PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 36(50), pp. 2934 (1984) 

NONEXISTENCE OF NONMOLECULAR GENERIC SETSDonald 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 ZermeloFraenkel settheoretical 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 settheoretical 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 Full text of the article:
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