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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 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.² 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 {\displaystyle \kappa } .

Notes

• Jensen, R. Björn (1972), "The fine structure of the constructible hierarchy", Annals of Mathematical Logic, 4 (3): 229–308, doi:10.1016/0003-4843(72)90001-0, MR 0309729

References

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. /wiki/Doi_(identifier) ↩

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. 978-3-540-44085-7 ↩