f β ( ξ ) and c β ( ξ ) = c β ( ξ ) ∩ f β ( ξ ). Since we have f γ ( ξ ) =sup { f β ( ξ ) | β ∈ Lim( C γ ) } and otp ( C γ ) < κ + ξ for all ξ γ ≤ ξ < δ , this shows that c γ ( ξ ) is a closed unbounded subset of f γ ( ξ ) of order-type less than κ + ξ for all ξ < δ ,and, if β ∈ Lim( C γ ) and ξ γ ≤ ξ < δ , then c γ ( ξ ) ∩ f β ( ξ ) = c β ( ξ ) holds. Finally,given β ∈ Lim ∩ γ and β ∈ Lim( C γ ) with β < β , our induction hypothesis yields ξ γ ≤ ζ < δ with f β ( ξ ) < f β ( ξ ) and c β ( ξ ) = c β ( ξ ) ∩ f β ( ξ ) for all ζ ≤ ξ < δ , andthis ensures that f β ( ξ ) < f γ ( ξ ) and c β ( ξ ) = c γ ( ξ ) ∩ f β ( ξ ) for all ζ ≤ ξ < δ .Given γ ∈ Lim ∩ κ + , the properties listed above ensure that [ c γ ] U is a closedunbounded subset of [ f γ ] U of order-type less than κ + . Moreover, if β, γ ∈ Lim ∩ κ + with β < γ , then [ c β ] U = [ c γ ] U ∩ [ f β ] U . These observations show that there is aclosed unbounded subset C of λ with C ∩ [ f γ ] U = [ c γ ] U for all γ ∈ Lim ∩ κ + andthis property directly implies that otp ( C ) = κ + < λ . Since λ is a regular cardinal LOSURE PROPERTIES OF MEASURABLE ULTRAPOWERS 15 in Ult(V , U ), this allows us to conclude that the set C is not contained in Ult(V , U )and hence it is fresh over Ult(V , U ). (cid:3) We end this section by using the above results to show that the validity of theequivalence stated in Theorem 1.4 has high consistency strength.
Proof of Theorem 1.8.
Let U be a normal ultrafilter on a measurable cardinal δ .(i) Assume that κ > δ + is a regular cardinal such that there exists θ ∈ { κ, κ + } with the property that there is a (cid:3) ( θ )-sequence. The regularity of θ then impliesthat j U [ θ ] is cofinal in j U ( θ ) and hence cof( j U ( θ )) = θ . Using Theorem 3.2, we nowfind an unbounded subset of j U ( θ ) that is fresh over Ult(V , U ).(ii) Now, assume that κ is a singular strong limit cardinal of cofinality δ suchthat κ δ = κ + and there exists a (cid:3) κ -sequence. Set θ = κ + and λ = ( θ + ) Ult(V ,U ) .Since | j U ( κ ) | ≤ κ δ = θ and elementarity implies that λ < j U ( κ ), we have λ < θ + . Claim. cof( λ ) = θ .Proof of the Claim. As in the proof of Theorem 5.2, we can find a monotone enu-meration h κ ξ | ξ < δ i of a closed unbounded subset of κ of order-type δ with theproperty that [ ξ κ ξ ] U = κ and [ ξ κ + ξ ] U = θ . Then [ ξ κ ++ ξ ] U = λ . Assume,towards a contradiction, that cof( λ ) = θ . Since κ is singular and λ < θ + , this showthat there is ζ < δ with cof( λ ) < κ ζ . Pick a sequence h f α | α < cof( λ ) i of functionswith domain δ such that f α ( ξ ) < κ ++ ξ holds for all α < cof( λ ) and all ξ < δ ,and the induced sequence h [ f α ] U | α < cof( λ ) i is strictly increasing and cofinal in λ . By our assumption, there is a function f with domain δ and the property that f α ( ξ ) < f ( ξ ) < κ ++ ξ holds for all α < cof( λ ) and all ζ ≤ ξ < δ . But then we have[ f α ] U < [ f ] U < λ for all α < cof( λ ), a contradiction. (cid:3) Since κ is a strong limit cardinal, the above claim now allows us to apply Theorem5.2 to find an unbounded subset of λ that is fresh over Ult(V , U ). (cid:3) Ultrapowers of canonical inner models
With the help of the results of the previous sections, we are now ready to provethe main result of this paper.
Proof of Theorem 1.6.
Fix a normal ultrafilter U on a measurable cardinal δ thatsatisfies the three assumptions listed in the statement of the theorem. By Proposi-tion 1.2, if λ is a limit ordinal with the property that the cardinal cof( λ ) is eithersmaller than δ + or weakly compact, then no unbounded subset of λ is fresh overUlt(V , U ). In the proof of the converse implication, we first consider two specialcases. Claim. If κ is a cardinal with the property that the cardinal cof( κ ) is greater than δ and not weakly compact, then there is an unbounded subset of κ that is fresh over Ult(V , U ) .Proof of the Claim. We start by noting that, if cof( κ ) = δ + , then the fact thatour assumptions imply that 2 δ = δ + holds allows us to use Proposition 1.3 find asubset of κ with the desired properties. Therefore, in the following, we may assumethat δ + < cof( κ ) ≤ κ . Let ν ≤ κ be minimal with ν δ ≥ κ . By the minimality of ν , we then have µ δ < ν for all µ < ν . In particular, the fact that 2 δ = δ + < κ implies that ν > δ > δ and therefore we know that ν + = ν ν ≥ ν δ ≥ κ ≥ ν . Thesecomputations show that either cof( ν ) > δ and κ = ν , or cof( ν ) ≤ δ and κ = ν + .First, assume that either cof( ν ) > δ and κ = ν , or cof( ν ) < δ and κ = ν + . ThenLemma 4.1 shows that j U ( κ ) = κ holds in both cases. Moreover, since cof( κ ) is aregular cardinal greater than δ + and j U (cof( κ )) = cof( j U ( κ )) Ult(V ,U ) = cof( κ ) Ult(V ,U ) , the fact that cof( κ ) is not weakly compact allows us to use Theorem 3.2 to find anunbounded subset of cof( κ ) Ult(V ,U ) that is fresh over Ult(V , U ). In this situation,we can then apply Proposition 5.1 to obtain an unbounded subset of κ that is freshover Ult(V , U ).Finally, assume that cof( ν ) = δ and κ = ν + . In this situation, we know that ν isa singular cardinal of cofinality δ with 2 ν = ν + and the property that µ δ < ν holdsfor all µ < ν . Since the assumptions of the theorem guarantee the existence of a (cid:3) ν -sequence, we can apply Theorem 4.3 to find an unbounded subset of κ that isfresh over Ult(V , U ). (cid:3) Claim.
Let λ be a limit ordinal with the property that the cardinal cof( λ ) is greaterthan δ and not weakly compact. If λ is a regular cardinal in Ult(V , U ) , then thereis an unbounded subset of λ that is fresh over Ult(V , U ) .Proof of the Claim. First, if λ is a cardinal, then we can use the above claim todirectly derive the desired conclusion. Hence, we may assume that λ is not acardinal. Subclaim.
There is a cardinal κ of cofinality δ such that λ ∈ ( κ + , j U ( κ )] ∪ { j U ( κ + ) } and κ > δ implies that µ δ < κ for all µ < κ .Proof of the Subclaim. Let θ = | λ | . Then our assumptions imply that δ < cof( λ ) ≤ θ < λ < θ + . Moreover, we have cof( θ ) = δ , because otherwise θ would be a singular strong limitcardinal of cofinality δ and our assumptions would allow us to repeat the argumentfrom the first part of the proof of Theorem 4.3 to show that θ + = ( θ + ) Ult(V ,U ) , con-tradict our assumption that λ is a cardinal in Ult(V , U ). In addition, we know thatthere is some ν < θ satisfying ν δ ≥ θ , because otherwise Lemma 4.1 would implythat j U ( θ ) = θ < λ < θ + = j U ( θ + ) = ( j U ( θ ) + ) Ult(V ,U ) , which again contradicts theassumption that λ is a cardinal in Ult(V , U ). Let κ < θ be the minimal cardinalwith the property that κ δ ≥ θ holds. Then the minimality of κ implies that µ δ < κ holds for all µ < κ .First, assume that κ = 2. Then δ < θ ≤ δ = δ + and therefore θ = δ + .Since Lemma 4.1 implies that j U ( δ ++ ) = δ ++ , we know that j U ( δ ++ ) = δ ++ = θ + > λ and, as above, we can conclude that λ is not contained in the interval( j U ( δ + ) , δ ++ ). Moreover, since our assumptions on λ directly imply that λ is notcontained in the interval ( j U ( δ ) , j U ( δ + )), we can conclude that λ is an element ofthe set ( δ + , j U ( δ )] ∪ { j U ( δ + ) } in this case.Next, assume that κ > δ . Then our cardinal arithmetic assumptions and theminimality of κ imply that cof( κ ) ≤ δ and θ = κ + . But then we already know thatcof( κ ) = δ , because otherwise we could apply Lemma 4.1 to conclude that j U ( θ ) = LOSURE PROPERTIES OF MEASURABLE ULTRAPOWERS 17 θ < λ < θ + = j U ( θ + ). Since our assumptions imply that ( κ + ) δ = κ + , Lemma 4.1implies that j U ( κ ++ ) = κ ++ = θ + and this shows that λ is not contained in theinterval ( j U ( κ + ) , κ ++ ). Since λ is also not contained in the interval ( j U ( κ ) , j U ( κ + )),we can conclude that λ is contained in the set ( κ + , j U ( κ )] ∪ { j U ( κ + ) } . (cid:3) First, assume that κ = δ holds. By our assumptions, Lemma 4.1 shows that δ ++ = j U ( δ ++ ) > j U ( δ + ). Since we know that δ + < λ ≤ j U ( δ + ) and cof( λ ) > δ ,this implies that cof( λ ) = δ + , and hence we can use Proposition 1.3 to find anunbounded subset of λ that is fresh over Ult(V , U ).Next, assume that κ > δ and λ = j U ( κ ). Then δ < cof( λ ) ≤ cof( λ ) Ult(V ,U ) = j U (cof( κ )) = j U ( δ ) < δ ++ and we can conclude that cof( λ ) = δ + . Another application of Proposition 1.3 nowyields the desired subset of λ .Now, assume that κ > δ and λ = j U ( κ + ). Then our assumptions ensure theexistence a (cid:3) ( κ + )-sequence and therefore we can apply Theorem 3.2 to find anunbounded subset of λ that is fresh over Ult(V , U ).Finally, we assume that κ > δ and κ + < λ < j U ( κ ). Then we know that µ δ < κ holds for all µ < κ . Subclaim. cof( λ ) = κ + .Proof of the Subclaim. Assume, towards a contradiction, that cof( λ ) = κ + . Since κ is singular and cof( λ ) < λ < j U ( κ ) < κ ++ , this implies that cof( λ ) < κ . In thissituation, we can repeat an argument from the first part of the proof of Theorem4.3 to find a monotone enumeration h κ ξ | ξ < δ i of a closed unbounded subset of κ of order-type δ such that κ > cof( λ ), [ ξ κ ξ ] U = κ and [ ξ κ + ξ ] U = κ + .Fix a function f with domain δ such that [ f ] U = λ holds and f ( ξ ) is a regularcardinal in the interval ( κ + ξ , κ ) for all ξ < δ . Pick a sequence h f α | α < cof( λ ) i of functions with domain δ such that f α ( ξ ) < f ( ξ ) holds for all α < cof( λ ) andall ξ < δ , and the induced sequence h [ f α ] U | α < cof( λ ) i is strictly increasing andcofinal in λ . Given ξ < δ , the fact that f ( ξ ) is a regular cardinal greater thancof( λ ) then yields an ordinal γ ξ < f ( ξ ) with f α ( ξ ) < γ ξ for all α < cof( λ ). Butthen [ f α ] U < [ ξ γ ξ ] U < λ for all α < cof( λ ), a contradiction. (cid:3) By the above computations, we now know that κ is a singular cardinal of cofi-nality δ with the property that µ δ < κ holds for all µ < κ , and λ is a limit ordinalof cofinality κ + with κ + < λ < j U ( κ ) that is a regular cardinal in Ult(V , U ). Sinceour assumptions guarantee the existence of a (cid:3) κ -sequence, we can use Theorem 5.2to show that there also exists an unbounded subset of λ that is fresh over Ult(V , U )in this case. (cid:3) To conclude the proof of the theorem, fix a limit ordinal λ with the propertythat the cardinal cof( λ ) is greater than δ and not weakly compact. Set λ =cof( λ ) Ult(V ,U ) . By [10, Lemma 3.7.(ii)], we then have cof( λ ) = cof( λ ). Hence, wecan use the previous claim to find an unbounded subset of λ that is fresh overUlt(V , U ). Using Proposition 5.1, we can conclude that there is an unboundedsubset of λ that is fresh over Ult(V , U ). (cid:3) We end this section by using famous results of Schimmerling and Zeman to showthat, in canonical inner models, the assumptions of Theorem 1.6 are satisfied forall measurable cardinals.
Proof of Theorem 1.7.
We argue that Jensen-style extender models without sub-compact cardinals satisfy the statements (a), (b) and (c) listed in Theorem 1.6.First, notice that the GCH holds in all of these models and hence statement (a)is satisfied. Next, recall that [25, Theorem 15] shows that, in Jensen-style exten-der models, a (cid:3) ν -sequence exists if and only if ν is not a subcompact cardinal.In particular, we know that, in Jensen-style extender models without subcompactcardinals, (cid:3) ( ν + )-sequences exist for all infinite cardinals ν . Since [29, Theorem0.1] yields the existence of (cid:3) ( κ )-sequences for inaccessible cardinals κ in the rele-vant models, we can conclude that statement (b) holds in these models. Finally,the validity of statement (c) in Jensen-style extender models without subcompactcardinals again follows from [25, Theorem 15]. (cid:3) Consistency strength
We end this paper by establishing the equiconsistency stated in Theorem 1.10.We start by showing that the existence of a weakly compact cardinal above a mea-surable cardinal is a lower bound for the consistency of the corresponding statement.
Theorem 7.1.
Assume that there is no inner model with a weakly compact cardinalabove a measurable cardinal. If U is a normal ultrafilter on a measurable cardinal δ , then there is an unbounded subset of δ ++ that is fresh over Ult(V , U ) .Proof. By our assumptions, we can use the results of [6] to show that 2 δ = δ + holds.Set κ = δ ++ . Then our assumptions imply that κ is not weakly compact in L[ U ]. Inthis situation, we can construct a tail of a (cid:3) ( κ )-sequence h C ν | ξ < ν < κ, ν ∈ Lim i in L[ U ] above some ordinal ξ > δ + with ξ < κ , using the argument in [11, Section6] for L. A consequence of this proof, published by Todorˇcevi´c in [28, 1.10], butprobably first noticed by Jensen (see [23, Theorem 2.5] for a modern account), isthat the sequence h C ν | ξ < ν < κ, ν ∈ Lim i remains a tail of a (cid:3) ( κ )-sequence inV. We can now easily extend this sequence to a (cid:3) ( κ )-sequence h C ν | ν ∈ Lim ∩ κ i in V. Since 2 δ = δ + holds, Lemma 4.1 shows that j U ( κ ) = κ and hence we can useTheorem 3.2 to find an unbounded subset of κ that is fresh over Ult(V , U ). (cid:3) We now use forcing to show that the above large cardinal assumption is also anupper bound for the consistency strength of the non-existence of fresh subsets at thedouble successor of a measurable cardinal. The following lemma is a reformulationand slight strengthening of [21, Lemma 3.5]. The notion of λ -strategically closedpartial orders and the corresponding game G λ ( P ) are introduced in [3, Definition5.15]. Lemma 7.2.
Let U be a normal ultrafilter on a measurable cardinal δ , let λ be alimit ordinal with cof( λ ) > δ and let A be an unbounded subset of λ that is freshover Ult(V , U ) . If P is a ( δ + 1) -strategically closed partial order, then P (cid:13) “ ˇ A / ∈ Ult(V , ˇ U ) ” . Proof.
Assume, towards a contradiction, that there is a condition p in P and a P -name ˙ f for a function with domain δ with the property that, whenever G is P -genericover V with p ∈ G , then [ ˙ f G ] U = A holds in V[ G ]. As P is ( δ + 1)-strategicallyclosed, there is a condition p in P below p and a subset X of δ with the propertythat, whenever G is P -generic over V with p ∈ G , then X = { ξ < δ | ˙ f G ( ξ ) ∈ V } . Schimmerling’s and Zeman’s notion of
Jensen core model in [25] agrees with our notion ofJensen-style extender model.
LOSURE PROPERTIES OF MEASURABLE ULTRAPOWERS 19
Claim. If ξ ∈ δ \ X and q ≤ P p , then there is γ < λ and conditions r and r in P below q such that r (cid:13) P “ ˇ γ ∈ ˙ f ( ˇ ξ ) ” and r (cid:13) P “ ˇ γ / ∈ ˙ f ( ˇ ξ ) ”.Proof of the Claim. If such a pair of conditions does not exist, then it is easy tocheck that the condition q forces ˙ f ( ˇ ξ ) to be equal to the set { γ < λ | ∃ r ≤ P q r (cid:13) P “ ˇ γ ∈ ˙ f ( ˇ ξ ) ” } , contradicting our assumption that ξ is not an element of X . (cid:3) Claim. X ∈ U .Proof of the Claim. Assume, towards a contradiction, that X is not an element of U . Fix a winning strategy σ for Player Even in the game G δ +1 ( P ), some sufficientlylarge regular cardinal θ and an elementary submodel M of H( θ ) of cardinality δ satisfying ( δ + 1) ∪ { λ, σ, ˙ f, p , A, U, X, P } ⊆ M and <δ M ⊆ M . We define η = sup( λ ∩ M ) < λ and fix a function h with domain δ such that [ h ] U = A ∩ η .Note that, given a partial run of G δ +1 ( P ) of even length less than δ that consistsof conditions in M and was played according to σ by Player Even , the given sequenceis an element of M and Player Even responds to it with a move in M . Therefore, if τ is a strategy for Player Odd in G δ +1 ( P ) that answers to sequences of conditions in M by playing a condition in M and h p ξ | ξ ≤ δ i is a run of G δ +1 ( P ) played accordingto σ and τ , then p ξ ∈ M for all ξ < δ . Moreover, the previous claim allows us touse elementarity to show for every ξ ∈ δ \ X and every condition q ∈ M ∩ P with q ≤ P p , there is γ ∈ M ∩ λ and a condition r ∈ M ∩ P with r ≤ P p and(1) γ ∈ h ( ξ ) ⇐⇒ r (cid:13) P “ ˇ γ / ∈ ˙ f ( ˇ ξ ) ” ⇐⇒ ¬ ( r (cid:13) P “ ˇ γ ∈ ˙ f ( ˇ ξ ) ”) . Now, pick a strategy τ for Player Odd in G δ +1 ( P ) with the following properties: • τ plays the condition p in move 1. • Given ξ ∈ X , if Player Even played a condition q ∈ M ∩ P in move (2+2 · ξ ),then τ responds by also playing the condition q in the next move. • Given ξ ∈ δ \ X , if Player Even played a condition q ∈ M ∩ P in move(2 + 2 · ξ ), then τ responds by playing a condition r ∈ M ∩ P with r ≤ P q such that the equivalences of (1) hold true for some γ ∈ M ∩ λ .Let h p ξ | ξ ≤ δ i be the run of G δ +1 ( P ) played according to σ and τ . By the aboveremarks, we then have p ξ ∈ M for all ξ < δ . In particular, for every ξ ∈ δ \ X ,there exists γ ξ < λ with γ ξ ∈ h ( ξ ) ⇐⇒ p δ (cid:13) P “ ˇ γ ξ / ∈ ˙ f ( ˇ ξ ) ” ⇐⇒ ¬ ( p δ (cid:13) P “ ˇ γ ξ ∈ ˙ f ( ˇ ξ ) ”) . Let G be P -generic over V with p δ ∈ G . Then the closure properties of P implythat [ h ] U = A ∩ η holds in V[ G ]. Since A = [ ˙ f G ] U holds in V[ G ], we know that theset Y = { ξ < δ | h ( ξ ) is an initial segment of ˙ f G ( ξ ) } is an element of U . But then there is some ξ ∈ Y \ X = Y ∩ ( δ \ X ) and ourconstruction ensures that the ordinal γ ξ is contained in the symmetric difference of h ( ξ ) and ˙ f G ( ξ ), a contradiction. (cid:3) Now, let G be P -generic over V with p ∈ G . By the previous claim and theclosure properties of P , we can find a function f with domain δ in V such that[ f ] U = [ ˙ f G ] U = A holds in V[ G ]. Since forcing with P adds no new functions from δ to the ordinals, we can conclude that [ f ] U = A also holds in V, a contradictionas A was chosen to be fresh over Ult(V , U ). (cid:3) The previous lemma now allows us to prove the following results that can beused to complete the proof of Theorem 1.10 by considering the case µ = δ + . Theorem 7.3.
Let U be a normal ultrafilter on a measurable cardinal δ , let µ > δ be a regular cardinal, let W be an inner model containing U and let κ > µ be weaklycompact in W . If V is a Col( µ, <κ ) W -generic extension of W , then no unboundedsubset of κ is fresh over Ult(V , U ) .Proof. Assume, towards a contradiction, that there is an unbounded subset A of κ that is fresh over Ult(V , U ). Note that, in V, our assumptions imply that µ δ = µ and hence Lemma 4.1 implies that j U ( κ ) = κ . In particular, for every γ < κ ,there is a function f ∈ H( κ ) with domain δ and [ f ] U = γ . By our assumptions,there exists G Col( µ, <κ ) W -generic over W with V = W[ G ] and hence we knowthat <µ W ⊆ W. Moreover, since Col( µ, <κ ) satisfies the κ -chain condition inW, there exist Col( µ, <κ )-nice names ˙ A and ˙ F in W such that ˙ A G = A and˙ F G is a function with domain κ and the property that for all γ < κ , the set˙ F G ( γ ) : δ −→ H( κ ) ∩ P ( κ ) is a function with [ ˙ F G ( γ )] U = A ∩ γ .Work in W and pick an elementary submodel M of H( κ + ) of cardinality κ such that <κ M ⊆ M and ( κ + 1) ∪ { ˙ A, ˙ F, U } ⊆ M . In this situation, the weakcompactness of κ yields a transitive set N with <κ N ⊆ N and an elementaryembedding j : M −→ N with critical point κ (see [9, Theorem 1.3]).Now, let H be Col( µ, [ κ, j ( κ )))-generic over V. Then there is H ∈ V[ H ] thatis Col( µ, We end this paper by stating two questions raised by the above results.Our first question is motivated by the fact that, in contrast to the proof of The-orem 5.2, our proof of Theorem 4.3 heavily makes use of the assumption that theGCH holds at the given singular cardinal. Therefore, it is not possible to use The-orem 4.3 to derive additional consistency strength from the existence of a normalultrafilter U on a measurable cardinal δ and a singular cardinal κ of cofinality δ with the property that no unbounded subset of κ + is fresh over Ult(V , U ), becausethe existence of a cardinal δ < µ < κ with 2 µ > κ + might prevent us from applyingTheorem 4.3, and this constellation can be realized by forcing over a model con-taining a measurable cardinal. In contrast, if it were possible to remove the GCHassumption from Theorem 4.3, then this would show that the above hypothesisimplies that at least one of the following statements holds true: • The GCH fails at a measurable cardinal. LOSURE PROPERTIES OF MEASURABLE ULTRAPOWERS 21 • The SCH fails. • There exists a countably closed singular cardinal κ with the property thatthere are no (cid:3) κ -sequences.Note that a combination of the main result of [6], [7, Theorem 1.4] and [24, Corollary6] shows that the disjunction of the above statements implies the existence of ameasurable cardinal κ with o ( κ ) = κ ++ in an inner model. These considerationsmotivate the following question: Question 8.1. Let U be a normal ultrafilter on a measurable cardinal δ and let κ be a singular cardinal of cofinality δ such that λ δ < κ holds for all λ < κ . Assumethat there exists a (cid:3) κ -sequence. Is there an unbounded subset of κ + that is freshover Ult(V , U )?Our second question addresses the fact that, in the models of set theory studiedin Theorems 1.4 and 1.7, the existence of fresh subsets only depends on the cor-responding measurable cardinal and the cofinality of the given limit ordinal, butnot on the specific normal ultrafilter used in the construction of the ultrapower.Therefore, it is natural to ask whether this is always the case. Question 8.2. Is it consistent there there exist normal ultrafilters U and U ona measurable cardinal δ such that there is a limit ordinal λ with the property thatno unbounded subset of λ is fresh over Ult(V , U ) and there exists an unboundedsubset of λ that is fresh over Ult(V , U )? References 1. 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