Closure properties of measurable ultrapowers

We study closure properties of measurable ultrapowers with respect to Hamkin's notion of"freshness"and show that the extent of these properties highly depends on the combinatorial properties of the underlying model of set theory. In one direction, a result of Sakai shows that, by collapsing a strongly compact cardinal to become the double successor of a measurable cardinal, it is possible to obtain a model of set theory in which such ultrapowers possess the strongest possible closure properties. In the other direction, we use various square principles to show that measurable ultrapowers of canonical inner models only possess the minimal amount of closure properties. In addition, the techniques developed in the proofs of these results also allow us to derive statements about the consistency strength of the existence of measurable ultrapowers with non-minimal closure properties.


Introduction
The present paper studies the structural properties of ultrapowers of models of set theory constructed with the help of normal ultrafilters on measurable cardinals. Two of the most fundamental properties of these ultrapowers are that these models do not contain the ultrafilter utilized in their construction and that they are closed under sequences of length equal to the relevant measurable cardinal. In the following, we want to further analyze the closure and non-closure properties of measurable 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). 1 Given a normal ultrafilter U on a measurable cardinal, we let Ult(V, U ) denote the (transitive collapse of the) induced ultrapower and we let j U : V −→ Ult(V, U ) denote the corresponding elementary embedding. For notational simplicity, we confuse Ult(V, U ) and its elements with their transitive collapses. In this paper, for a given normal ultrafilter U , we aim to determine the class of limit ordinals containing an unbounded subset that is fresh over the ultrapower Ult(V, U ). For the images of regular cardinals under the embedding j U , this question was already studied by Shani in [26]. Moreover, Sakai investigated closure properties of measurable ultrapowers that imply the non-existence of unbounded fresh subsets at many ordinals in [21].
The following proposition lists the obvious closure properties of measurable ultrapowers with respect to the non-existence of fresh subsets. Note that the second part of the second statement also follows directly from [21,Corollary 3.3]. The proof 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.
In the other direction, the fact that normal ultrafilters are not contained in the corresponding ultrapowers directly yields the following non-closure properties of these 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 an unbounded 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 unbounded subset of λ that is fresh over Ult(V, U ). (iii) If 2 δ = δ + holds and λ is a limit ordinal with cof(λ) = δ + , then there is an unbounded subset of λ that is fresh over Ult(V, U ).
In the following, we will present results that show that the above propositions already cover all provable closure and non-closure properties of measurable ultrapowers, in the sense that there are models of set theory in which fresh subsets exist at all limit ordinals that are not ruled out by Proposition 1.2 and models in which fresh subsets only exist at limit ordinals, where their existence is guaranteed by Proposition 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 with minimal non-closure properties can be constructed by collapsing a strongly compact cardinal to become the double successor of a measurable cardinal. Note that we phrase the following result in a non-standard way to clearly distinguish between the ground model of the forcing extension and the model used in the corresponding ultrapower construction. Theorem 1.4. Let δ be a measurable cardinal and let W be an inner model such that 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, 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 ).
Proof. For one direction, our assumptions directly imply that the GCH holds and therefore 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 other direction, 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 ). Then Proposition 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 that cof(λ) ≥ δ ++ > |a| Ult(V,U) ≥ |A ∩ a| implies that A ∩ a is bounded in λ and hence the assumption that A is fresh over Ult(V, U ) allows us to conclude that A ∩ a is an element of Ult(V, U ), a contradiction.
For the other direction, we will prove results that show that canonical inner models provide measurable ultrapowers with closure properties that are minimal in the above sense. These arguments make use of the validity of various combinatorial principles in these models. In particular, they heavily rely on the existence of suitable square sequences. For specific types of cardinals, similar constructions have already been done in [21,Section 3.3] and [26, Section 3]. Definition 1.5.
(i) Given an uncountable regular cardinal κ, a sequence C γ | γ ∈ Lim ∩ κ is a (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 (κ + )-sequence C γ | γ ∈ Lim ∩ κ + is a 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 for measurable ultrapowers exist at all limit ordinals that are not ruled out by the conclusions of Proposition 1.  [29], the above result will allow us to show that measurable ultrapowers of a large class of canonical inner models, so called Jensen-style extender models, possess the minimal amount of closure properties with respect to freshness. These inner models go back to Jensen in [12], following a suggestion of S. Friedman, and can have various large cardinals below supercompact cardinals. As for example in [29], we demand that they satisfy classical consequences of iterability such as solidity and condensation. The theorem below holds for Mitchell-Steel extender models with the same properties constructed as in [20] as well. But it turned out that Jensen-style constructions are more natural in the proof of -principles in canonical inner models, so this is what Schimmerling and 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 have a subcompact cardinal. Then the statements (i) and (ii) listed in Theorem 1.6 are equivalent for every normal ultrafilter U on a measurable cardinal δ and every limit ordinal λ.
We restrict ourselves to inner models without subcompact cardinals in the statement of Theorem 1.7, as the non-existence of κ -sequences in Jensen-style extender models is equivalent to κ being subcompact (see [25]). Results of Kypriotakis and Zeman in [15] show that (κ + )-sequences can exists even if κ is subcompact, but we decided to not discuss this further here.
The techniques developed in the proof of Theorem 1.6 also allows us to derive large 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 + } with the property that there exists a (θ)-sequence, then there exists a limit ordinal 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 2 holds at κ and there exists a κ -sequence, then there exists 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 the statements (i) and (ii) listed in 1.4 are equivalent for every limit ordinal λ, then the first part of the above theorem shows that κ = δ ++ is a countably closed 3 regular cardinal that is greater than max{2 ℵ0 , ℵ 3 } and has the property that there are no (κ)-and no κ -sequences. By [23,Theorem 5.6], the existence of such a cardinal implies Projective Determinacy. In addition, [13, Theorem 0.1] derives the existence of a sharp for a proper class model with a proper class of strong cardinals and a proper 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 limit cardinal κ with the property that there are no κ -sequences implies that AD holds in L(R), and even stronger consequences of this assumptions are given by the results of [22]. Finally, note that work of Gitik and Mitchell (see [7] and [19]) shows that a failure of SCH at a singular strong limit cardinal κ of uncountable cofinality implies that 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 strength of the existence of a measurable ultrapower that has the property that no unbounded subsets of the double successor of the corresponding measurable cardinal 2 Remember that the SCH states that κ cof(κ) = κ + holds for every singular cardinal κ with 2 cof(κ) < κ.

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 directly 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 γ α | α < κ 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 its transitive collapse). Let c : [κ] 2 −→ U denote the unique function with the property that } holds for all α < β < κ. In this situation, since κ > δ is weakly compact, we know that |U | = 2 δ < κ and hence the weak compactness of κ yields an unbounded subset H of κ and an element X of U with the property that c[H] 2 = {X}. Pick a function g with domain δ and the property that g(ξ) = {g α (ξ) | α ∈ H} holds for 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 fresh over Ult(V, U ).
Pick a function g with domain δ and is a cofinal subset of λ of order-type δ + . In particular, the closure of Ult(V, U ) under δ-sequences implies that 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 (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 γ α | α < δ + be the monotone enumeration of 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 of order-type δ + . Then the closure of Ult(V, U ) under δ-sequences implies that A is fresh over Ult(V, U ).
Note that, in the situation of Proposition 1.3, we have cof(λ) = δ + for every limit ordinal λ with cof(λ) Ult(V,U) = j U (δ + ). In particular, if κ is a strong limit cardinal of cofinality δ + , then the fact that j U (κ) = κ holds allows us to use the second part of the above proposition to conclude that there is an unbounded subset of κ that is fresh over Ult(V, U ). Moreover, the results of Cummings in [1] discussed in the first section show that the cardinal arithmetic assumption in the third part of the proposition can, in general, not be omitted.

Fresh subsets at image points of ultrapower embeddings
In this section, we will use a modified square principle introduced in [16] to show that the existence of a (κ)-sequence allows us to construct a fresh subset of j U (κ). The principle defined in the next definition is a variation of the indexed square principles studied in [4] and [5].
satisfying the following statements: The main result of [17] now shows that for all infinite regular cardinals δ < κ, the existence of a (κ)-sequence implies the existence of a ind (κ, δ)-sequence. The proof 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 (κ)-sequence, then there is a closed unbounded subset of j U (κ) that is fresh over Ult(V, U ).

Fresh subsets of successors of singular cardinals
We now aim to construct fresh subsets of cardinals that are not contained in the image of the corresponding ultrapower embedding, e.g. successors of singular cardinals whose cofinality is equal to the relevant measurable cardinal. Our arguments will rely on two standard observations about measurable ultrapowers and κ -sequences that we present first. A proof of the following lemma is contained in the 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 The next lemma contains a well-known construction (see [2, Section 4]) that shows that, in the situations relevant for our proofs, the existence of some κsequence already implies the existence of such a sequence with certain additional structural properties.
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( This completes the proof of the lemma. We are now ready to prove the main result of this section that will allow us to handle successors of singular cardinals of measurable cofinality in the proof of Theorem 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 2 κ = κ + and the property that λ δ < κ holds for all λ < κ. If there exists a κ -sequence, then there is a closed unbounded subset of κ + that is fresh over Ult(V, U ).
Proof. By our assumptions, we can apply Lemma 4.2 to obtain a κ -sequence C γ | γ ∈ Lim ∩ κ + and a stationary subset E of S κ + δ such that otp (C γ ) < κ and 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 the monotone enumeration κ ξ | ξ < δ of a closed unbounded subset of κ of ordertype δ with the property that j U (κ ξ ) = κ ξ holds for all ξ < δ. In this situation, In the following, we inductively construct a sequence The idea behind this construction is that these functions represent a cofinal subset of κ + and thereby in particular witness that [ξ → κ + ξ ] U = κ + . We identify each f γ ∈ ξ<δ κ + ξ with a function with domain δ in the obvious way and define: Proof of the Claim. (i) We prove the statement by induction on 0 < γ < κ + , where the successor step follows trivially from our induction hypothesis. Now, assume that γ ∈ Lim ∩ κ + with Lim(C γ ) bounded in γ. Since δ is an uncountable regular cardinal, our induction hypothesis allows us to find ζ < δ with the property that f β0 (ξ) < f β1 (ξ) holds for all β 0 , β 1 ∈ C γ \ max(Lim(C γ )) with β 0 < β 1 and all ζ ≤ ξ < δ. By definition, we now have ) is a cofinal subset of γ, the desired statement 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.
(iii) This statement is a direct consequence of the definition of the sequence f γ | γ < κ + and statement (ii).
Note that the first part of the above claim in particular shows that we have Next, notice that the fact that 2 κ = κ + holds allows us to fix an enumeration h α | α < κ + of ξ<δ P(κ + ξ ) of order-type κ + . In addition, let γ α | α < κ + denote the monotone enumeration of E. We now inductively define a sequence c γ | γ ∈ Lim ∩ κ + 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.
This completes the proof of the theorem.

Regular cardinals in Ult(V, U )
We now turn to the construction of fresh subsets of limit ordinals that are not cardinals in V. We first observe that we can restrict ourselves to ordinals that are regular cardinals in the corresponding ultrapower. Proof. Set λ 0 = cof(λ) Ult(V,U) . Let A be an unbounded subset of λ 0 that is fresh over Ult(V, U ) and let γ η | η < λ 0 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 unbounded in λ and fresh over Ult(V, U ).
In the proof of the following result, we modify techniques from the proof of Theorem 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 be a 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 cardinal in Ult(V, U ). If there is a κ -sequence, then there is an unbounded subset of λ that is fresh over Ult(V, U ).
Proof. As in the proof of Theorem 4.3, we can apply Lemma 4.1 to find the monotone enumeration κ ξ | ξ < δ 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 implies that [ξ → κ ξ ] U = κ and we can repeat arguments from the first part of the proof of Theorem 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 interval (κ + ξ , κ) for all ξ < δ. Fix a sequence h γ ∈ ξ<δ h(ξ) | γ < κ + such that the sequence [h γ ] U | γ < κ + is strictly increasing and cofinal in λ.
In particular, this shows that the sequence [f γ ] U | γ < κ + is strictly increasing. Since the above definition ensures that [h γ ] U < [f γ+1 ] U holds for all γ < κ + , we also know that this sequence is cofinal in λ.
Given γ ∈ Lim ∩ κ + , the properties listed above ensure that [c γ ] U is a closed unbounded 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 a closed unbounded subset C of λ with C ∩ [f γ ] U = [c γ ] U for all γ ∈ Lim ∩ κ + and this property directly implies that otp (C) = κ + < λ. Since λ is a regular cardinal 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 ).
We end this section by using the above results to show that the validity of the equivalence 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 (θ)-sequence. The regularity of θ then implies that j U [θ] is cofinal in j U (θ) and hence cof(j U (θ)) = θ. Using Theorem 3.2, we now find an unbounded subset of j U (θ) that is fresh over Ult(V, U ).
Since κ is a strong limit cardinal, the above claim now allows us to apply Theorem 5.2 to find an unbounded subset of λ that is fresh over Ult(V, U ).

Ultrapowers of canonical inner models
With the help of the results of the previous sections, we are now ready to prove the main result of this paper.
Proof of Theorem 1.6. Fix a normal ultrafilter U on a measurable cardinal δ that satisfies the three assumptions listed in the statement of the theorem. By Proposition 1.2, if λ is a limit ordinal with the property that the cardinal cof(λ) is either smaller than δ + or weakly compact, then no unbounded subset of λ is fresh over Ult(V, U ). In the proof of the converse implication, we first consider two special cases.
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 ).
First, assume that either cof(ν) > δ and κ = ν, or cof(ν) < δ and κ = ν + . Then Lemma 4.1 shows that j U (κ) = κ holds in both cases. Moreover, since cof(κ) is a regular cardinal greater than δ + and the fact that cof(κ) is not weakly compact allows us to use Theorem 3.2 to find an unbounded 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 fresh over Ult(V, U ).
Finally, assume that cof(ν) = δ and κ = ν + . In this situation, we know that ν is a singular cardinal of cofinality δ with 2 ν = ν + and the property that µ δ < ν holds for all µ < ν. Since the assumptions of the theorem guarantee the existence of a ν -sequence, we can apply Theorem 4.3 to find an unbounded subset of κ that is fresh over Ult(V, U ).
Claim. Let λ be a limit ordinal with the property that the cardinal cof(λ) is greater than δ and not weakly compact. If λ is a regular cardinal in Ult(V, U ), then there is 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 to directly derive the desired conclusion. Hence, we may assume that λ is not a cardinal.
Moreover, we have cof(θ) = δ, because otherwise θ would be a singular strong limit cardinal of cofinality δ and our assumptions would allow us to repeat the argument from the first part of the proof of Theorem 4.3 to show that θ + = (θ + ) Ult(V,U) , contradict our assumption that λ is a cardinal in Ult(V, U ). In addition, we know that there is some ν < θ satisfying ν δ ≥ θ, because otherwise Lemma 4.1 would imply that j U (θ) = θ < λ < θ + = j U (θ + ) = (j U (θ) + ) Ult(V,U) , which again contradicts the assumption that λ is a cardinal in Ult(V, U ). Let κ < θ be the minimal cardinal with the property that κ δ ≥ θ holds. Then the minimality of κ implies that µ δ < κ holds for all µ < κ.
Next, assume that κ > δ and λ = j U (κ). Then and we can conclude that cof(λ) = δ + . Another application of Proposition 1.3 now yields the desired subset of λ. Now, assume that κ > δ and λ = j U (κ + ). Then our assumptions ensure the existence a (κ + )-sequence and therefore we can apply Theorem 3.2 to find an unbounded subset of λ that is fresh over Ult(V, U ).
By the above computations, we now know that κ is a singular cardinal of cofinality δ with the property that µ δ < κ holds for all µ < κ, and λ is a limit ordinal of cofinality κ + with κ + < λ < j U (κ) that is a regular cardinal in Ult(V, U ). Since our assumptions guarantee the existence of a κ -sequence, we can use Theorem 5.2 to show that there also exists an unbounded subset of λ that is fresh over Ult(V, U ) in this case.
To conclude the proof of the theorem, fix a limit ordinal λ with the property that the cardinal cof(λ) is greater than δ and not weakly compact. Set λ 0 = cof(λ) Ult(V,U) . By [10, Lemma 3.7.(ii)], we then have cof(λ 0 ) = cof(λ). Hence, we can use the previous claim to find an unbounded subset of λ 0 that is fresh over Ult(V, U ). Using Proposition 5.1, we can conclude that there is an unbounded subset of λ that is fresh over Ult(V, U ).
We end this section by using famous results of Schimmerling and Zeman to show that, in canonical inner models, the assumptions of Theorem 1.6 are satisfied for all measurable cardinals.
Proof of Theorem 1.7. We argue that Jensen-style extender models without subcompact 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] 4 shows that, in Jensen-style extender models, a ν -sequence exists if and only if ν is not a subcompact cardinal. In particular, we know that, in Jensen-style extender models without subcompact cardinals, (ν + )-sequences exist for all infinite cardinals ν. Since [29, Theorem 0.1] yields the existence of (κ)-sequences for inaccessible cardinals κ in the relevant models, we can conclude that statement (b) holds in these models. Finally, the validity of statement (c) in Jensen-style extender models without subcompact cardinals again follows from [25, Theorem 15].

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 measurable 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 cardinal above 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 ]. In this situation, we can construct a tail of a (κ)-sequence C ν | ξ < ν < κ, ν ∈ Lim in L[U ] above some ordinal ξ > δ + with ξ < κ, using the argument in [11,Section 6] for L. A consequence of this proof, published by Todorčević in [28, 1.10], but probably first noticed by Jensen (see [23, Theorem 2.5] for a modern account), is that the sequence C ν | ξ < ν < κ, ν ∈ Lim remains a tail of a (κ)-sequence in V. We can now easily extend this sequence to a (κ)-sequence C ν | ν ∈ Lim ∩ κ in V. Since 2 δ = δ + holds, Lemma 4.1 shows that j U (κ) = κ and hence we can use Theorem 3.2 to find an unbounded subset of κ that is fresh over Ult(V, U ).
We now use forcing to show that the above large cardinal assumption is also an upper bound for the consistency strength of the non-existence of fresh subsets at the double successor of a measurable cardinal. The following lemma is a reformulation and slight strengthening of [21,Lemma 3.5]. The notion of λ-strategically closed partial orders and the corresponding game G λ (P) are introduced in [3, Definition 5.15].
Lemma 7.2. Let U be a normal ultrafilter on a measurable cardinal δ, let λ be a limit ordinal with cof(λ) > δ and let A be an unbounded subset of λ that is fresh over Ult(V, U ). If P is a (δ + 1)-strategically closed partial order, then Proof. Assume, towards a contradiction, that there is a condition p in P and a Pnameḟ for a function with domain δ with the property that, whenever G is P-generic over V with p ∈ G, then [ḟ G ] U = A holds in V[G]. As P is (δ + 1)-strategically closed, there is a condition p 1 in P below p and a subset X of δ with the property that, whenever G is P-generic over V with p 1 ∈ G, then X = {ξ < δ |ḟ G (ξ) ∈ V}.
Proof of the Claim. If such a pair of conditions does not exist, then it is easy to check that the condition q forcesḟ (ξ) to be equal to the set {γ < λ | ∃r ≤ P q r P "γ ∈ḟ (ξ) "}, contradicting our assumption that ξ is not an element of X.
Proof of the Claim. Assume, towards a contradiction, that X is not an element of U . Note that, given a partial run of G δ+1 (P) of even length less than δ that consists of conditions in M and was played according to σ by Player Even, the given sequence is 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 p ξ | ξ ≤ δ is a run of G δ+1 (P) played according to σ and τ , then p ξ ∈ M for all ξ < δ. Moreover, the previous claim allows us to use elementarity to show for every ξ ∈ δ \ X and every condition q ∈ M ∩ P with q ≤ P p 1 , there is γ ∈ M ∩ λ and a condition r ∈ M ∩ P with r ≤ P p and (1) γ ∈ h(ξ) ⇐⇒ r P "γ / ∈ḟ (ξ) " ⇐⇒ ¬(r P "γ ∈ḟ (ξ) "). Now, pick a strategy τ for Player Odd in G δ+1 (P) with the following properties: • τ plays the condition p 1 in move 1.
Let G be P-generic over V with p δ ∈ G. Then the closure properties of P imply is an initial segment ofḟ G (ξ)} is an element of U . But then there is some ξ ∈ Y \ X = Y ∩ (δ \ X) and our construction ensures that the ordinal γ ξ is contained in the symmetric difference of h(ξ) andḟ G (ξ), a contradiction. Now, let G be P-generic over V with p 1 ∈ G. By the previous claim and the closure properties of P, we can find a function f with domain δ in V such that [f ] U = [ḟ 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 contradiction as A was chosen to be fresh over Ult(V, U ).
The previous lemma now allows us to prove the following results that can be used 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 weakly compact in W. If V is a Col(µ, <κ) W -generic extension of W, then no unbounded subset 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 know that <µ W ⊆ W. Moreover, since Col(µ, <κ) satisfies the κ-chain condition in W, there exist Col(µ, <κ)-nice namesȦ andḞ in W such thatȦ G = A anḋ F G is a function with domain κ and the property that for all γ < κ, the seṫ So, given γ < κ, elementarity implies that the set {ξ < δ |Ḟ G (γ)(ξ) is an initial segment of f (ξ)} is an element of U since j * (Ḟ G (γ)) =Ḟ G (γ). But this implies that A is an initial segment of [f ] U in V[H 0 ] and hence A is not fresh over Ult(V, U ) in V[H 0 ], contradicting Lemma 7.2.

Open Questions
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 Theorem 5.2, our proof of Theorem 4.3 heavily makes use of the assumption that the GCH holds at the given singular cardinal. Therefore, it is not possible to use Theorem 4.3 to derive additional consistency strength from the existence of a normal ultrafilter 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 ), because the existence of a cardinal δ < µ < κ with 2 µ > κ + might prevent us from applying Theorem 4.3, and this constellation can be realized by forcing over a model containing a measurable cardinal. In contrast, if it were possible to remove the GCH assumption from Theorem 4.3, then this would show that the above hypothesis implies that at least one of the following statements holds true: • The GCH fails at a measurable cardinal.
• The SCH fails.
• There exists a countably closed singular cardinal κ with the property that there are no κ -sequences.
Note that a combination of the main result of [6], [7, Theorem 1.4] and [24,Corollary 6] shows that the disjunction of the above statements implies the existence of a measurable cardinal κ with o(κ) = κ ++ in an inner model. These considerations motivate 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. Assume that there exists a κ -sequence. Is there an unbounded subset of κ + that is fresh over Ult(V, U )?
Our second question addresses the fact that, in the models of set theory studied in Theorems 1.4 and 1.7, the existence of fresh subsets only depends on the corresponding measurable cardinal and the cofinality of the given limit ordinal, but not 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 0 and U 1 on a measurable cardinal δ such that there is a limit ordinal λ with the property that no unbounded subset of λ is fresh over Ult(V, U 0 ) and there exists an unbounded subset of λ that is fresh over Ult(V, U 1 )?