SET MAPPING REFLECTION

JUSTIN TATCH MOORE

Abstract. In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2^{ω} = ω_{2} and that L(𝒫(ω_{1})) satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that □(κ) fails for all regular κ > ω_{1}.

1. Introduction

The notion of properness was introduced by Shelah in [14] and is a weakening of both the countable chain condition and the property of being countably closed. Its purpose was to provide a property of forcing notions which implies that they preserve ω_{1} and which is preserved under countable support iterations. With the help of a supercompact cardinal, one can prove the consistency of the following statement (see [6]).

PFA: If 𝒫 is a proper forcing notion and 𝒟 is a family of dense subsets of 𝒫 of size ω_{1}, then there is a filter G ⊆ 𝒫 which meets every element of 𝒟.

The Proper Forcing Axiom (PFA) is therefore a strengthening of the better known and less technical MA_{ω_{1}}[16]. It has been extremely useful, together with the stronger Martin’s Maximum (MM) [7], in resolving questions left unresolved by Martin’s Axiom.

Early on it was known that it was not possible to replace ω_{1} by ω_{2} and get a consistent statement (see [3]). The stronger forcing axiom MM was known to already imply that the continuum is ω_{2} [7]. Later Todorcevic and Velickovic showed that PFA also implies that the continuum is ω_{2} (see [23] and [26]). This proof used a deep analysis of the gap structure of ω^{ω}/fin [22] and of the behavior of the oscillation map [21]. Their proof, however, was less generous than some of the proofs that the continuum was ω_{2} from Martin’s Maximum. In particular, while MM implies that L(𝒫(ω_{1})) satisfies AC [27], the same was not known for PFA (compare to the final remark section 3 of [26]).

2000 Mathematics Subject Classification. 03E05, 03E10, 03E47, 03E65.

Key words and phrases. BPFA, continuum, MRP, PFA, reflection, square, definable well ordering.

This paper served as a cornerstone for the proposal for my NSF grant DMS– 0401893; revisions were made to the paper while I was supported by this grant.

MM was also shown to have a variety of large cardinal consequences. Many of these are laid out in [7]. Much of this was proved via stationary reflection principles which seemed to typify the consequences of MM which do not follow from PFA. Todorcevic showed that PFA implies that the combinatorial principle □(κ) fails for all κ > ω_{1} [19]. This, combined with modern techniques in inner model theory [12], gives a considerable lower bound on the consistency strength of PFA.^{1} Both the impact on the continuum and the large cardinal strength of these forcing axioms have figured prominently in their development.

The purpose of this note is to introduce a new reflection principle, MRP, which follows from PFA. The reasons are threefold. First, this axiom arose as a somewhat natural abstraction of one its consequences which in turn implies that there is a well ordering of ℝ which is Σ_{1}- definable over (H(ω_{2}), ∈). A corollary of the proof will be that the Bounded Proper Forcing Axiom implies that there is such a well ordering of ℝ, thus answering a question from [8].

Second, this principle seems quite relevant in studying consequences of PFA which do not follow from the ω-Proper Forcing Axiom. The notion of ω-properness was introduced by Shelah in the course of studying preservation theorems for not adding reals in countable support iterations (see [15]). For our purpose it is sufficient to know that both c.c.c. and countably closed forcings are ω-proper and that ω-proper forcings are preserved under countable support iterations. While I am not aware of the ω-PFA having been studied in the literature, nearly all of the studied consequences of PFA are actually consequences of the weaker ω-PFA.^{2} It is my hope and optimism that the Mapping Reflection Principle will be useful tool in studying the consequences of PFA which do not follow from the ω-PFA in much the same way that the Strong Reflection Principle has succeeded in implying the typical consequences of MM which do not follow from PFA [17] (see also [5]).

Finally, like the Open Coloring Axiom, the Ramsey theoretic formulation of Martin’s Axiom, and the Strong Reflection Principle, this principle can be taken as a black box and used without knowledge of forcing. The arguments using it tend to be rather elementary in nature and require only some knowledge of the combinatorics of the club filter on [X]^{ω} and Löwenheim-Skolem arguments.

1It is not known if “□(κ) fails for all regular κ > ω_{1}” is equiconsistent with the existence of a supercompact cardinal.

2For example: MA_{ω_{1}}, the non-existence of S-spaces [18], all ω_{1}-dense sets of reals are isomorphic [4], the Open Coloring Axiom [22], the failure of □(κ) for all regular κ > ω_{1} [19], the non-existence of Kurepa trees [3].

The main results of this note are summarized as follows.

Theorem 1.1. The Proper Forcing Axiom implies the Mapping Reflection Principle.

Theorem 1.2. The Mapping Reflection Principle implies that 2^{ω} = 2^{ω_{1}} = ω_{2} and that L(𝒫(ω_{1})) satisfies the Axiom of Choice.

Theorem 1.3. The Bounded Proper Forcing Axiom implies that 2^{ω} = ω_{2} and that L(𝒫(ω_{1})) satisfies the Axiom of Choice.

Theorem 1.4. The Mapping Reflection Principle implies that □(κ) fails for every regular κ > ω_{1}.

The notation used in this paper is more or less standard. If θ is a regular cardinal, then H(θ) is the collection of all sets of hereditary cardinality less than θ. As is common, when I refer to H(θ) as a structure I will actually mean (H(θ), ∈, ◁) where ◁ is some well order of H(θ) which can be used to compute Skolem functions and hence generate the club E ⊆ [H(θ)]^{ω} of countable elementary submodels of H(θ). If X is a set of ordinals, then otp(X) represents the ordertype of (X, ∈) and π_{X} is the unique collapsing isomorphism from X to otp(X). While an attempt has been made to keep parts of this paper self contained, a knowledge of proper forcing is assumed in Section 3. The reader is referred to [3], [15], and [22] for more reading on proper forcing and PFA. Throughout the paper the reader is assumed to have a familiarity with set theory ([9] and [11] are standard references).

2. The Mapping Reflection Principle

The following definition will be central to our discussion. Recall that for an uncountable set X, [X]^{ω} is the collection of all countable subsets of X.

Definition 2.1. Let X be an uncountable set, M be a countable elementary submodel of H(θ) for some regular θ such that [X]^{ω} ∈ M. A subset Σ of [X]^{ω} is M-stationary if whenever E ⊆ [X]^{ω} is a club in M there is an N in E ∩ Σ ∩ M.

Example 2.2. If M is a countable elementary submodel of H(ω_{2}) and A ⊆ M ∩ ω_{1} has order type less than δ = M ∩ ω_{1}, then δ \ A is M-stationary.

The set [X]^{ω} is equipped with the Ellentuck topology obtained by declaring the sets

[x, N] = {Y ∈ [X]^{ω} : x ⊆ Y ⊆ N}

to be open for all N in [X]^{ω} and finite x ⊆ N. In this paper “open” will always refer to this topology. It should be noted that the sets which are closed in the Ellentuck topology and cofinal in the order structure generate the closed unbounded filter on [X]^{ω}.

For ease of reading I will make the following definition.

Definition 2.3. A set mapping Σ is said to be open stationary if, for some uncountable set X and regular cardinal θ with X in H(θ), it is the case that elements of the domain of Σ are elementary submodels of H(θ) which contain M and Σ(M) ⊆ [X]^{ω} is open and M-stationary for all M in the domain of Σ. If necessary, the parameters X and θ will be referred to as X_{Σ} and θ_{Σ}.

The following is among the simplest example of an open stationary set mapping.

Example 2.4. Let r : ω_{1} → ω_{1} be regressive on the limit ordinals. If Σ is defined by putting Σ(N) = [{r(δ)}, δ] for N a countable elementary submodel of H(ω_{2}), then Σ is open and stationary.

This motivates the following reflection principle which asserts that this example is present inside any open stationary set mapping.

MRP: If Σ is an open stationary set mapping whose domain is a club, then there is a continuous ∈-chain ⟨N_{ν} : ν < ω_{1}⟩ in the domain of Σ such that for all limit 0 < ν < ω_{1} there is a ν_{0} < ν such that N_{ξ} ∩ X_{Σ} ∈ Σ(N_{ν}) whenever ν_{0} is in N_{ξ} and ξ < ν.

If ⟨N_{ν} : ν < ω_{1}⟩ satisfies the conclusion of MRP for Σ, then it is said to be a reflecting sequence for Σ. We now continue with the first example.

Example 2.5. An immediate consequence of MRP is that if C_{δ} is a cofinal ω-sequence in δ for each countable limit ordinal δ, then there is a club E ⊆ ω_{1} such that E ∩ C_{δ} is finite for all δ. To see this, let C_{δ} (δ < ω_{1}) be given, and if N is a countable elementary submodel of H(ω_{2}), let Σ(N) be the complement of C_{δ} ∪ {δ}. It is easily checked that Σ is an open-stationary set mapping and that if ⟨N_{ν} : ν < ω_{1}⟩ is a reflecting sequence for Σ, then E = {ν < ω_{1} : N_{ν} ∩ ω_{1} = ν} is closed unbounded set with the desired properties.^{3}

3It is easy to verify, however, that this consequence of MRP can not be forced with an ω-proper forcing over a model of ♣. That this statement follows from PFA appears in [15]. It is essentially the only example in the literature that I am aware of which is a combinatorial consequence of PFA but not of ω-PFA.

3. PFA implies MRP

The purpose of this section is to prove the following theorem. Recall that a forcing notion 𝒫 is proper if whenever M is a countable elementary submodel of H(|2^{𝒫} |^{+}) containing 𝒫 and p is in 𝒫 ∩ M, there is a p̄ ≤ p which is (M, 𝒫)-generic.^{4}

Theorem 3.1. PFA implies MRP.

Proof. Let Σ be a given open stationary set mapping defined on a club of models and abbreviate X = X_{Σ} and θ = θ_{Σ}. Let 𝒫_{Σ} denote the collection of all continuous ∈-increasing maps p : α+1 → dom(Σ) such that for all 0 < ν ≤ α there is a ν_{0} < ν with p(ξ) ∩ X ∈ Σ(p(ν)) whenever ν_{0} is in p(ξ) and ξ < ν. 𝒫_{Σ} is ordered by extension. I will now prove that 𝒫_{Σ} is proper. Notice that if this is the case, then the sets 𝒟_{α} = {p ∈ 𝒫_{Σ} : α ∈ dom(p)} must be dense. This is because 𝒟_{x}^{∗} = {p ∈ 𝒫_{Σ} : ∃ν ∈ dom(p)(x ∈ p(ν))} is clearly dense for all x in X and therefore, after forcing with 𝒫_{Σ}, there is always a surjection from {α : ∃p ∈ G(α ∈ dom(p))} onto the uncountable set X.

To see that 𝒫_{Σ} is proper, let p be in 𝒫_{Σ} and M be an elementary submodel of H(λ) for regular λ sufficiently large such that Σ, 𝒫_{Σ}, p, and H(|𝒫_{Σ} |^{+}) are all in M. Let {D_{i} : i < ω} enumerate the dense subsets of 𝒫_{Σ} which are in M. We will now build a sequence of conditions p_{0} ≥ p_{1} ≥ … by recursion. Set p_{0} = p and let p_{i} be given. Let E_{i} be the collection of all intersections of the form N = N^{∗} ∩ X where N^{∗} is a countable elementary submodel of H(|𝒫_{Σ} |^{+}) containing H(θ), D_{i}, 𝒫_{Σ}, and p_{i} as elements. Then E_{i} ⊆ [X]^{ω} is a club in M. Since Σ(M ∩ H(θ)) is open and M ∩ H(θ)-stationary, there is an N_{i} in E_{i} ∩ Σ(M ∩ H(θ)) ∩ M and an x_{i} in [N_{i}]^{<ω} such that [x_{i}, N_{i}] ⊆ Σ(M ∩ H(θ)). Extend p_{i} to

q_{i} = p_{i} ∪ {(ζ_{i} + 1, hull({p_{i}} ∪ x_{i}))}

where ζ_{i} is the last element of the domain of p_{i} and hull({p_{i}} ∪ x_{i}) is the Skolem hull taken in H(θ). The function q_{i} : ζ_{i} +1 → dom(Σ) is ∈- increasing since it extends p_{i} by a single entry which, by arrangement, contains the range of p_{i} as a subset. Since p_{i} has successor length and the domain of q_{i} contains no new limit ordinals, q_{i} is continuous and is in 𝒫_{Σ}.

Notice that q_{i} is in N_{i}^{∗} since N_{i}^{∗} contains p_{i} and H(θ). Working in N_{i}^{∗}, find an extension p_{i+1} of q_{i} which is in N_{i}^{∗} ∩ D_{i}. The key observation here is that everything in the range of p_{i+1} which is not in the range of p_{i} has an intersection with X which is in the interval [x_{i}, N_{i}] and therefore in Σ(M ∩ H(θ)) by the virtue x_{i}, N_{i} witnessing that Σ(M ∩ H(θ)) is a neighborhood of N_{i}. Define p_{∞}(ξ) = p_{i}(ξ) if ξ ≤ ζ_{i} and p_{∞}(sup_{i} ζ_{i}) = M ∩ H(θ). It is easily checked that p_{∞} is well defined and that p_{∞} is a condition which is moreover (M, 𝒫_{Σ})- generic. □

4Here condition p̄ is (M, 𝒫)-generic if whenever r is an extension of p̄ and D ⊆ 𝒫 is a predense set in M, there is a s in D ∩ M such that s is compatible with r.

4. MRP and the continuum.

In this section we will see that MRP can be used to code reals in a way which is somewhat reminiscent of the SRP style coding methods (e.g. ϕ_{AC}, ψ_{AC} of [27]). Before we begin, we first need to introduce some notation.

Fix a sequence ⟨C_{ξ} : ξ ∈ lim(ω_{1})⟩ such that C_{ξ} is a cofinal subset of ξ of ordertype ω for each limit ξ < ω_{1}. Let N, M be countable sets of ordinals such that N ⊆ M, otp(M) is a limit, and sup(N) < sup(M). Define

w(N, M) = | sup(N) ∩ π_{M}^{−1}[C_{α}]|

where α is the ordertype of M. A trivial but important observation is that w is left monotonic in the sense that w(N_{1}, M) ≤ w(N_{2}, M) whenever N_{1} ⊆ N_{2} ⊆ M and sup(N_{2}) < sup(M). Also, if N ⊆ M are countable sets of ordinals with sup(N) < sup(M) and ι is an order preserving map from M into the ordinals, then w(N, M) = w(ι′′N, ι′′M).

We will now consider the following statement about a given subset A of ω_{1}:

υ_{AC}(A): There is an uncountable δ < ω_{2} and an increasing sequence ⟨N_{ξ} : ξ < ω_{1}⟩ which is club in [δ]^{ω} such that for all limit ν < ω_{1} there is a ν_{0} < ν such that if ν_{0} < ξ < ν, then N_{ν} ∩ ω_{1} ∈ A is equivalent to w(N_{ξ} ∩ ω_{1}, N_{ν} ∩ ω_{1}) < w(N_{ξ}, N_{ν}).

The formula υ_{AC} is the assertion that υ_{AC}(A) holds for all A ⊆ ω_{1}.

Proposition 4.1. υ_{AC} implies that in L(𝒫(ω_{1})) there is a well ordering of 𝒫(ω_{1})/NS in length ω_{2}. In particular υ_{AC} implies both 2^{ω_{1}} = ω_{2} and that L(𝒫(ω_{1})) satisfies the Axiom of Choice.

Proof. For each [A] in 𝒫(ω_{1})/NS let δ_{[A]} be the least uncountable δ < ω_{2} such that there is a club N_{ξ} (ξ < ω_{1}) in [δ]^{ω} such that for all limit ν < ω_{1} there is a ν_{0} < ν such that for all ν_{0} < ξ < ν we have that

N_{ν} ∩ ω_{1} ∈ A iff w(N_{ξ} ∩ ω_{1}, N_{ν} ∩ ω_{1}) < w(N_{ξ}, N_{ν}).

It is easily checked that this definition is independent of the choice of representative of [A]. It is now sufficient to show that if A and B are subsets of ω_{1} and δ_{[A]} = δ_{[B]}, then [A] = [B]. To this end suppose that δ < ω_{2} is an uncountable ordinal such that for clubs N_{ξ}^{A} (ξ < ω_{1}) and N_{ξ}^{B} (ξ < ω_{1}) in [δ]^{ω} we have for every limit ν < ω_{1} there is a ν_{0} < ν such that for all ν_{0} < ξ < ν we have both

w(N_{ξ}^{A} ∩ ω_{1}, N_{ν}^{A} ∩ ω_{1}) < w(N_{ξ}^{A}, N_{ν}^{A}) iff ξ ∈ A
w(N_{ξ}^{B} ∩ ω_{1}, N_{ν}^{B} ∩ ω_{1}) < w(N_{ξ}^{B}, N_{ν}^{B}) iff ξ ∈ B.

Now there is a closed unbounded set C ⊆ ω_{1} such that if ξ is in C, then N_{ξ}^{A} = N_{ξ}^{B}. It is easily seen that if ν is a limit point of C, then ν is in A iff ν is in B. □

Proposition 4.2. υ_{AC} implies that 2^{ω} = 2^{ω_{1}}.

Proof. This is virtually identical to the proof that the statement θ_{AC} of Todorcevic implies 2^{ω} = 2^{ω_{1}} as proved in [25]. □

The reason for formulating υ_{AC} is that it is a consequence of MRP.

Theorem 4.3. MRP implies υ_{AC}.

This is a consequence of the following fact.

Lemma 4.4. If M is a countable elementary submodel of H((2^{ω_{1}})^{+}), then the following sets are open and M-stationary:

Σ_{<}(M) = {N ∈ M ∩ [ω_{2}]^{ω} : w(N ∩ ω_{1}, M ∩ ω_{1}) < w(N, M ∩ ω_{2})}

Σ_{≥}(M) = {N ∈ M ∩ [ω_{2}]^{ω} : w(N ∩ ω_{1}, M ∩ ω_{1}) ≥ w(N, M ∩ ω_{2})}.

To see how to prove Theorem 4.3 from the lemma, define

Σ_{A}(M) = \{ \begin{matrix}Σ_{<}(M) & if M ∩ ω_{1} ∈ A \\ Σ_{≥}(M) & if M ∩ ω_{1} ∉ A. \end{matrix}

Now let N_{ξ}^{∗} (ξ < ω_{1}) be a reflecting sequence for Σ_{A}. Let

δ = ⋃_{ξ<ω_{1}} N_{ξ} ∩ ω_{2}

N_{ξ} = N_{ξ}^{∗} ∩ ω_{2}.

It is easily verified that δ is an ordinal, taken together with ⟨N_{ξ} : ξ < ω_{1}⟩, satisfies the conclusion of υ_{AC}(A). We will now return our attention to the proof of the lemma.

Proof. To see that Σ_{<}(M) is M-stationary, let E ⊆ [ω_{2}]^{ω} be a club in M. By the pigeonhole principle, there is a γ < ω_{1} such that

{sup(N) : N ∈ E and N ∩ ω_{1} ⊆ γ}

is unbounded in ω_{2}. By elementarity of M there is a γ < M ∩ ω_{1} such that

{sup(N) : N ∈ E ∩ M and N ∩ ω_{1} ⊆ γ}

is unbounded in M ∩ ω_{2}. Pick an N in E ∩ M such that

w(N ∩ ω_{2}, M ∩ ω_{2}) = | sup(N) ∩ π_{M∩ω_{2}}^{−1}[C_{otp(M∩ω_{2})}]| > |C_{M∩ω_{1}} ∩ γ|.

Since N is in M and N is countable, N ⊆ M and sup(N) < sup(M). It now follows from the definition of Σ_{<}(M) that N is in E ∩Σ_{<}(M)∩M.

To see that Σ_{<}(M) is open, let N be in Σ_{<}(M). If N does not have a last element, let ξ be the least element of N greater than

max(sup(N) ∩ π_{M∩ω_{2}}^{−1}[C_{otp(M∩ω_{2})}]).

If N has a greatest element, set ξ = max(N). Define x to be the finite set (N ∩ C_{M∩ω_{1}}) ∪ {ξ}. It is easy to see that

w(x ∩ ω_{1}, M ∩ ω_{1}) = w(N ∩ ω_{1}, M ∩ ω_{1})
w(x ∩ ω_{2}, M ∩ ω_{2}) = w(N ∩ ω_{2}, M ∩ ω_{2})

and hence by left monotonicity of w we have that [x, N] ⊆ Σ_{<}(M).

In order to see that Σ_{≥}(M) is M-stationary, let E ⊆ [ω_{2}]^{ω} be a club in M and let γ < ω_{2} be uncountable such that E ∩ [γ]^{ω} is a club in [γ]^{ω}. By elementarity of M, such a γ can be found in M. Working in M it is possible to find an N in E ∩ [γ]^{ω} such that

|N ∩ ω_{1} ∩ C_{M∩ω_{1}}| ≥ |γ ∩ π_{M∩ω_{2}}^{−1}[C_{otp(M∩ω_{2})}]|

and γ ∩ π_{M∩ω_{2}}^{−1}[C_{otp(M∩ω_{2})}] ⊆ N. Then N is in E ∩ Σ_{≥}(M) ∩ M. The proof that Σ_{≥}(M) is open is similar to the corresponding proof for Σ_{<}(M). □

5. The Bounded Proper Forcing Axiom and the continuum

In this section we will see that the Bounded Proper Forcing Axiom implies υ_{AC}. Before arguing this, I will first give a little context to the result. The Bounded Proper Forcing Axiom is equivalent to the assertion that

(H(ω_{2}), ∈) ≺_{Σ_{1}} V^{𝒫}

for every proper forcing 𝒫. That is if ϕ is a Σ_{1}-formula with a parameter in H(ω_{2}), then ϕ is true iff it can be forced to be true by some proper forcing. The original statement of BPFA is due to Goldstern and Shelah [8] and is somewhat different, though equivalent. The above formulation and its equivalence to the original is due to Bagaria [2]. The consistency strength of BPFA is much weaker than that of PFA — it is exactly that of a reflecting cardinal^{5} [8] [24]. It should be noted though that many of the consequences of PFA — MA_{ω_{1}}, the non-existence of S-space and Kurepa trees, the assertion that all ω_{1}- dense sets of reals are order isomorphic — are actually consequences of BPFA.

5For instance, if κ is Mahlo, then V_{κ} satisfies that there is a proper class of reflecting cardinals.

Recently there has been a considerable amount of work on bounded forcing axioms and well orderings of the continuum. Woodin was the first to give such a proof from the assumption of Bounded Martin’s Maximum and “there is a measurable cardinal” [27]. The question of whether Bounded Martin’s Maximum alone sufficed remained an intriguing question. In light of Woodin’s result, a reasonable approach was to show that BMM has considerable large cardinal strength and use this to synthesize the role of the measurable cardinal. Asperó showed that BMM implies the dominating number 𝔡 is ω_{2} [1]. Soon after, Todorcevic showed that BMM implies that 𝔠 = ω_{2} and that, moreover, there is a well ordering of ℝ which is Σ_{1}-definable from an ω_{1}-sequence of reals [25]. Very recently Schindler showed that BMM does have considerable consistency strength[13]. It should be noted, however, that it is still unclear whether the measurable cardinal can be removed from Woodin’s argument.

Now we will see that the Bounded Proper Forcing Axiom is already sufficient to give a definable well ordering of ℝ from parameters in H(ω_{2}). Notice that for a fixed A, υ_{AC}(A) is a Σ_{1}-sentence which takes the additional parameter ⟨C_{ξ} : ξ ∈ lim(ω_{1})⟩ (in order to define w). Further examination of the above proof reveals that for each A, there is a proper forcing which forces υ_{AC}(A). Hence the Bounded Proper Forcing Axiom implies υ_{AC}. Asperó pointed out that, unlike statements such as ψ_{AC} and ϕ_{AC}, the statement υ_{AC} can be forced over any model with an inaccessible cardinal.

6. MRP and □(κ)

Recall the following combinatorial principle — defined for a regular κ > ω_{1} — which was introduced by Todorcevic in [20]:^{6}

□(κ): There is a sequence ⟨C_{α} : α < κ⟩ such that:

(1) C_{α+1} = {α} and C_{α} ⊆ α is closed and cofinal if α is a limit ordinal.

(2) If α is a limit point of C_{β}, then C_{α} = C_{β} ∩ α.

(3) There is no club C ⊆ κ such that for all limit points α in

C the equality C_{α} = C ∩ α holds.

In this section we will prove Theorem 1.4 — that MRP implies that □(κ) fails for all regular κ > ω_{1}. The reason for proving such a result is that it shows MRP has considerable consistency strength. The essence of Theorem 1.4 is contained in the following lemma.

6This seems to be the first place □(κ) was defined explicitly. A similar principle □_{κ} was defined and studied by Jensen in [10].

Lemma 6.1. If M is a countable elementary submodel of H(κ^{+}) containing a □(κ)-sequence ⟨C_{α} : α < κ⟩, then the set Σ(M) of all N ⊆ M ∩ κ such that sup(N) is not in C_{sup(M∩κ)} is open and Mstationary.

Remark 6.2. Given the lemma, let N_{ν} (ν < ω_{1}) be a reflecting sequence for Σ and set E = {sup(N_{ν} ∩ κ) : ν < ω_{1}}. Then E is closed and of order type ω_{1}. Let β be the supremum of E. Now there must be a limit point α in E ∩ C_{β}. Let ν be such that α = sup(N_{ν} ∩ κ). But now there is a ν_{0} < ν such that sup(N_{ξ} ∩ κ) is not in C_{α} = C_{β} ∩ α whenever ν_{0} < ξ < ν. This means that α is not a limit point of E, a contradiction.

Proof. First we will check that Σ(M) is open. To see this, let N be in Σ(M). If N has a last element γ, then [{γ}, N] ⊆ Σ(M). If N does not have a last element, then, since C_{sup(M∩κ)} is closed, there is a γ in N such that if ξ < sup(N) is in C_{sup(M∩κ)}, then ξ < γ. Again [{γ}, N] ⊆ Σ(M).

Now we will verify that Σ(M) is M-stationary. To this end, let E ⊆ [κ]^{ω} be a club in M. Let S be the collection of all sup(N) such that N is in E. Clearly S has cofinally many limit points in κ. If S ∩M is contained in C_{sup(M∩κ)}, then we have that whenever α < β are limit points in S ∩ M,

C_{α} = C_{sup(M∩κ)} ∩ α
C_{β} = C_{sup(M∩κ)} ∩ β

and hence C_{α} = C_{β} ∩ α. But, by elementarity of M, this means that for all limit points α < β in S, C_{α} = C_{β} ∩ α. This would in turn imply that the union C of C_{α} for α a limit point of S is a closed unbounded set such that C_{α} = C ∩ α for all limit points α of C, contradicting the definition of ⟨C_{α} : α < κ⟩. Hence there is an N in E such that sup(N) is not in C_{sup(M∩κ)}. □

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