Global square

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Global square is an important concept in set theory, a branch of mathematics. It has been introduced by Ronald Jensen in his analysis of the fine structure of the constructible universe L. According to Ernest Schimmerling and Martin Zeman, Jensen's square principle and its variants are ubiquitous in set theory.^[1]

Definition

Define Sing to be the class of all limit ordinals which are not regular. Global square states that there is a system $(C_{\beta })_{{\beta \in Sing}}$ satisfying:

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

Variant relative to a cardinal

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

Notes

↑ Ernest Schimmerling and Martin Zeman, Square in Core Models, The Bulletin of Symbolic Logic, Volume 7, Number 3, Sept. 2001
↑ 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.

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