Generic Coding with Help and Amalgamation Failure
Sy-David Friedman, Dan Hathaway

TL;DR
This paper explores the limitations of generic coding in set theory, demonstrating how certain models and reals can be constructed to exhibit specific amalgamation failures and degree-theoretic properties.
Contribution
It introduces new results on generic coding and amalgamation failure, including constructions involving models of ZF and ZFC, and analyzes the Turing degree structure of certain sets of reals.
Findings
Existence of a generic over a model that codes a real outside the model.
Construction of models with non-embeddable unions in ZFC extensions.
Cofinal sets in Turing degrees formed by non-constructible reals and generic reals.
Abstract
We show that if is a countable transitive model of ZF and if are reals not in , then there is a generic over such that . We then present several applications such as the following: if is any countable transitive model of ZFC and is another countable transitive model of ZFC of the same ordinal height , then there is a forcing extension of such that is not included in any transitive model of ZFC of height . Also, assuming exists, letting be the set of reals generic over , although is disjoint from the Turing cone above , we have that for any non-constructible real , is cofinal in the Turing degrees.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Computability, Logic, AI Algorithms · Limits and Structures in Graph Theory
