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Forcing Closed Unbounded Subsets of א ω1+1

  • M. C. Stanley (a1)

Abstract

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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[1] Beller, A., Jensen, R. B., and Welch, P., Coding the universe, London Mathematical Society Lecture Note Series 47, Cambridge University Press, 1982.
[2] Cummings, J., Foreman, M., and Magidor, M., Canonical, structure in the universe of, set theory: part two, Annals of Pure and Applied Logic, vol. 142 (2006), pp. 5575.
[3] Devlin, K. J., Constructibility, Springer-Verlag, 1984.
[4] Shelah, S., Cardinal arithmetic, Oxford Science Publications, 1994.
[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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