Stationary sets added when forcing squares

Current research in set theory raises the possibility that (cid:2) κ,<λ can be made compatible with some stationary reﬂection, depending on the parameter λ . The purpose of this paper is to demonstrate the difﬁculty in such results. We prove that the poset S (κ, < λ) , which adds a (cid:2) κ,<λ -sequence by initial segments, will also add non-reﬂecting stationary sets concentrating in any given coﬁnality below κ . We also investigate the CMB poset, which adds (cid:2) ∗ κ in a slightly different way. We prove that the CMB poset also adds non-reﬂecting stationary sets, but not necessarily concentrating in any coﬁnality.


Introduction
The κ principle, pronounced "square kappa," was introduced by Jensen in order to study the fine structure of Gödel's constructible universe L [3]. κ and similar principles imply the failure of many of the compactness properties entailed by large cardinals. For our purposes we focus on stationary reflection.
In the first section we will define the hierarchy of principles κ,<λ for 1 < λ ≤ κ + and summarize some of the existing work on the extent to which κ,<λ impedes stationary reflection for different values of λ. Then we will define two posets: one denoted S(κ, < λ) that forces κ,<λ for any fixed λ, and another called the CMB poset, denoted C, that forces * κ . In the second section we will show that S(κ, < λ) adds non-reflecting stationary sets, and in the third section we will show that C adds a non-reflecting stationary set as well. In the last section, we will show that in the case of singular cardinals of uncountable cofinality, C is compatible with some stationary reflection.
Here lim C denotes the accumulation points in a club C. Although we discuss only limit ordinals α here, indexing the sequence for all α < κ + is a standard abuse of notation.
κ,λ is defined similarly, but where the first bullet point is replaced by 1 ≤ |C α | ≤ λ. To say that κ,<λ holds is, naturally, to assert the existence of a κ,<λ -sequence. Jensen's original κ is κ,1 while κ,κ , which is usually denoted * κ , is called "weak square". The generalized definition for different values of λ is due to Schimmerling [6].
The intuitive purpose for κ,<λ is to define a way of comparing a given model of set theory and L-the smaller the value of λ, the more a given model resembles L. We are interested in the tension between κ,<λ for various λ and stationary reflection, which is a key ingredient to Jensen's proof that L contains Suslin trees and has numerous other applications.

Definition 2
If κ is a regular cardinal and S ⊂ κ is stationary in κ, then we say S reflects at α ∈ κ if α has uncountable cofinality and S ∩ α is stationary as a subset of α.
Usually the assumption that α has uncountable cofinality is tacit. In general, κ,<λ clashes with stationary reflection. Some results along these lines follow: (1) If κ holds then every stationary S ⊂ κ + contains a non-reflecting stationary subset. (2) If κ is singular and κ,λ holds for λ < κ, then for every stationary S ⊂ κ + , there is a sequence T i : i < cf κ of stationary subsets of S so that for every α < κ + , there is some i < cf κ such that T i does not reflect at α. (3) Assuming the existence of countably many supercompact cardinals, it is consistent for κ to be a singular cardinal of countable cofinality and for κ,ω to hold while every stationary S ⊂ κ + reflects to some α < κ + .
(4) Again, assuming the existence of countably many supercompact cardinals: It is consistent for κ to be a singular cardinal of countable cofinality and for * κ to hold while for λ < κ every sequence S i : i < λ of stationary subsets of κ + containing ordinals of cofinality less than some τ < κ reflects simultaneously to some α < κ + , i.e. such that S i ∩ α is stationary for every i < λ.
These consistency results use variations on the following poset, which is used to force general square sequences.
The ordering is direct extension: p ≤ q if p dom q = q. S(κ, λ) is defined similarly.
A similar poset was recently studied by the author.

Definition 4
Let κ be a singular strong limit of cofinality μ and let κ i : i < μ be a sequence of regular cardinals converging to κ. C i is the poset of closed bounded subsets of κ + of order-type less than κ i , where p ≤ C i q if p end-extends q, meaning that max p ≥ max q and p ∩ (max q) = q.
We let We use [ f ] to refer to the equivalence class of f .
[ f ] ≤ [g] will refer to ordering modulo the equivalence relation, and f (i) ≤ g(i) will refer to the ordering of C i . We call C the collapses-mod-bounded poset, or more briefly the CMB poset.
Facts 3 [5] Let C refer to the CMB poset.
then V C contains no very good scales on κ.
Although the isomorphism-type of C depends on the sequence κ i : i < μ used in its definition, the properties discussed will not depend on the particular sequence.

Lemma 1 Suppose κ is a cardinal and X
Then there is a ρ < κ + and a collection C α : α ∈ X of clubs in ρ such that for every α ∈ X, ot C α = α. If μ = cf κ or if X is bounded in κ then we can find ρ ≤ κ such that this works.
Let γ ξ : ξ < μ be continuous, increasing, and cofinal in α such that γ 0 = 0 and γ ξ is a successor ordinal if ξ is a successor ordinal. Any ordinal γ < α is in the It remains to show that f is continuous. Suppose γ < α is a limit. If γ = γ ξ , then by construction ξ is a limit ordinal for some ξ < μ and so Otherwise γ = γ ξ + β for some limit β > 0 and so Observe that if X ⊂ (κ + 1) ∩ cof(μ) is unbounded in κ and μ = cof κ, then it is impossible to obtain the conclusion of this lemma for any ρ ≤ κ. Since sup X = κ in this case, it follows that ρ ≥ κ, and then since cf ρ = cf κ we have ρ > κ.
Proof Apply Lemma 1 to X in order to obtain δ ≤ κ and clubs C α ⊂ δ such that ot C α = α for all α ∈ X . Let q ∈ S(κ, < λ) be a condition forcing thatĊ is a club in κ + . We will construct a descending sequence p ξ : ξ ≤ δ below q where γ ξ = max dom p ξ as follows: • Find α 0 and r ≤ q such that r "α 0 ∈Ċ". Then find p 0 ≤ r such that max dom p 0 > α 0 .
We must show that p ξ is a condition in S(κ, < λ). The only nontrivial consideration is to show that if E ∈ p ξ (γ ξ ) and β ∈ lim E, then E ∩ β ∈ p ξ (β). If β ∈ lim E then β = γ ζ such that ζ is a limit. If E = {γ η : η < ξ} then this means E ∩ β = {γ η : η < ζ}, so this club would have been included at step ζ in the definition of p ζ . If E = {γ η : η ∈ C α ∩ ξ } for some α ∈ Y , then this means that {γ η : η ∈ C α ∩ ζ } is unbounded in γ ζ and therefore that C α is unbounded in ζ , so the club E ∩ β = {γ η : η ∈ C α ∩ ζ } would have been included at step ζ . • Suppose ξ = δ. This is the same as the other limit cases except that p δ (γ δ ) = {D α : α ∈ X } and we know that {α ∈ X : C α is unbounded in δ} = X . Showing that p δ is a condition is strictly simpler than in the previous case.
When using a generalized κ,<λ -sequence C α : α < κ + to prove a non-reflection result, it is typical to make use of the function F : α → {ot C : C ∈ C α } and to find a stationary set on which F is constant. The significance of the above theorem is that S(κ, < λ) adds a κ,<λ -sequence such that for every possible value X that F can take, F −1 (X ) is stationary.

The non-reflecting stationary set added by the CMB poset
For the remainder of this paper, fix a singular strong limit cardinal κ of cofinality μ and a sequence κ i : i < μ of regular cardinals converging to κ, and define the CMB poset C in terms of this sequence as in Definition 4. Using similar methods to the previous section, we prove: Theorem 2 C adds a non-reflecting stationary set in κ + ∩ cof(μ).
Observe that the definition of X does not depend on the representative. We will make use of the following fact, that was earlier established by the author [5]: The non-reflecting stationary set will be Note that Fact 4 implies that S ⊂ κ + ∩ cof(μ).
Claim 2 S does not reflect at any α < κ + of uncountable cofinality. In other words, S is a non-reflecting stationary set.
Proof Work in V [G] and fix some α < κ + of uncountable cofinality. We will show that S cannot reflect at α. Suppose for contradiction that it does. Since X is unbounded in every point in S, X is unbounded in α. The fact that S ⊂ κ + ∩ cof(μ) implies that cf α > μ. It follows from Fact 4 that α ∈ X . Now let j < μ be such that α ∈ lim f (i) for all i ≥ j and let C = α ∩ i≥ j f (i). Since cf α > μ, C is a club in α. Moreover, if β ∈ lim C, then clearly β ∈ X , and since X ∩ S = ∅ by definition, we have lim C ∩ S = ∅.
This completes the proof.

Reflection in extensions by the CMB poset
In this section we demonstrate that the CMB poset C does not necessarily add nonreflecting stationary sets in all cofinalities. We must draw on several facts from the literature. The first is widely known:
The second fact is a result of Laver [4].

Fact 6
If λ is supercompact, then there is a forcing poset P such that if P "Q is λ-directed closed", then V P * Q | "λ is supercompact".
A supercompact cardinal having this property is called "indestructibly supercompact".
Finally, we use a fact drawn from the theory of the approachability ideal, namely that * κ implies the approachability property at κ, which in turn implies that stationary subsets of S ⊂ κ + ∩ cof(τ ) are preserved by τ + -closed forcing posets [2].
Fact 7 If * κ holds for a singular cardinal κ and S ⊂ κ + ∩ cof(τ ) is stationary for some regular τ < κ, then S is stationary in any τ + -closed forcing extension.
Proof Fix a C-generic filter G and a stationary set S ⊂ κ + ∩ cof(τ ) such that S ∈ V [G]. Let j be such that τ, λ < κ j and let D := j≤i<μ C i , where C i is again the set of closed bounded sets in κ + of order-type less than κ i , and C i is ordered by end-extension. Observe that D is κ j -directed closed since the same is true of each C i for i ≥ j. If e(i) = ∅ for i < j let π( f ) = [e f ] ∈ C. Observe that π : D → C is a projection of forcing posets in the sense that if D/G := { f ∈ D : π( f ) ∈ G} and H is D/G-generic, then there is a D-generic K such that V [G * H ] = V [K ].
If H is D/G-generic, it follows from Fact 7 that V [G * H ] | "S is stationary in ν". Then, since D is κ j -directed closed and λ < κ j , Fact 6 implies that V [G * H ] | "λ is supercompact. And so Fact 5 tells us that V [G * H ] | "S reflects". Reflection is downward absolute, so V [G] | "S reflects".
In this paper we provided a poset S(κ, < λ) that adds non-reflecting stationary subsets of κ + in any cofinality, and then a poset C that adds non-reflecting stationary subsets of κ + in the critical cofinality cf κ but not necessarily in cofinalities less than cf κ. But what about cofinalities greater than cf κ? We close with a question.

Question 1
If the CMB poset is defined in terms of a singular strong limit κ with cf κ = μ, then does it add non-reflecting stationary subsets of κ + ∩ cof(τ ) for τ > μ?