Weak compactness and no partial squares

We present a characterization of weakly compact cardinals in terms of generalized stationarity. We apply this characterization to construct a model with no partial square sequences. One of the striking features of weak compactness in the large cardinal hierarchy is the wide variety of characterizations of the concept. These characterizations meet many disparate areas of set theory, including metamathematics, combinatorics, and elementary embeddings, among others. In this paper we expand on this theme by characterizing weak compactness in terms of generalized stationarity. The property of Mahloness illustrates a way to define or characterize a large cardinal property: specify a naturally defined set, and then assert that the large cardinal property holds iff the set is stationary. For example, a cardinal κ is Mahlo iff the set of strongly inaccessible cardinals below κ is stationary. The first goal of the paper is to characterize weakly compact cardinals along the same lines. Let κ > ω be a regular cardinal. Let S be the set of N in Pκ(H(κ )) satisfying: (1) N∩κ is strongly inaccessible, (2) N<N∩κ ⊆ N , and (3) the transitive collapse of N is a 1-elementary substructure of H((N ∩ κ)). We will prove that κ is weakly compact iff S is stationary in Pκ(H(κ )). The second goal of the paper is to demonstrate how this characterization of weak compactness can be applied in forcing arguments. The property that a forcing poset P is λ-distributive, for a regular cardinal λ, is Π1 over H(δ), where P ∈ H(δ) and λ ≤ δ, and thus this property is preserved under 1-elementarity. This fact will enable us to use the above characterization of weak compactness to prove that a certain forcing iteration is distributive. Specifically, we would like to construct a model in which there are no partial square sequences on any stationary subset of μ ∩ cof(μ), where μ is a regular uncountable cardinal. A partial square sequence on a set A ⊆ μ ∩ cof(μ) is a sequence 〈cα : α ∈ A〉 satisfying that each cα is a club subset of α with order type μ, and if γ is a common limit point of cα and cβ , then cα ∩ γ = cβ ∩ γ. If there is such a sequence with domain A, we say that A carries a partial square. It is known to be consistent that there are no stationary subsets of μ ∩ cof(μ) which carry a partial square sequence. This follows from a strong stationary set reflection property which holds in a model of Magidor [4]. We would like to construct a model with no partial squares more directly. Specifically, we iterate forcing to destroy the stationarity of any subset of μ ∩ cof(μ) which carries a partial square. In general, such an iteration will collapse cardinals. So first we prepare the ground model by Lévy collapsing a weakly compact cardinal κ to become μ. Then we use the above characterization of weak compactness to show that the iteration of club adding is κ-distributive and thus preserves