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Backwards easton forcing and 0#

Published online by Cambridge University Press:  12 March 2014

M. C. Stanley
M. C. Stanley*
Affiliation:
Department of Mathematics and Computer Science, San Jose State University, San Jose, California 95192
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Abstract

It is shown that if κ is an uncountable successor cardinal in L[0#], then there is a normal tree T ϵ L[0#] of height κ such that 0# ∉ L[T], yet T is < κ-distributive in L[0#]. A proper class version of this theorem yields an analogous L[0#]-definable tree such that distinct branches in the presence of 0# collapse the universe. A heretofore unutilized method for constructing in L[0#] generic objects for certain L-definable forcings and “exotic sequences”, combinatorial principles introduced by C. Gray, are used in constructing these trees.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 3 , September 1988 , pp. 809 - 833
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

[ASS]Abraham, U., Shelah, S., and Solovay, R., Squares with diamonds and Souslin trees with special squares, Fundamenta Mathematicae, vol. 127 (1986), pp. 133–162.CrossRefGoogle Scholar
[BJW]Beller, A., Jensen, R. B., and Welch, P., Coding the universe, London Mathematical Society Lecture Note Series, vol. 47, Cambridge University Press, Cambridge, 1982.CrossRefGoogle Scholar
[D]Devlin, K., Aspects of constructibility, Lecture Notes in Mathematics, vol. 354, Springer-Verlag, Berlin, 1973.CrossRefGoogle Scholar
[FMS]Foreman, M., Magidor, M., and Shelah, S., 0# and some forcing principles, this Journal, vol. 51 (1986), pp. 39–46.Google Scholar
[F]Friedman, S., An immune partition of the ordinals, Recursion theory week (Oberwolfach, 1984), Lecture Notes in Mathematics, vol. 1141, Springer-Verlag, Berlin, 1985, pp. 141–147.CrossRefGoogle Scholar
[G]Gray, C., Iterated forcing from the strategic point of view, Ph.D. thesis, University of California, Berkeley, California, 1981.Google Scholar
[Jc]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[J]Jensen, R. B.. The fine structure of the constructihle hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.CrossRefGoogle Scholar
[P]Paris, J., Patterns of indiscernibles, Bulletin of the London Mathematical Society, vol. 6 (1974), pp. 183–188.CrossRefGoogle Scholar
[V ]Veličković, B., Jensen's □ principles and the Novak number of partially ordered sets, this Journal, vol. 51 (1986), pp. 47–58.Google Scholar