Computable structures in generic extensions

Julia Knight, Antonio Montalban, Noah Schweber

Published 2014-05-29, updated 2014-12-10Version 2

In this paper, we investigate connections between structures present in every generic extension of the universe $V$ and computability theory. We introduce the notion of {\em generic Muchnik reducibility} that can be used to to compare the complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of {\em generic presentability}, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making $\omega_2$ countable generically presents some countable structure with no copy in the ground model; and that every structure generically presentble by a forcing notion that does not make $\omega_2$ countable has a copy in the ground model. We also show that any countable structure $\mathcal{A}$ that is generically presentable by a forcing notion not collapsing $\omega_1$ has a countable copy in $V$, as does any structure $\mathcal{B}$ generically Muchnik reducible to a structure $\mathcal{A}$ of cardinality $\aleph_1$. The former positive result yields a new proof of Harrington's result that counterexamples to Vaught's conjecture have models of power $\aleph_1$ with Scott rank arbitrarily high below $\omega_2$. Finally, we show that a rigid structure with copies in all generic extensions by a given forcing has a copy already in the ground model.