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Forcing Closed Unbounded Subsets of אω1+1

Published online by Cambridge University Press:  12 August 2016

M. C. Stanley
M. C. Stanley*
Affiliation:
Math Department San Jose State, San Jose, Ca 95192, USA, E-mail: stanley@math.sjsu.edu
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Abstract

Using square sequences, a stationary subset ST of אω1+1 is constructed from a tree T of height ω1, 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 ω1, אω1, and אω1+1. An application is that in ZFC there is no parameter free definition of the family of subsets of אω1+1 that have a closed unbounded subset in some ω1, אω1, and אω1+1 preserving outer model.

Keywords

03E0503E3503E4003E45stationary setclosed unbounded settreesquarestrong coveringpatternforcingclass forcing
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 3 , September 2013 , pp. 681 - 707
Copyright
Copyright © Association for Symbolic Logic 2013

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References

REFERENCES

[1] Beller, A., Jensen, R. B., and Welch, P., Coding the universe, London Mathematical Society Lecture Note Series 47, Cambridge University Press, 1982.Google Scholar
[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. 55–75.Google Scholar
[3] Devlin, K. J., Constructibility, Springer-Verlag, 1984.CrossRefGoogle Scholar
[4] Shelah, S., Cardinal arithmetic, Oxford Science Publications, 1994.CrossRefGoogle Scholar
[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. 365–382.CrossRefGoogle Scholar
[6] Stanley, M. C., Forcing closedunboundedsubsets of ω 2 , Annals of Pure and Applied Logic, vol. 110 (2001), pp. 23–87.CrossRefGoogle Scholar