Generic Coding with Help and Amalgamation Failure

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 $M$ is a countable transitive model of ZF and if $a,b$ are reals not in $M$, then there is a $G$ generic over $M$ such that $b \in L[a,G]$. We then present several applications such as the following: if $J$ is any countable transitive model of ZFC and $M \not\subseteq J$ is another countable transitive model of ZFC of the same ordinal height $\alpha$, then there is a forcing extension $N$ of $J$ such that $M \cup N$ is not included in any transitive model of ZFC of height $\alpha$. Also, assuming $0^\#$ exists, letting $S$ be the set of reals generic over $L$, although $S$ is disjoint from the Turing cone above $0^\#$, we have that for any non-constructible real $a$, $\{ a \oplus s : s \in S \}$ is cofinal in the Turing degrees.