Square principle

Definition

Define Sing to be the class of all limit ordinals which are not regular. Global square states that there is a system ( C β ) β ∈ S i n g satisfying:

1. C β is a club set of β .
2. ot ( C β ) < β
3. If γ is a limit point of C β then γ ∈ S i n g and C γ = C β ∩ γ

Variant relative to a cardinal

Jensen introduced also a local version of the principle. If κ is an uncountable cardinal, then ◻ κ asserts that there is a sequence ( C β ∣ β  a limit point of  κ + ) satisfying:

1. C β is a club set of β .
2. If c f β < κ , then | C β | < κ
3. If γ is a limit point of C β then C γ = C β ∩ γ

Jensen proved that this principle holds in the constructible universe for any uncountable cardinal κ.