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Ladder gaps over stationary sets

Published online by Cambridge University Press:  12 March 2014

Uri Abraham
Affiliation:
Department of Mathematics, Ben-Gurion University, Beér-Sheva, 84105, Israel, E-mail: abraham@math.bgu.ac.il
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel, E-mail: shelah@math.huji.ac.il

Abstract.

For a stationary set S ⊆ ω1, 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 ω1S 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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2004

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References

[1]Hechler, S. H., Short nested sequences in βN ∖ N and small maximal almost disjoint families, General Topology and its Applications, vol. 2 (1972), pp. 139149.CrossRefGoogle Scholar
[2]Martin, D. A. and Solovay, R., Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), pp. 143178.CrossRefGoogle Scholar
[3]Scheepers, M., Gaps in ωω, Set Theory of the Reals, Israel Mathematical Conference Proceedings (Ramat Gan, 1991), vol. 6, Bar-Ilan University, Ramat Gan, 1993, pp. 439561.Google Scholar
[4]Talayaco, D., Applications of cohomology to set theory I. Hausdorff gaps, Annals of Pure and Applied Logic, vol. 71 (1995), no. 1, pp. 69106.CrossRefGoogle Scholar
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