Square principle

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.^[1] They were introduced by Ronald Jensen in his analysis of the fine structure of the constructible universe L.

Definition

Define Sing to be the

. Global square states that there is a system ${\displaystyle (C_{\beta })_{\beta \in \mathrm {Sing} }}$ satisfying:

1. ${\displaystyle C_{\beta }}$ is a club set of ${\displaystyle \beta }$.
2. ot${\displaystyle (C_{\beta })<\beta }$
3. If ${\displaystyle \gamma }$ is a limit point of ${\displaystyle C_{\beta }}$ then ${\displaystyle \gamma \in \mathrm {Sing} }$ and ${\displaystyle C_{\gamma }=C_{\beta }\cap \gamma }$

Variant relative to a cardinal

Jensen introduced also a local version of the principle.^[2] If ${\displaystyle \kappa }$ is an uncountable cardinal, then ${\displaystyle \Box _{\kappa }}$ asserts that there is a sequence ${\displaystyle (C_{\beta }\mid \beta {\text{ a limit point of }}\kappa ^{+})}$ satisfying:

1. ${\displaystyle C_{\beta }}$ is a club set of ${\displaystyle \beta }$.
2. If ${\displaystyle cf\beta <\kappa }$, then ${\displaystyle |C_{\beta }|<\kappa }$
3. If ${\displaystyle \gamma }$ is a limit point of ${\displaystyle C_{\beta }}$ then ${\displaystyle C_{\gamma }=C_{\beta }\cap \gamma }$

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

Notes

• Jensen, R. Björn (1972), "The fine structure of the constructible hierarchy", Annals of Mathematical Logic, 4 (3): 229–308,
  MR 0309729

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