Square principle
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In mathematical set theory, a square principle is a combinatorial principle asserting the existence of a cohering sequence of short closed unbounded (club) sets so that no one (long) club set coheres with them all. As such they may be viewed as a kind of incompactness phenomenon. They were introduced by Ronald Jensen in his analysis of the fine structure of the constructible universe L.
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 {\displaystyle (C_{\beta })_{\beta \in \mathrm {Sing} }} satisfying:
- C β {\displaystyle C_{\beta }} is a club set of β {\displaystyle \beta }.
- ot( C β ) < β {\displaystyle (C_{\beta })<\beta }
- If γ {\displaystyle \gamma } is a limit point of C β {\displaystyle C_{\beta }} then γ ∈ S i n g {\displaystyle \gamma \in \mathrm {Sing} } and C γ = C β ∩ γ {\displaystyle C_{\gamma }=C_{\beta }\cap \gamma }
Construction of κ {\displaystyle \kappa } -Suslin trees
In the proof of construction of κ {\displaystyle \kappa } or κ + {\displaystyle \kappa ^{+}}-Suslin trees in L, one might want to construct said tree purely via recursion on the levels. On a stationary set of levels, we must have that all antichains must be "killed off", but at a limit stage α {\displaystyle \alpha } later in the construction, we might have T ↾ α {\displaystyle T\upharpoonright \alpha } "resemble" being Aronszajn. To counteract this, we can use ◻ κ {\displaystyle \Box _{\kappa }}, which allows us to split up the construction of the tree into two cases. At some stages, we might kill off some antichains using ◊ {\displaystyle \Diamond }, but at later stages (such as α {\displaystyle \alpha } in the example), ◻ κ {\displaystyle \Box _{\kappa }} is used to refine the construction.
Variant relative to a cardinal
Jensen introduced also a local version of the principle. If κ {\displaystyle \kappa } is an uncountable cardinal, then ◻ κ {\displaystyle \Box _{\kappa }} asserts that there is a sequence ( C β ∣ β a limit point of κ + ) {\displaystyle (C_{\beta }\mid \beta {\text{ a limit point of }}\kappa ^{+})} satisfying:
- C β {\displaystyle C_{\beta }} is a club set of β {\displaystyle \beta }.
- If c f β < κ {\displaystyle cf\beta <\kappa }, then | C β | < κ {\displaystyle |C_{\beta }|<\kappa }
- If γ {\displaystyle \gamma } is a limit point of C β {\displaystyle C_{\beta }} then C γ = C β ∩ γ {\displaystyle C_{\gamma }=C_{\beta }\cap \gamma }
Jensen proved that this principle holds in the constructible universe for any uncountable cardinal κ {\displaystyle \kappa }.