Forcing as a Local Method of Accessing Small Extensions

TL;DR

This paper explores small extensions of a set-theoretic universe as generalized degrees of computability, analyzing the complexity of defining subclasses of degrees and characterizing forcing complexity within this framework.

Contribution

It introduces a novel perspective on small extensions as generalized degrees of computability and formalizes the complexity of methods to define subclasses of these degrees.

Findings

Characterization of forcing complexity in the framework

Formalization of methods to define subclasses of degrees

Analysis of small extensions as generalized degrees of computability

Abstract

Fix a set-theoretic universe $V$. We look at small extensions of $V$ as generalised degrees of computability over $V$. We also formalise and investigate the complexity of certain methods one can use to define, in $V$, subclasses of degrees over $V$. Finally, we give a nice characterisation of the complexity of forcing within this framework.