Global square

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 Ernest Schimmerling and Martin Zeman, Jensen's square principle and its variants are ubiquitous in set theory.[1]

Definition[edit]

Define Sing to be the class of all limit ordinals which are not regular. Global square states that there is a system (C_\beta)_{\beta \in Sing} satisfying:

  1. C_\beta is a club set of \beta.
  2. ot(C_\beta) < \beta
  3. If \gamma is a limit point of C_\beta then \gamma \in Sing and C_\gamma = C_\beta \cap \gamma

Variant relative to a cardinal[edit]

Jensen introduced also a local version of the principle.[2] If \kappa is an uncountable cardinal, then \Box_\kappa asserts that there is a sequence (C_\beta|\beta \text{ a limit point of }\kappa^+) satisfying:

  1. C_\beta is a club set of \beta.
  2. If  cf \beta < \kappa , then |C_\beta| < \kappa
  3. If \gamma is a limit point of C_\beta then C_\gamma = C_\beta \cap \gamma

Notes[edit]

  1. ^ Ernest Schimmerling and Martin Zeman, Square in Core Models, The Bulletin of Symbolic Logic, Volume 7, Number 3, Sept. 2001
  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.