Generic Coding with Help and Amalgamation Failure

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.