Square principle

In mathematical set theory, the global square principle is a combinatorial principle 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.^[1] 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.

Notes

1. Jech, Thomas (2003), Set Theory: Third Millennium Edition, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44085-7, p. 443.

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