Square principle


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 club set coheres with them all. As such they may be viewed as a kind of
incompactness phenomenon. They were introduced by Ronald Jensen in his analysis of the fine structure of the constructible universe L.

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

Construction of \kappa-Suslin trees

In the proof of construction of or -Suslin trees in L, one might want to construct said tree purely via recursion on the levels. On a stationary set of levels, we must have that all antichains must be "killed off", but at a limit stage later in the construction, we might have "resemble" being Aronszajn. To counteract this, we can use, which allows us to split up the construction of the tree into two cases. At some stages, we might kill off some antichains using, but at later stages, is used to refine the construction.

Variant relative to a cardinal

Jensen introduced also a local version of the principle. 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.