Forcing as a Local Method of Accessing Small Extensions

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.

37 pages, 2 figures. Figure 2 is added to provide a better intuition of our main reducibility relation on local method definitions. Section 4 recapitulates some definitions and theorems from arXiv:2402.01213. Certain short proofs of these theorems are replicated for greater self-containment of the article