Definition 1.1. Suppose that λ ≤ κ are cardinals and Γ is a subset of (κ, κ+). By , we mean the principle asserting that there is a sequence 〈Fν | ν ∈ lim(Γ)〉 such that for every ν ∈ lim(Γ), the following hold.(1) 1 ≤ card(Fν) < λ.(2) The following hold for every C ∈ Fν.(a) C ⊆ ν ∩ Γ,(b) C is club in ν,(c) o.t.(C) ≤ κ,By we mean . If Γ = (κ, κ+), then we write for and for .These weak square principles were introduced in [Sch2, 5.1]. They generalize Jensen's principles □κ and , which are equivalent to and respectively. Jensen's global □ principle implies □κ for all κ.Theorem 1.2. Suppose that is a core model. Assume that every countable premouse M which elementarily embeds into a level of is (ω1 + 1)-iterable. Then, for every κ, holds in .The minimal non-1-small mouse is essentially a sharp for an inner model with a Woodin cardinal. We originally proved Theorem 1.2 under the assumption that is 1-small, building on [MiSt] and [Sch2]. Some generalizations followed by combining our methods with those of [St2] and [SchSt2]. (For example, the tame countably certified core model Kc satisfies .) In order to eliminate the smallness assumption all together, one replaces our use of the Dodd-Jensen lemma in proofs of condensation properties for with the weak Dodd-Jensen lemma of [NSt].
Weak square sequences and special Aronszajn trees