Transferring Compactness

We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal. As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a cardinal $\kappa$ that is $n$-$d$-stationary for all $n\in \omega$ but not weakly compact. This is in sharp contrast to the situation in the constructible universe $L$, where $\kappa$ being $(n+1)$-$d$-stationary is equivalent to $\kappa$ being $\mathbf{\Pi}^1_n$-indescribable. We also show that it is consistent that there is a cardinal $\kappa\leq 2^\omega$ such that $P_\kappa(\lambda)$ is $n$-stationary for all $\lambda\geq \kappa$ and $n\in \omega$, answering a question of Sakai.


Introduction
In general, compactness refers to the phenomenon that if some property holds for all small substructures then it holds for the structure itself.For example, a compact topological space asserts that any collection of closed sets with the finite intersection property, has a non-empty intersection; The compactness theorem for first order Logic states that any first order theory such that all of its finite subsets are consistent must also be consistent; In cardinal arithmetic, Silver's theorem [29] asserts that if 2 ℵα = ℵ α+1 for any α < ω 1 , then necessarily 2 ℵω 1 = ℵ ω 1 +1 .This compactness phenomenon does not occur at the level of ℵ ω , as Magidor proves [21] that it is consistent that 2 ℵn = ℵ n+1 for every n < ω while 2 ℵω > ℵ ω+1 .In Graph Theory, König's Lemma asserts that if G has an infinite, locally finite, and connected graph, then there is an infinite simple path.This lemma ensures for example that ω has the tree property which is a paradigmatic compactness principle which says that any countably infinite tree, such that every level is finite must have a branch.
The dual notion of compactness is reflection i.e. if some property holds at some mathematical structure, then there must be a small substructure for which it was true.So compactness of some property φ is equivalent to the reflection of ¬φ.It turns out that many important instances of compactness such as free of abelian groups, Metrizable topological spaces [23] and others [28,30] boil down to a specific reflection principle known as stationary reflection.Recall that a subset C ⊆ κ is a closed unbounded (club) if C is closed in the order topology of κ and in unbounded below κ.A subset S ⊆ κ is stationary if it intersects any club.Definition 0.1.We say that a cardinal κ satisfies sationary reflection if for any stationary set S ⊆ κ, there is α < κ of uncountable cofinality such that S ∩ α is stationary at α.
Usually, reflection principles require assumptions beyond ZF C i.e. large cardinals.In fact, some large cardinal notions are tailored to satisfy reflection and compactness properties e.g.weakly/strongly/super-compact cardinals.One specific hierarchy of large cardinals which this paper considers is the Π 1 n -indescribable cardinals (see definition 0.2) which was discovered by Hanf and Scott [15].These large cardinals turned out to form a yardstick hierarchy in the landscape of large cardinals and provide a nice characterization of other large cardinal notions in terms of their ability to reflect formulas of higher complexity.Due to lack of technologies, a few implications among certain compactness principles around the region of "moderate large cardinals" are not well understood.
It is important that we insist κ is a strongly inaccessible cardinal in the last 3 items since these properties are consistent with κ being weakly inaccessible but not strong limit.Since if κ is weakly compact, then it is necessarily a strong limit cardinal, we can cheat and declare these principles are separated.However, if we insist that κ is strongly inaccessible, then these problems become much harder.In fact, they are open.
In this paper, we explore the possibility of "fixing the cheat" by transferring compactness principles at a weakly inaccessible cardinal to a strongly inaccessible cardinal.The technology we employ is Radin forcing [26], denoted R Ū , which is defined using a measure sequence Ū on a cardinal κ.Radin forcing has already turned out useful in order to tune the large cardinal properties and compactness principles holding at κ in the model V R Ū .For example, (1) If cf (lh( Ū )) = ρ < κ then V R Ū |= cf (κ) = cf (ρ) ([14, Section 5.1]).
(3) If Ū has a repeat point then V R Ū |= κ is measurable ( [25]).(4) If κ + ≤ cf (lh( Ū )) ≤ lh( Ū ) < 2 κ , then V R Ū |= ¬♦ κ (Woodin, see [7]).( 5) If Ū has a weak repeat point then V R Ū |= κ is weakly compact ( [7]).(6) If cf (lh( Ū )) ≥ κ ++ then V R Ū |= κ satisfy stationary reflection ( [7]).(7) If Ū satisfy the local repeat point then V R Ū |= κ is almost inaffable ([8]).The common idea in those results is that we isolate some property of the length lh( Ū ) of the measure sequence Ū which guarantees that κ has some large cardinal property in V R Ū .Let us just mention that most of the implications above are reversible.In [8], Ben-Neria and the second author tighten the connection between compactness principles in the Radin extension and properties.In this paper, the length of the sequence is usually lh( Ū ) < (2 κ ) + .The very rough idea is that if we force using a measure sequence such that the length of the measure sequence satisfies suitable compactness principles, then the Radin forcing transfers these compactness principles to actually hold at κ, which is strongly inaccessible in the generic extension.
As an application, relative to the existence of large cardinals, we construct a model where higher order stationary reflections hold at a strongly inaccessible cardinal which is not weakly compact.To properly state the theorem, we need the following definitions.Bagaria [2, Definition 4.1] used generalized logic to extend the indescribable cardinal hierarchy of Hanf and Scott to Π 1 ξ -formulas for ξ ≥ ω.Definition 0.2 (Hanf-Scott for ξ ∈ ω, Bagaria [2] for ξ ≥ ω).Let ξ be an ordinal.A set Definition 0.3 (Bagaria [2]).Recursively define that a set A is: ( We say that α is ξ-stationary if α is ξ-stationary as a subset of α. Bagaria's motivation for the notions comes from a result in [2], where these higher-order stationary reflection properties characterize the non-isolated points in the ordinal topology interpretation of generalized provability logics (see [5] or [6] for more information regarding this motivation).Note that: (1) A is 1-stationary iff A is stationary, (2) α is 1-stationary iff α has uncountable cofinality, (3) α is 2-stationary iff every stationary subset of α reflects.
Let us define the two variations of Bagaria's higher order stationarity central to this paper.Loosely speaking, one is obtained by varying the degree of simultaneous reflection and the other one is the diagonal version.Definition 0.4.Let 2 ≤ χ < κ be any regular cardinal and κ be a limit ordinal. (1) stationary iff every less than χ-many stationary subsets of cof(≥ χ) ∩ κ reflect simultaneously.
Definition 0.7.Let κ be a ordinal. (1) • κ is 2-d-stationary iff for any T i : i < κ where each T i is a stationary subset of κ, there exists a regular α < κ such that for all i < α, T i ∩α is stationary in α.
In terms of the consistency strength of these principles, Magidor [22] showed that the existence of a 2-s-stationary is equiconsistent with the existence of a weakly compact cardinal.Surprisingly, Mekler and Shelah [24] showed that the consistency strength of κ being 2-stationary is strictly in between a greatly Mahlo cardinal and a weakly compact.They isolated reflection cardinals and showed κ being a reflection cardinal is equiconsistent with κ being a 2-stationary cardinal.Generalizing their results and methods, Bagaria-Magidor-Mancilla [3] showed that the consistency strength of a ξ + 1-stationary cardinal is strictly in between a ξ-greatly-Mahlo cardinal and a Π 1 ξ -indescribable cardinal.We refer the readers to [3] for relevant definitions.To achieve this, they isolate the notion of a ξ-reflection cardinal and show that • there are many ξ-reflection cardinals below any Π 1 ξ -indescribable cardinal, • no ξ-reflection cardinal can be ≤ the first ξ-greatly Mahlo cardinal.Note that by definition, if κ is a ξ-reflection cardinal, then κ is ξ-stationary.In L, even more is true: it is ξ + 1-stationary.
It is therefore a natural question to clarify the relationship between higher order stationary reflections and indescribable cardinals.For example, for any given ζ, is it true that there exists a large enough ξ such that whenever κ is ξ-d-stationary (or ξ-stationary), then κ is Π 1 ζ -indescribable?The main result of this paper is that in general the answer is negative.
Another reason for this investigation is to expose another way of establishing higher order stationary reflection principles, fundamentally different from the Mekler-Shelah approach.Aside from the papers mentioned previously, variations of the Mekler-Shelah method have been used to study the extent of the weakly compact reflection principle by Cody and Sakai [9].
The following are the main results for this paper.
In particular, we have a way of producing a non strong limit weakly inaccessible cardinal λ that is λ-d-stationary.The next theorem "transfers" this compactness to a strongly inaccessible cardinal, using the technology of Radin forcing.
Theorem 0.11.Relative to the existence of a H(λ ++ )-hypermeasurable cardinal 1 κ where λ > κ is a measurable cardinal, it is consistent that a strongly 1 A cardinal κ is an H(θ)-hypermeasurable cardinal if there is an elementary embedding j : V → M with crit(j) = κ and H(θ) ∈ M .inaccessible cardinal κ is n-d-stationary for all n ∈ ω, but κ is not weakly compact.
The organization of this paper is: (1) In Section §1, we record some preliminary facts regarding higher order stationary sets.(2) In Section §2, we prove Theorem 0.10 and its 2-cardinal generalization.
(3) In Section §3 we prapare the ground model and present the relevant background for Radin forcing.(4) In Section §4, we present a proof of Theorem 0.11.(5) In Section §5, we conclude with some open questions.0.1.Notations.Given a function f : A → B and X ⊆ A, the pointwise image of X by f is the set f ′′ X := {f (x) | x ∈ X}.Given a set X and a cardinal λ, we denote by P λ X = {Y ⊆ X | |Y | < λ}.For a sequence X i | i < λ consisting of subsets of λ we denote by the diagonal intersection ∆ i<λ X i := {ν < λ | ∀α < ν, ν ∈ X i }.For a set of ordinals A, sup(A) = ∪A and we say that A is bounded in λ if sup(A ∩ λ) < λ.We say that A is closed if it is closed in the order topology of the ordinals.A set C is a club at λ is it is closed and unbounded, and the club filter is A set S is called stationary in λ is S∩C = ∅ for every club C in λ.We assume familiarity with forcing theory and refer the reader to [12] for background and standard notations.An elementary embedding j is always a function j : V → M where M is a transitive model, crit(j) denoted the minimal ordinal which is moved by j.If U is a σ-complete ultrafilter then j U : V → M U denoted the ultrapower by U .Given two finite sequence x 1 , ..., x n and y 1 , ..., y n we denote by x 1 , ..., x n y 1 , ..., y n = x 1 , ..., x n , y 1 , ..., y n .
Lemma 1.1.Fix a regular cardinal λ, T ⊂ λ, χ < λ and n < λ. ( For (2), fix β ∈ Tr χ n (A).Let S i : i < χ ′ for some χ ′ < χ and S i ⊂ β being (k i , χ)-s-stationary where k i < k be given.Since A ∩ β is (n, χ)-s-stationary, there exists In other words, T ∩ β is (k, χ)-s-stationary.For (3), fix β ∈ Tr d n (A).Let S i : i < β where each S i ⊂ β is k i -dstationary where for some k i < k.Since A ∩ β is n-d-stationary, there exists In other words, T ∩ β is k-d-stationary.To see the "furthermore" part, fix α ∈ T r d n (A) and i < α, we know that A − (i + 1) ⊂ T r d k (T i ).By the previous argument, we have that α ∈ T r d n (A − (i + 1)) ⊂ T r d k (T i ).The combinatorial properties of the n-stationary sets are best expressed in the language of ideals and filters.Ideals are the standard absractization of the notion of "smallness".Recall that a set I ⊆ P (X) is an ideal on X if ∅ ∈ I, I is downward closed with respect to "⊆" and closed under finite unions.We say that an ideal I is proper if X / ∈ I.The dual notion of an ideal is a filter, i.e. given an ideal I we define the dual filter We extend the definition of I * to any set I ⊆ P (X).The set of positive sets with respect to some ideal I is denoted by I + := P (X) \ I.For more information about ideal and filters we refer the reader to [17,Ch. 7 , Cub d λ be the corresponding dual filters.Fact 1.3.
(1) If T / ∈ N S n λ , then T r n (T ) ∈ Cub m λ for any m > n.Indeed, λ \ T r n (T ) is not m-stationary as witnessed by the n-stationary set T .
(2) Conversely, if N S m λ is proper, then for every set C ∈ Cub m λ there is an n-stationary set T for some n < m such that T r n (T ) ⊆ C. To see this, since λ \ C ∈ N S m λ , there is some n < m and a n-stationary for any n ≤ m < λ.This follows from the fact that whenever S is mstationary, by Definition 0.3, it is also n-stationary.(4) N S n λ is always upward closed with respect to ⊆.
λ and Y ⊆ X, then and m-stationary set S for m < n which witnesses that X is not n-stationary will also witness that Y is not n-stationary.
(5) If N S n λ is an ideal, then it is proper iff λ is an n-stationary cardinal.Lemma 1.4.Let λ be regular and n < λ.Fix any S ⊂ λ.Then S is n + 1stationary iff λ is n-stationary and for any n-stationary T ⊂ λ, T r n (T )∩S = ∅.
Proof.We prove the non-trivial direction (←).Given any m ≤ n and mstationary W , we need to show T r m (W ) ∩ S = ∅.If m = n, then we are done by the hypothesis.So assume m < n.
We record the following fact for the other ideals.The proof is similar to that of Lemma 1.4.
(1) Given T i : i < χ ′ such that each T i is (k i , χ)-s-stationary for some The proof is similar to the previous one, except that we apply Lemma 1.1(3) instead.
(3) Suppose that κ is (n, χ)-s-stationary such that χ is an infinite cardinal and either n is a successor ordinal or cf(n) ≥ χ.For any (1) Immediate.
(2) Closure under subsets is immediate.To see that it is χ-complete, suppose that By the hypothesis about n, we can find some k < n such that (1) Immediate.(2) Closure under subsets is immediate.To see that it is normal, suppose that A i | i < κ ⊆ N S ξ,d κ , then for each i < κ there is a sequence T j,i | j < κ with η i j < ξ such that each T i is η i j -d-stationary and (3) By definition, there is a sequence 1.2.The relationship between different ideals in the constructible universe.
Lemma 1.8.Fix a cardinal κ and n < κ.If N S n κ is normal, and for any k < n, {α < κ : Let us clarify that "{α < κ : κ is not proper.In particular, this is the case when κ is singular as N S n κ is assumed to be normal.We need to show N S n,d κ is also not proper.If N S k κ is not proper for some k < n, then by the hypothesis, N S k,d κ is not proper, which in turn implies that N S n,d κ is not proper.Hence, we may assume N S k κ is proper for all k < n.By the assumption, there is some k-stationary T ⊂ κ such that T r k (T ) = ∅.Suppose for the sake of contradiction that N S n,d κ is proper.As T is k-dstationary by the hypothesis, there is some α such that T ∩α is k-d-stationary and N S k α = N S k,d α .In particular, T ∩ α is k-stationary.This contradicts with the fact that T r k (T ) = ∅.
We may now assume that N S n κ is proper.
κ , as N S n κ and N S n,d κ are normal and in particular κ-complete.
First we show that if A is n-stationary, then A is n-d-stationary.In particular, this implies N S n,d κ is proper.Let T i : i < κ be given such that each T i is k i -d-stationary for some k i < n.By the hypothesis, we know that Next we show that if A is n-d-stationary, then A is n-stationary.Let T be a k-stationary subset of κ for some k < n.By the hypothesis, Proposition 1.9.Suppose for any cardinal κ and any n < κ, N S n κ is normal, then for all κ and n < κ, N S n κ = N S n,d κ .Proof.Suppose otherwise for the sake of contradiction.Fix the least cardinal κ and then the least n < κ such that N S n κ = N S n,d κ .Note that κ is regular and n > 1.We will reach a contradiction by verifying that the hypotheses of Lemma 1.8 are satisfied.For k < n, • N S n κ is normal by the assumption, κ by the minimality of n and Similar proofs to those in Lemma 1.8 and Proposition 1.9 give the following: Lemma 1.10.Fix a cardinal κ, n < κ and χ < κ.If N S n κ is χ-complete, and for any k < n, {α < κ : Proposition 1.11.Suppose for any cardinal κ, any n < κ and χ < κ, N S n κ is χ-complete, then for all κ and n < κ, N S n κ = N S (n,χ) κ .
Corollary 1.12.If V = L, then all the following are equivalent for ξ < κ: In fact, in the theorem above, all the ideals corresponding to each clause are the same.Namely, Proof.This follows from [4, Corollary 2.5], [2, Theorem 5.1] and Propositions 1.9, 1.11.
1.3.2-cardinal higher order stationarity.Sakai [27] generalized the higher order stationarity notions to the two-cardinal setting.
Definition 1.13 (Sakai [27]).For a regular cardinal κ, a set A ⊃ κ and n ∈ κ, The collection of B satisfying the above is called the m-trace of T , written as Tr m (T ) (this is slight abuse of notation but there should be no difficulty inferring from the context).
• P κ A is n-stationary if P κ A is n-stationary as a subset of P κ A.
Remark 1.14.In the original definition, only n ∈ ω was considered.Generalizing that to n ∈ κ poses no difficulty.
Remark 1.15.If T ⊂ P κ A and B ∈ Tr m (T ), then for any 2. Higher order stationary reflection at a non strong limit cardinal Definition 2.1.Let λ be a regular cardinal and κ be a cardinal.An ideal Fact 2.2.Let I be uniform normal κ-saturated ideal on λ where κ < λ.Let G ⊂ P (λ)/I be generic over V .Then in V [G], (1) [13, Chapter 2] there is an elementary embedding j : (3) for any Ẋ such that P (λ)/I Ẋ ∈ Ī * , there exists X ∈ I * such that P (λ)/I X ⊂ Ẋ (this follows from the fact that P (λ)/I is κ-c.c and Theorem 2.3.Let λ be a regular cardinal carrying a uniform normal κsaturated ideal I for some κ < λ.Fix also some χ < λ.Then for all k < λ, N S k λ , N S (k,χ) λ , N S k,d λ are all proper ideals on λ.In particular, λ is λ-stationary, (λ, χ)-s-stationary and λ-d-stationary.
Proof.We prove the following statement ( * ) n by induction on n ∈ Ord.For any λ > κ, n such that λ carries a uniform normal κ-saturated ideal I, for any T ⊂ λ, (1) if T is n-stationary, then T r n (T ) ∈ I * .
Proof of the Claim.
(1) Since λ is λ-stationary iff λ is k-stationary for all k < λ, the first clause is immediate.
(2) Given χ ′ < χ and T i : i < χ ′ such that each T i is (η i , χ)-s-stationary for some η i < λ, by the hypothesis we know there are ) Given T i : i < λ such that each T i is η i -d-stationary for some η i < λ, by the hypothesis, there are Base case n = 1.Recall that in this case, Let G ⊂ P (λ)/I be generic over V .If T is a stationary subset of λ (stationary relative to λ ∩ cof(≥ χ)), then P (λ)/I T ⊂ λ is stationary since the forcing satisfies κ-c.c.Note that j(T ) ∩ κ = T where j : V → M is the elementary embedding from fact 2.2 item (1).In particular, M |= κ ∈ j(T r 1 (T )) and by fact 2.2 (4) T r 1 (T ) ∈ I * (T r χ 1 (T ) ∈ I * , or T r d 1 (T ) ∈ I * ).Suppose we have proved ( * ) i for all i < n, let us show ( * ) n .Let G ⊂ P (λ)/I be generic over V and let j : V → M ≃ Ult(V, G) be an ultrapower embedding in V [G].Let us first assume T ⊂ λ is n-stationary.It suffices to show that M |= T is n-stationary, as the conclusion follows from the elementarity of j.Suppose for the sake of contradiction that As the proof for the case where T is (n, χ)-s-stationary and n-d-stationary is similar to the above, we only sketch the differences.Let us assume T is nd-stationary for concreteness.Proceed as above and assume M |= T is not nd-stationary for the sake of contradiction.There exists S i : i < λ such that each S i is η i -d-stationary for some η i < n, and ∆ i<λ (T , apply the induction hypothesis and the normality of Ī, for each k < n, we can get C k ∈ I * such that By Fact 2.2 (3), we may assume that , which gives the desired contradiction.Fix i < α and η i = k.By the definition of C k , we have that Proof of Theorem 0.10.Let λ be a measurable cardinal and a κ-c.c forcing P be given where κ < λ.A theorem of Kunen ([20, Lemma 2]) gives that in V P , there exists a λ-complete normal κ-saturated ideal on λ.Then we apply Theorem 2.3 to get the conclusion as desired.
Let us turn our attension to the 2-cardinal higher order stationary reflection principles (see Definition 1.13).Sakai [27] posed the following question: For n ≥ 3, is it consistent that there is a cardinal κ ≤ 2 ω such that P κ (λ) is n-stationary for all λ ≥ κ?
We answer this question positively in the following, adapting the proof of Theorem 2.3 to the 2-cardinal setting.Recall that a cardinal κ is called λsupercompact if there is a fine normal measure U over P κ (λ).Equivalently, if there is an elementary embedding j : V → M such that crit(j) = κ, M is transitive and M λ ⊆ M .κ is called supercomapct if it is λ-supercompact for every λ.The following is a 2-cardinal version of the classic Mitchell order on normal ultrafilters on measurable cardinals.Definition 2.5.We define an ordinal function o on the set N F of all normal fine measures on P κ λ recursively as follows: for all U ∈ N F , (1) o(U ) ≥ 0, (2) o(U ) ≥ ξ if for any ξ ′ < ξ, there exists some Observation 2.6.If κ is a supercompact cardinal, then for any λ ≥ κ and ξ < λ, there is a normal fine measure U on P κ λ such that o(U ) ≥ ξ.
Proof.Recall that for a normal fine ultrafilter U on P κ λ and η, o(U ) ≥ η if for any ξ < η, there exists some normal fine ultrafilter W on P κ λ with o(W ) ≥ ξ which belongs to M U ≃ Ult(V, U ). Fix λ ≥ κ.Suppose for the sake of contradiction that sup{o(U ) + 1 : U is a normal fine ultrafilter on P κ λ} = η < λ.
Let j : V → M witness that κ is λ + -supercompact.Let W be the normal fine ultrafilter on P κ λ derived from j.Let i : V → N ≃ Ult(V, W ) and let k : N → M be defined such that k([f ] W ) = j(f )(j ′′ λ).The following facts are standard (see [19]): (1) both i and k are elementary and j As a result, by elementarity, N |= sup{o(U ) + 1 : U is a normal fine ultrafilter on P κ λ} = η < λ.
Note that since N is sufficiently closed, any normal fine ultrafilter on P κ λ in N is a normal fine ultrafilter on P κ λ in V .But then by the definition, o(W ) ≥ η, which is a contradiction.Theorem 2.7.Let κ be a supercompact cardinal and P be a forcing satisfying ν-c.c for some ν < κ.In V P , P κ λ is n-stationary for any n < κ and λ ≥ κ.
Proof.The proof is similar to before, so we only highlight the modifications.We prove the following statement (⋆) n by induction on n ∈ Ord: for any ν, any λ > κ > max{n, ν}, any forcing P satisfying ν-c.c and any normal fine measure U on P κ λ such that o(U ) ≥ n, the following holds in V P : for any n-stationary T ⊂ P κ λ, there exists A ∈ U such that T r n (T ) ⊃ A. As (⋆) 1 is easy to be seen to hold, let us focus on the inductive step.
Suppose we have proved (⋆) i for all i < n and let us show (⋆) k .Let λ > κ > {n, ν}, forcing P and normal fine ultrafilter U on P κ λ with o(U ) ≥ n be given.Let j : V → M ≃ Ult(V, U ) be the supercompact ultrapower embedding, in particular, λ M ⊂ M .Let G ⊂ P be generic over V and T ⊂ P κ λ ∈ V [G] be n-stationary.We can continue to force and find G * ⊂ j(P ) generic over V extending j ′′ G, such that we can lift j to j In V [G], standard arguments show that the ideal generated by the dual of U is the same as {X ⊂ P κ λ : j(P )/j ′′ G j ′′ λ ∈ j+ (X)}.It suffices to show that j + ′′ T remains n-stationary in P κ j ′′ λ in M [G * ].Then we finish by the elementarity of j.
Suppose for the sake of contradiction that j +′′ T is not n-stationary in P κ j ′′ λ.In M [G * ], let S ⊂ P κ j ′′ λ be some m-stationary set such that T r m (S) ∩ j +′′ T = ∅ where m < n.Since o(U ) ≥ n, we can find some normal fine ultrafilter W ∈ M on P κ λ such that o(W ) ≥ m.We may identify W as a normal fine ultrafilter W ′ on P κ j ′′ λ induced by j ↾ λ ∈ M .
Applying the induction hypothesis (⋆) m in M with respect to W ′ , S and j(P ), we know that there is B ∈ W ′ such that M [G * ] |= B ⊂ T r m (S).Since in P κ λ and P κ j ′′ λ are isomorphic, we know that Let µ = D ∩κ and we know µ is a regular cardinal in κ.By the elementarity of j + , we have j 16.This contradicts with the fact that T r m (S)

Preparing the ground model and Radin forcing
Start with a model of GCH where κ is an H(λ ++ )-hypermeasurable cardinal where λ is the least measurable cardinal greater than κ.Our goal is to produce a universe V where 2 κ = λ + and there exists an elementary embedding j : V → M such that (1) H(λ ++ ) ⊂ M , (2) for every X ⊂ λ, there is g ∈ V such that j(g)(κ) = X, (3) for any n ∈ ω, λ is (n, κ + )-s-stationary.Let r : κ → κ be the function that takes any α to the minimal measurable cardinals α < r(α).Since κ is an H(λ ++ )-hypermeasurable cardinal, r : κ → κ.Since the preparation is standard, we will only sketch the proof and refer the readers to the relevant literature for more details.Specifically, we follow largely [10], [16] and [11].
We will use some standard facts about term-space forcing.
Definition 3.1.Let P be a forcing and Q be a P -name for a forcing.Define Q/P to be the poset consisting of terms σ such that P σ ∈ Q.The order on Q/P is: σ ≤ τ iff P σ ≤ τ .
(1) [12, Proposition 22.3] Fix a forcing P and a P -name for a forcing Q.Let G ⊂ P be generic over V and let H ⊂ Q/P be generic over V .Then (2) [10, Fact 2] Let κ be such that κ <κ = κ and P be a κ-c.c forcing.
Let Q be a P -name for Add(κ, γ).Then in V , Add(κ, γ) is forcing equivalent to Q/P .

3.1.
Step One.The first stage is to ensure that there is a universe V 0 in which (1) κ is an H(λ ++ )-hypermeasurable cardinal, (2) GCH holds at all inaccessible α ≤ κ and β ≥ λ, (3) there is an elementary embedding j : • j(r)(κ) = λ, and • κ M ⊂ M along with i : V 0 → N being the ultrapower by the normal measure derived from j, there is F ∈ V 0 that is generic for i(Add(κ, λ + )) over N .(4) V 0 is a κ ++ -c.c forcing extension of V .For the construction, see [16,Corollary 2.7].Apter and Cummings [1] independently, in some unpublished work, has an alternative way of achieving the above.

3.2.
Step Two.We may take V 0 from the previous subsection as our ground model in this subsection.The second step is to perform the Easton support iteration P β , Qα : α ≤ κ, β ≤ κ + 1 such that for any α < κ, Qα is trivial unless Pα α is inaccessible, in which case Qα is a P α -name for Add(α, r(α) + ).Finally, let Qκ be the P κ -name for Add(κ, λ + ).Let G = G κ * g κ be V -generic for P κ+1 = P κ * Qκ .Let j : V → M be the embedding from Step One.We may without loss of generality assume that j = j E where E is a short (κ, λ ++ )-extender on κ.Let i : V → N be the ultrapower by the normal ultrafilter on κ derived from j.
From now on, let us denote by V * = V and j * = j the model and the elementary embedding of the previous corollary.Let us give a brief description of the notations we use for Radin forcing [26] and relevant background for our main result.For full detailed definitions and proofs consult [14].
3.3.Radin Forcing.Let Ū be a measure sequence derived from j of length λ i.e.Ū = κ U (ξ) | ξ < λ such that for every ξ < λ we define recursively and U (0) is just the normal measure derived from j using κ.More generally, a measure sequence is any sequence of ultrafilter w of any length, denoted by lh( w), which is derived from some elementary embedding j w in the same way Ū was derived, always starting with the seed crit(j) which we denote by κ( w) = crit(j w).We denote by MS the class of all measure sequences.It is well-known (see for example [14]) that we may only consider w ∈ MS which concentrate on MS i.e MS ∩ V κ( w) ∈ ξ<lh( w) w(ξ) =: w.Remark 3.5.We will always assume that given a set A ∈ w, for every v ∈ A, A ∩ V κ(v) ∈ v.Such a v is said to be addable to A. For generally, we say that η = (v 1 , ..., vn ) is addable to ( w, A), and denote it by η << A, if for every 1 Let us follow the description of Radin forcing from [14] with the exception that q ≤ p means q is a stronger condition.Definition 3.6.The Radin forcing with Ū , denoted by R Ū consist of all finite sequences p = d i | i ≤ k such that each d i is either an ordinal κ i < κ (which we identify as the measure sequence ūp i = κ i ) or a pair (ū p i , A p i ), such that: (1) We denote the length of the condition lh(p) = k, the lower part p 0 = d i | i < lh(p) , the upper part ( Ū , A) (so we may write p = p ⌢ 0 ( Ū , A)), Let κ 0 (p) = κ(d k−1 ), and We say that p is a direct extension of q and p ≤ * q if: (1) lh(p) = lh(q).
(2) For every i < lh(p), ūp i = ūq i and A p i ⊆ A q i .Definition 3.9.Let p ∈ R ū.A one-step extension of p is obtained by choosing i ≤ lh(p) ūp i and v ∈ A p i which is addable to (ū p i , A p i ) and forming the condition We define recursively, p v1 , . . ., vn+1 = (p v1 , . . ., vn ) vn+1 .
Remark 3.10.We clarify the notations regarding concatenation and one-step extension.
• p ⌢ ( w, A) is the string concatenation in the usual sense.In particular, in order for this to make sense, κ( w) > κ(v) for all v ∈ p and we always need to specify the large set A. • p w is the one step extension as in the definition above.In particular, κ( w) < κ(v) for some v ∈ p and we only specify the measure sequence since the measure one set for w is already determined.
We define the order by p ≤ q if there are v1 , . . .vn , such that vi ∈ A q i and p ≤ * q v1 , . . .vn .Let us list some basic properties of R Ū (for the proof see [14]): Proposition 3.11. ( and any i < lh(p), we can factor (4) R Ū satisfies the Prikry property: For any sentence in the forcing language σ, and any condition p ∈ R Ū there is a direct extension p * ≤ * p such that p * σ ∨ p * ¬σ3 .(5) R Ū satisfies the strong Prikry property: For every dense open set D ⊆ R Ū and any condition p, there is p * ≤ * p and a p * -fat tree 4 T such that for every maximal branch t ∈ T , p * t ∈ D.
We will need the following proposition that reduces R Ū -names to names which depends on bounded information: Proposition 3.12.Let γ α | α < κ ∈ V be any sequence of ordinals below κ and p = p 0 ( Ū , A) ∈ R Ū and ẋα | γ α < κ be a sequence of R Ū -names such that p ẋα ⊆ γ α .Then there is q ≤ p, q 0 = p 0 and a function f : A q → V κ such that for every w ∈ A q , f ( w) is an R w-name forced by q 0 ( w, A q ∩ V κ( w) ) to be a subset of κ( w) and q w f ( w) = ẋκ( w) .
Proof.The proof is exactly as in [8, Lemma 2.13] exploiting the κ-closure of ≤ * of the upper part to determine Ṫ ∩ γ κ( w) .Definition 3.13.Let G ⊆ R Ū be V -generic.We denote by We say that a set A is generated by a set in the ground model if there is B ∈ V such that A = O(B ∩ MS G ).Other useful lemmas concerning Radin forcing can also be found in [7] and [8].
3.4.Compactness and stationarity in Radin Extensions.Let G ⊆ R Ū be V -generic.It turns out that some large cardinal properties of κ in the generic extension V [G] correspond to combinatorial properties of lh( Ū ) (see the discussion in the introduction for some examples).One which is relevant to us the that of a weakly compact cardinal which is due to the second author and Ben-Neria: Lemma 3.14 ([8, Lemma 3.14]).Suppose that (2 κ ) M does not divide lh( Ū ), then in V R Ū |= κ is not weakly compact.
In particular, the measure sequence Ū which we have prepared satisfies that lh( Ū ) = λ < (2 κ ) M .To show that κ has some reflection properties in the generic extension, we will need to analyze stationary sets and higherorder stationary sets in Radin forcing extensions.
For the sake of convenience, let us denote Z e 0 \∅ as Z e 0 .This provided the main ingredient in [7] to guarantee stationary reflection in the generic extension: Theorem 3.18 ([7]).Suppose that Ū is a measure sequence such that cf (lh( Ū )) ≥ κ ++ , then V R Ū |= every stationary set at κ reflects and moreover every κ-sequence of stationary subsets of κ reflects diagonally (or in our terminology, κ is 2-d-stationary).
Our intention is to generalize this characterization to higher levels of stationarity and show that the measure sequence we produced in the preparation, guarantees that Definition 3.20.Fix n < κ and a measure sequence Ū on κ of length λ.Let χ < λ.We say that A ⊂ V κ is: If in addition λ is (n, κ + )-s-stationary, then the following proposition holds: Proposition 4.5 (φ 4,n ).In V R Ū , for any sequence of n-d-stationary sets Strictly speaking, we should decorate these propositions with Ū , namely φ j,n should be φ Ū j,n for j = 0, 1, 2, 3, 4, since for each n ∈ ω, we quantify over all good measure sequences.In the following, we suppress the superscript if the measure sequence we are dealing with is Ū .Otherwise, we will always decorate with the superscript to make precise which good measure sequence the induction hypothesis is applied to.
We focus on the inductive case n > 1.The argument is, to some extent, a generalization of that in [7].
For each i < κ, by our assumption about the embedding j, let f i be such that j(f i )(κ) = A i and let Γ We would like to prove that O(Γ i ∩ M S G ) | i < κ witness that κ is not n-d-stationary.Indeed We may assume without loss of generality that ).By our assumption on S, we have that Hence S is n-d-stationary.• (→): Suppose there exists a set Γ ⊂ MS that is Ū -(n − 1, κ + )-s-club such that S ∩ O(Γ ∩ MS G ) = ∅, we will cook up a n − 1-d-stationary set H such that T r d n−1 (H)∩ S is bounded.This implies that S is not n-d-stationary. 5Since λ is assumed to be (n−1, κ + )-s-stationary, by Lemma 1.6 (3), we can find T 0 ⊂ λ which is (n − 2, κ + )-s-stationary such that T r κ + n−2 (T 0 ) ⊆ C = Ind Ū (Γ).By our assumption on j, there is Proof of the Claim.Let ū ∈ Γ * ∩ MS G .Note that ū is good and lh(ū) is (n − 2, κ(ū))-s-stationary.We can then apply the induction hypothesis φ ū as the upper part of the forcing does not add subsets to κ(ū).It follows that κ(ū) ∈ T r d n−1 (H).For the other direction, clearly each α Proof of φ 2,n .Let p = p ⌢ 0 ( Ū , A).By Proposition 3.12, we may assume that for each w ∈ A, there is an R w-name f ( w) such that p ⌢ w forces Ṫ ∩ (κ( w) + 1) = f ( w).Furthermore, we may assume there exists a measure function b satisfying the following: for each w ∈ A, and any r ∈ R <κ( w) , there exists a direct extension r ′ of r in R <κ( w) such that r ′ ⌢ ( w, b( w)) decides the statement κ( w) ∈ f ( w).Split A into two sets: Proof.Otherwise, Ind Ū (A 1 ) is not (n − 1, κ + )-s-stationary, and thus A 2 is a Ū − (n − 1, κ + )-s-club.Note that by our construction of p and by definition of A 2 , for every ū ∈ A 2 and any t ∈ R <κ(ū) /p 0 , there is a direct extension t ′ of t such that Let H be any generic with p ∈ H. Since p Ṫ is n-d-stationary, we have Hence we can find ū ∈ A 2 and a condition p . For each ξ ∈ S 1 , U ↾ ξ ∈ j(A 1 ).By elementarity of j and the definition of A 1 , find some t ξ ∈ R <κ such that t ξ ≤ R<κ p 0 and t ⌢ ξ ( Ū ↾ ξ, j(b)( Ū ↾ ξ)) ⌢ (j( Ū ), j(A) \ V κ+1 ) j(R Ū ) κ ∈ j( Ṫ ).By the κ + -completeness of NS (n−1,κ + ) λ , we can find e 0 ∈ R <κ /p 0 such that Z e 0 = def {ū ∈ A : e ⌢ 0 (ū, b(ū)) ⌢ ( Ū , A) κ(ū) ∈ Ṫ } is Ū -(n − 1, κ + )s-stationary.Then e 0 is the desired lower part.All that is left to do is to shrink the measure one set.
Let us say that − → η ∈ A <ω is nice, if Z e 0 \ η is Ū -(n − 1, κ + )-s-stationary.We next show that we can find a Ū -measure one set B such any − → η ∈ B <ω is nice.We achieve the task in steps by inducting on the length of the finite sequence of measure sequences.
The argument is similar to the previous step, by looking at the set Z e 0 \ − → η , and again applying the κ + -completeness of NS Then e * ≤ e ′ and also e * is compatible with e 0 w, b( w) Ū , A .So there exists an extension of e * that forces the following: (1) Ṫ ∩ O(Γ ∩ Ṁ S G ) = ∅.
For each i, let A i be a maximal antichain subset of {e ∈ R Ū : ∃ a measure function b, (e, b) is an n-d-stationary witness for Ṡi }.
Theorem 0.11 now follows easily from the proof in this section and Lemma 3.14.
Remark 4.10.Here is a comment on the necessity of the goodness assumption on the measure sequence in the proof above.More precisely, without the hypothesis that for any X ⊂ λ, there is f ∈ V such that j(f )(κ) = X, the statement φ 0,n may not be true.For example, if κ is strong in the ground [19, Theorem 17.1]  the ideal Ī generated by I is uniform normal and κ-saturated.
Fact 2.2 (2), the ideal Ī generated by I is uniform normal and κ-saturated in V [G].Therefore, we can apply the induction hypothesis ( * ) k in V [G] to conclude that there exists C ∈ Ī * such that C ⊂ T r V [G] k (S).Since Ī is generated by I, we may assume that C ∈ I * .In particular, C ∈ V .Apply ( * ) k in V , we know that C is k-stationary.As a result, T r k (C) ∩ T = ∅.Fix α ∈ T r k (C) ∩ T .Apply j to see that α
stationary, where η << b( w) means for any measure sequence v appearing in η, it is the case that v ∈ b( w) and b( w) ∩ V κ(v) ∈ v, namely, v can be added below ( w, b( w)).We call such (e,b) an n-d-stationary witness for Ṫ Proposition 4.4 (φ 3,n ).If Ṫ is a R Ū -name such that an n-d-stationary witness (e, b) for Ṫ exists, then e Ṫ is n-d-stationary.