Forcing as a Local Method of Accessing Small Extensions

TL;DR

The paper develops a framework to study degrees of small extensions of a universe $V$ by viewing outer models $W$ generated over a base CTM $U$ as small extensions $U[x]$ and organizing them into a degree structure isomorphic to $(\mathbf{M}_S(U),\subset)$. It introduces TCIs (theories with constraints in interpretation) to define local methods for accessing such extensions and defines a corresponding complexity hierarchy that mimics classical hierarchies. A central result is that set forcing aligns with the local-method level $\mathsf{Σ}^M_1$ (and equivalently $\mathsf{Π}^M_2$) in this hierarchy, with a Cantor–Bendixson-type analysis yielding a trichotomy for the size of $\mathrm{Eval}^V(\mathfrak{T})$ for $\Pi_2$ TCIs: either none, exactly one (the ground model $V$), or continuum-many small extensions. The paper also demonstrates a strengthening that provides a canonical correspondence $F_{\mathfrak{T}^*}$ between $\Pi_2$ TCIs and forcing notions, clarifying the role of forcing as a maximal local method in this setting and outlining directions for separating levels in the local method hierarchy.

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.

Figures (2)

• Figure 1: comparison between conventional notions of relative computability (left) and our generalised notions (right).
• Figure 2: Visual representation of a function $F$ witnessing $X \leq^M Y$, where $X$ and $Y$ are local method definitions. Here, $\mathfrak{T}$ is an arbitrary consistent member of $X$, and $B = \mathrm{Eval}^V(F(\mathfrak{T})) \subset \mathrm{Eval}^V(\mathfrak{T}) = A$.

Theorems & Definitions (106)

• Remark 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.6
• proof
• Proposition 2.7
• proof
• Definition 2.8
• Proposition 2.10
• proof