Grigorieff Forcing on Uncountable Cardinals Does Not Add a Generic of Minimal Degree
Brooke M. Andersen and Marcia J. Groszek
Source: Notre Dame J. Formal Logic
Volume 50, Number 2
(2009), 195-200.
Abstract
Grigorieff showed that forcing to add a subset of ~ using partial functions
with suitably chosen domains can add a generic real of minimal degree. We
show that forcing with partial functions to add a subset of an uncountable ~ without
adding a real never adds a generic of minimal degree. This is in contrast to
forcing using branching conditions, as shown by Brown and Groszek.
Primary Subjects: 03E35
Secondary Subjects: 03E45
Keywords: forcing; Grigorieff forcing; degrees of constructiblity; kappa degrees
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1242067710
Digital Object Identifier: doi:10.1215/00294527-2009-006
Mathematical Reviews number (MathSciNet):
MR2535584
References
[1] Brown, E. T., and M. J. Groszek, "Uncountable superperfect forcing and minimality", Annals of Pure and Applied Logic, vol. 144 (2006), pp. 73--82.
[2] Grigorieff, S., "Combinatorics on ideals and forcing", Annals of Pure and Applied Logic, vol. 3 (1971), pp. 363--94.
[3] Groszek, M. J., "Combinatorics on ideals and forcing with trees", The Journal of Symbolic Logic, vol. 52 (1987), pp. 582--93.
Mathematical Reviews (MathSciNet):
MR902978
[4] Hajnal, A., "On a consistency theorem connected with the generalized continuum problem", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 2 (1956), pp. 131--36.
[5] Jech, T., Set Theory. Pure and Applied Mathematics, Academic Press, New York, 1978.
Mathematical Reviews (MathSciNet):
MR506523
