Square principle

Define Sing to be the class of all limit ordinals which are not regular. 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 ${\displaystyle C_{\gamma }=C_{\beta }\cap \gamma }$

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.

• Jensen, R. Björn (1972), "The fine structure of the constructible hierarchy", Annals of Mathematical Logic, 4 (3): 229–308, doi:10.1016/0003-4843(72)90001-0, MR 0309729