Global Square

In mathematical set theory, a square principle is a combinatorial principle asserting the existence of a cohering sequence of
short club set, 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
compactness theorem, 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 (set theory), class of all limit ordinals which are not regular ordinal, regular. ''Global square'' states that there is a system $(C_\beta)_$ satisfying:

# $C_\beta$ is a club set of $\beta$.
# order type, ot$(C_\beta) < \beta$
# If $\gamma$ is a limit point of $C_\beta$ then $\gamma \in \mathrm$ and $C_\gamma = C_\beta \cap \gamma$

Variant relative to a cardinal

Jensen introduced also a local version of the principle., p. 443. If
$\kappa$ is an uncountable cardinal,
then $\Box_\kappa$ asserts that there is a sequence $(C_\beta\mid\beta \text\kappa^+)$ satisfying:

# $C_\beta$ is a club set of $\beta$.
# If $cf \beta < \kappa$, then $, C_\beta, < \kappa$
# 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 $\kappa$.

Notes

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Set theory
Constructible universe

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