### Ladder gaps over stationary sets

by Abraham and Shelah. [AbSh:598]

J Symbolic Logic, 2004

For a stationary set S subseteq omega_1 and a ladder system
C over S, a new type of gaps called C-Hausdorff is introduced
and investigated. We describe a forcing model of ZFC in which, for
some stationary set S, for every ladder C over S, every gap
contains a subgap that is C-Hausdorff. But for every ladder E
over omega_1 setminus S there exists a gap with no subgap that is
E-Hausdorff. A new type of chain condition, called polarized chain
condition, is introduced. We prove that the iteration with finite
support of polarized c.c.c posets is again a polarized c.c.c
poset.

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