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Square principle

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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.

Definition

Define Sing to be the class of all limit ordinals which are not regular. Global square states that there is a system ( C β ) β ∈ S i n g {\displaystyle (C_{\beta })_{\beta \in \mathrm {Sing} }} satisfying:

  1. C β {\displaystyle C_{\beta }} is a club set of β {\displaystyle \beta } .
  2. ot ( C β ) < β {\displaystyle (C_{\beta })<\beta }
  3. If γ {\displaystyle \gamma } is a limit point of C β {\displaystyle C_{\beta }} then γ ∈ S i n g {\displaystyle \gamma \in \mathrm {Sing} } and C γ = C β ∩ γ {\displaystyle C_{\gamma }=C_{\beta }\cap \gamma }

Variant relative to a cardinal

Jensen introduced also a local version of the principle. If κ {\displaystyle \kappa } is an uncountable cardinal, then ◻ κ {\displaystyle \Box _{\kappa }} asserts that there is a sequence ( C β ∣ β  a limit point of  κ + ) {\displaystyle (C_{\beta }\mid \beta {\text{ a limit point of }}\kappa ^{+})} satisfying:

  1. C β {\displaystyle C_{\beta }} is a club set of β {\displaystyle \beta } .
  2. If c f β < κ {\displaystyle cf\beta <\kappa } , then | C β | < κ {\displaystyle |C_{\beta }|<\kappa }
  3. If γ {\displaystyle \gamma } is a limit point of C β {\displaystyle C_{\beta }} then C γ = C β ∩ γ {\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.


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