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Square principle

From Wikipedia, the free encyclopedia

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.

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Transcription

Definition

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

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.

Notes

  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.
This page was last edited on 12 April 2020, at 08:25
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