# Square principle

(Redirected from Global square)

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 $(C_{\beta })_{\beta \in \mathrm {Sing} }$  satisfying:

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

## Variant relative to a cardinal

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

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

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