Square principle

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In mathematical set theory, the global square principle is a combinatorial principle introduced by Ronald Jensen in his analysis of the fine structure of the constructible universe L.

Definition[edit]

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 cardinal[edit]

Jensen introduced also a local version of the principle.[1] 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.

Notes[edit]

  1. ^ 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.