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Combinatorics on ideals and forcing with trees

Published online by Cambridge University Press:  12 March 2014

Marcia J. Groszek
Marcia J. Groszek*
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
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Abstract

Classes of forcings which add a real by forcing with branching conditions are examined, and conditions are found which guarantee that the generic real is of minimal degree over the ground model. An application is made to almost-disjoint coding via a real of minimal degree.

Keywords

Forcing, treesidealsminimal degreesdegrees of constructibilityalmostdisjoint coding.
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 3 , September 1987 , pp. 582 - 593
Copyright
Copyright © Association for Symbolic Logic 1987

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References

REFERENCES

[1] Baumgartner, J., Iterated forcing, Surveys in set theory (Mathias, A. R. D., editor), London Mathematical Society Lecture Note Series, no. 87, Cambridge University Press, Cambridge, 1983, pp. 1–59.Google Scholar
[2] Beller, A., Jensen, R., and Welch, P., Coding the universe, London Mathematical Society Lecture Note Series, no. 47, Cambridge University Press, Cambridge, 1982.CrossRefGoogle Scholar
[3] Carlson, T., Kunen, K., and Miller, A., A minimal degree which collapses ω1 , this Journal, vol. 49 (1984), pp. 298–300.Google Scholar
[4] Friedman, S., The coding method in set theory (book, in preparation).Google Scholar
[5] Gray, C., Iterated forcing from the strategic point of view, Ph.D. Thesis, University of California, Berkeley, California, 1980.Google Scholar
[6] Grigorieff, S., Combinatorics on ideals and forcing, Annals of Mathematical Logic, vol. 3 (1971), pp. 363–394.CrossRefGoogle Scholar
[7] Jensen, R., Definable sets of minimal degree, Mathematical logic and foundations of set theory (Bar-Hillel, Y., editor), North-Holland, Amsterdam, 1970, pp. 122–128.Google Scholar
[8] Jensen, R. and Solovay, R., Some applications of almost disjoint sets, Mathematical logic and foundations of set theory (Bar-Hillel, Y., editor), North-Holland, Amsterdam, 1970, pp. 84–104.Google Scholar
[9] Laver, R., On the consistency of Borel's conjecture, Acta Mathematica, vol. 137 (1976), pp. 151–169.CrossRefGoogle Scholar
[10] Miller, A., Rational perfect set forcing, Axiomatic set theory (Baumgartner, J. et al., editors), Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, Rhode Island, 1984, pp. 143–159.CrossRefGoogle Scholar
[11] Sacks, G., Countable admissible ordinals and hyperdegrees, Advances in Mathematics, vol. 19 (1976), pp. 213–262.CrossRefGoogle Scholar
[12] Sacks, G., Forcing with perfect closed sets, Axiomatic set theory (Scott, D., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 331–355.CrossRefGoogle Scholar