Characterizations of pretameness and the Ord-cc |
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Authors: | Peter Holy Regula Krapf Philipp Schlicht |
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Institution: | 1. Math. Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany;2. Math. Institut, Universität Koblenz-Landau, Universitätsstraße 1, 56070 Koblenz, Germany |
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Abstract: | It is well known that pretameness implies the forcing theorem, and that pretameness is characterized by the preservation of the axioms of , that is without the power set axiom, or equivalently, by the preservation of the axiom scheme of replacement, for class forcing over models of . We show that pretameness in fact has various other characterizations, for instance in terms of the forcing theorem, the preservation of the axiom scheme of separation, the forcing equivalence of partial orders and their dense suborders, and the existence of nice names for sets of ordinals. These results show that pretameness is a strong dividing line between well and badly behaved notions of class forcing, and that it is exactly the right notion to consider in applications of class forcing. Furthermore, for most properties under consideration, we also present a corresponding characterization of the Ord-chain condition. |
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Keywords: | 03E40 03E70 Class forcing Pretameness Ord-cc Forcing theorem Nice names |
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