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Forcing Closed Unbounded Subsets of א ω1+1

  • M. C. Stanley (a1)


Using square sequences, a stationary subset ST of א ω1+1 is constructed from a tree T of height ω 1, uniformly in T. Under suitable hypotheses, adding a closed unbounded subset to ST requires adding a cofinal branch to T or collapsing at least one of ω 1, א ω1, and א ω1+1. An application is that in ZFC there is no parameter free definition of the family of subsets of א ω1+1 that have a closed unbounded subset in some ω 1, א ω1, and א ω1+1 preserving outer model.



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[5] Stanley, M. C., Forcing closed unbounded subsets of, Sets and proofs (Cooper, S. B. and Truss, J. K., editors), London Mathematical Society Lecture Note Series, vol. 258, Cambridge University Press, 1999, pp. 365382.
[6] Stanley, M. C., Forcing closedunboundedsubsets of ω 2 , Annals of Pure and Applied Logic, vol. 110 (2001), pp. 2387.



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