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.[1] They were introduced by Ronald Jensen in his analysis of the fine structure of the constructible universe L.

DefinitionEdit

Define Sing to be the class of all limit ordinals which are not regular. Global square states that there is a system   satisfying:

  1.   is a club set of  .
  2. ot 
  3. If   is a limit point of   then   and  

Variant relative to a cardinalEdit

Jensen introduced also a local version of the principle.[2] If   is an uncountable cardinal, then   asserts that there is a sequence   satisfying:

  1.   is a club set of  .
  2. If  , then  
  3. If   is a limit point of   then  

Jensen proved that this principle holds in the constructible universe for any uncountable cardinal κ.

NotesEdit

  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.