PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.)
Vol. 36(50), pp. 29--34 (1984)
NONEXISTENCE OF NONMOLECULAR GENERIC SETS
Donald D. Steiner and Alexander AbianMCC, 9430 Research Blvd., Austin, Texas 78759, USA and Department of Mathematics, Iowa State University, Ames, Iowa 50011, USA.
Abstract: Generic subsets of partially ordered sets play an important role in giving significant examples of Zermelo-Fraenkel set-theoretical models. The significance of these models lies in the fact that a generic subset $G$ of a partially ordered set $P$, in general, does not exist in a model $M$ in which $P$ exists. Thus, by adjoining $G$ to $M$ an interesting extended model may ensue which has properties not shared by $M$. Thus, in considering generic extensions of set-theoretical models it is quite relevant to know whether or not a generic subset of a partially ordered set $P$ exists in the same model in which $P$ exists. In this paper, we give a necessary and sufficient condition for $P$ to have a generic subset in the same model.
Classification (MSC2000): 06A10
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