Computable structures in generic extensions

TL;DR

This paper explores the relationship between structures in generic extensions and computability, introducing generic Muchnik reducibility and analyzing how structures are present across different forcing extensions.

Contribution

It introduces the concept of generic Muchnik reducibility and studies its properties, connecting generic presentability with the existence of structures in the ground model.

Findings

Forcing notions making  countable generically present some countable structures not in the ground model.

Structures generically presentable by certain forcings have copies in the ground model.

Rigid structures with copies in all extensions already have a copy in the ground model.

Abstract

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.