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Milds # 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. 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 $(C_{\beta })_{\beta \in \mathrm {Sing} }$ 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.

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