The model of set theory generated by countably many generic reals

Journal of Symbolic Logic 46 (4):732-752 (1981)

Abstract
Adjoin, to a countable standard model M of Zermelo-Fraenkel set theory (ZF), a countable set A of independent Cohen generic reals. If one attempts to construct the model generated over M by these reals (not necessarily containing A as an element) as the intersection of all standard models that include M ∪ A, the resulting model fails to satisfy the power set axiom, although it does satisfy all the other ZF axioms. Thus, there is no smallest ZF model including M ∪ A, but there are minimal such models. These are classified by their sets of reals, and there is one minimal model whose set of reals is the smallest possible. We give several characterizations of this model, we determine which weak axioms of choice it satisfies, and we show that some better known models are forcing extensions of it
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DOI 10.2307/2273223
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References found in this work BETA

Set Theory and the Continuum Hypothesis.Paul J. Cohen - 1966 - Journal of Symbolic Logic 35 (4):591-592.
Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
The Axiom of Choice.Thomas J. Jech - 1976 - Journal of Symbolic Logic 41 (4):784-785.
The Theory of Semisets.[author unknown] - 1984 - Journal of Symbolic Logic 49 (4):1422-1423.

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Citations of this work BETA

Symmetric Submodels of a Cohen Generic Extension.Claude Sureson - 1992 - Annals of Pure and Applied Logic 58 (3):247-261.

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