Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-06T10:16:39.975Z Has data issue: false hasContentIssue false

Combinatorics on ideals and forcing with trees

Published online by Cambridge University Press:  12 March 2014

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

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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 159.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. 298300.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. 363394.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. 122128.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. 84104.Google Scholar
[9] Laver, R., On the consistency of Borel's conjecture, Acta Mathematica, vol. 137 (1976), pp. 151169.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. 143159.CrossRefGoogle Scholar
[11] Sacks, G., Countable admissible ordinals and hyperdegrees, Advances in Mathematics, vol. 19 (1976), pp. 213262.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. 331355.CrossRefGoogle Scholar