Book contents
- Frontmatter
- Contents
- Introduction
- Colloquium Program
- TUTORIAL
- RESEARCH ARTICLES
- Indestructibility and strong compactness
- Some applications of regularmarkers
- Has the continuumhypothesis been settled?
- Geometry of interaction IV: the feedback equation
- On local modularity in homogeneous structures
- Descriptive set theory and uncountable model theory
- Decidable properties of logical calculi and of varieties of algebras
- Stabilization — an alternative to double-negation translation for classical natural deduction
- Definability and reducibility in higher types over the reals
- Predicativity problems in point-free topology
- Rank inequalities in the theory of differentially closed fields
- Consistency and games— in search of new combinatorial principles
- Realizability for constructive Zermelo-Fraenkel set theory
- On long EF-equivalence in non-isomorphic models
- The theory of is undecidable
- Cocovering and set forcing
- References
Cocovering and set forcing
from RESEARCH ARTICLES
Published online by Cambridge University Press: 30 March 2017
- Frontmatter
- Contents
- Introduction
- Colloquium Program
- TUTORIAL
- RESEARCH ARTICLES
- Indestructibility and strong compactness
- Some applications of regularmarkers
- Has the continuumhypothesis been settled?
- Geometry of interaction IV: the feedback equation
- On local modularity in homogeneous structures
- Descriptive set theory and uncountable model theory
- Decidable properties of logical calculi and of varieties of algebras
- Stabilization — an alternative to double-negation translation for classical natural deduction
- Definability and reducibility in higher types over the reals
- Predicativity problems in point-free topology
- Rank inequalities in the theory of differentially closed fields
- Consistency and games— in search of new combinatorial principles
- Realizability for constructive Zermelo-Fraenkel set theory
- On long EF-equivalence in non-isomorphic models
- The theory of is undecidable
- Cocovering and set forcing
- References
Summary
Abstract. Let be an uncountable regular cardinal and let be a cardinal of cofinality greater than in the model V of ZFC. It is shown that if certain combinatorial properties hold between V and an outer model W, then every subset of in W is set generic for a forcing of V-cardinality less than. This leads to a combinatorial characterization of those outer models W that are set generic extensions of V.
Introduction. Let V[G] be the result of adding a Cohen real and a Cohen subset of to a model V of the GCH. Every infinite cardinal gets a new subset in V[G], namely, the Cohen real. Yet, in some sense, only and get new subsets. One way to capture this is to say that a sort of dual to Covering holds between V and V[G], namely, “cocovering” for other than and.
We need some definitions.
If is a regular cardinal in V, say that cocovering holds between V and the outer model W if, given any unbounded that lies in W, there exists such that and b is unbounded in.
If V is a standard transitive model of ZFC, say that is an outer model of V if W is also a standard transitive model of ZFC and. In this paper we are only concerned with outer models such that (W;V) satisfies ZFC (in a language with a predicate symbol for V).
Ultimately, ZFC is our metatheory; talk of models can be understood as talk of set, or even countable set models. The reader will note that our results usually can be paraphrased in more traditional terminology, perhaps at the cost of some generality.
This paper proves a sort of converse to the observation with which we began. The general case is not quite so simple as this example suggests. We show that if W is any outer model of V, and sufficient cocovering (and maybe a little covering) hold around, then every subset of that lies in W is set generic over V for a forcing of V-cardinality less than.
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- Logic Colloquium '03 , pp. 345 - 376Publisher: Cambridge University PressPrint publication year: 2006