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# Square principle

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.[1] They were introduced by Ronald Jensen in his analysis of the fine structure of the constructible universe L.

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## Definition

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 ${\displaystyle C_{\gamma }=C_{\beta }\cap \gamma }$

## Variant relative to a cardinal

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.

## Notes

1. ^ Cummings, James (2005), "Notes on Singular Cardinal Combinatorics", Notre Dame Journal of Formal Logic, 46 (3): 251–282, doi:10.1305/ndjfl/1125409326 Section 4.
2. ^ 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.