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.<ref>{{Citation | last1=Cummings | first1=James | title=Notes on Singular Cardinal Combinatorics | journal=Notre Dame Journal of Formal Logic | volume=46 | number=3 | year=2005 | pages=251–282| doi=10.1305/ndjfl/1125409326 | doi-access=free }} Section 4.</ref> 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 <math>(C_\beta)_{\beta \in \mathrm{Sing}}</math> satisfying:
# <math>C_\beta</math> is a club set of <math>\beta</math>. # ot<math>(C_\beta) < \beta </math> # If <math>\gamma</math> is a limit point of <math>C_\beta</math> then <math>\gamma \in \mathrm{Sing}</math> and <math>C_\gamma = C_\beta \cap \gamma</math>
== Construction of <math>\kappa</math>-Suslin trees == In the proof of construction of <math>\kappa</math> or <math>\kappa^+</math>-Suslin trees in L, one might want to construct said tree purely via recursion on the levels. On a stationary set of levels, we must have that all antichains must be "killed off", but at a limit stage <math>\alpha</math> later in the construction, we might have <math>T \restriction \alpha</math> "resemble" being Aronszajn. To counteract this, we can use <math>\Box_\kappa</math>, which allows us to split up the construction of the tree into two cases. At some stages, we might kill off some antichains using <math>\Diamond</math>, but at later stages (such as <math>\alpha</math> in the example), <math>\Box_\kappa</math> is used to refine the construction.<ref>{{Cite book |last=Devlin |first=Keith |title=Constructibility |publisher=Springer-Verlag |isbn=9780387132587 |edition=1st |publication-date=July 16, 1984 |language=EN}}</ref>
==Variant relative to a cardinal== Jensen introduced also a local version of the principle.<ref>{{Citation | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory: Third Millennium Edition | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003}}, p. 443.</ref> If <math>\kappa</math> is an uncountable cardinal, then <math>\Box_\kappa</math> asserts that there is a sequence <math>(C_\beta\mid\beta \text{ a limit point of }\kappa^+)</math> satisfying:
# <math>C_\beta</math> is a club set of <math>\beta</math>. # If <math> cf \beta < \kappa </math>, then <math>|C_\beta| < \kappa </math> # If <math>\gamma</math> is a limit point of <math>C_\beta</math> then <math>C_\gamma = C_\beta \cap \gamma</math>
Jensen proved that this principle holds in the constructible universe for any uncountable cardinal <math>\kappa</math>.
==Notes== {{Reflist}} {{refbegin}} *{{Citation | last1=Jensen | first1=R. Björn | title=The fine structure of the constructible hierarchy | doi=10.1016/0003-4843(72)90001-0 | mr=0309729 | year=1972 | journal=Annals of Mathematical Logic | volume=4 | issue=3 | pages=229–308| doi-access=free }} {{refend}}
Category:Set theory Category:Constructible universe
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