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| '''Global square''' is an important concept in [[set theory]], a branch of [[mathematics]]. It has been introduced by [[Ronald Jensen]] in his analysis of the fine structure of the [[constructible universe]] '''L'''. According to | | I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. The favorite pastime for him and his kids is style and he'll be beginning some thing else along with it. Alaska is where he's always been residing. She functions as a journey agent but soon she'll be on her personal.<br><br>My page - real psychic - [http://galab-work.Cs.Pusan.ac.kr/Sol09B/?document_srl=1489804 internet site], |
| Ernest Schimmerling and Martin Zeman, ''Jensen's square principle and its variants are ubiquitous in set theory''.<ref>Ernest Schimmerling and Martin Zeman, Square in Core Models, The Bulletin of Symbolic Logic, Volume 7, Number 3, Sept. 2001</ref>
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| ==Definition==
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| Define '''Sing''' to be the [[class (set theory)|class]] of all [[limit ordinal]]s which are not [[regular ordinal|regular]]. ''Global square'' states that there is a system <math>(C_\beta)_{\beta \in Sing}</math> satisfying:
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| # <math>C_\beta</math> is a [[club set]] of <math>\beta</math>.
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| # [[order type|ot]]<math>(C_\beta) < \beta </math>
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| # If <math>\gamma</math> is a limit point of <math>C_\beta</math> then <math>\gamma \in Sing</math> and <math>C_\gamma = C_\beta \cap \gamma</math>
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| ==Variant relative to a cardinal==
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| 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
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| <math>\kappa</math> is an uncountable cardinal,
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| then <math>\Box_\kappa</math> asserts that there is a sequence <math>(C_\beta|\beta \text{ a limit point of }\kappa^+)</math> satisfying:
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| # <math>C_\beta</math> is a [[club set]] of <math>\beta</math>.
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| # If <math> cf \beta < \kappa </math>, then <math>|C_\beta| < \kappa </math>
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| # If <math>\gamma</math> is a limit point of <math>C_\beta</math> then <math>C_\gamma = C_\beta \cap \gamma</math>
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| ==Notes==
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| {{Reflist}}
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| [[Category:Set theory]]
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| {{settheory-stub}}
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