Continuous Craig Interpolation

TL;DR

This work extends Craig interpolation to the continuous model theory of metric structures by proving a continuous Robinson Consistency analogue and two interpolation theorems. The Weak Interpolant Theorem follows from the Robinson consistency framework, guaranteeing that for any $\varepsilon>0$ there exists a $V\cap W$-sentence $\theta$ with $T_V\models\varphi\ge(\theta)$ and $T_W\models(\theta\ge(\psi-\varepsilon))$ whenever $\varphi$ and $\psi$ satisfy $\varphi=0$ and $\psi=0$. The Strong Interpolant Theorem is then established by a constructive aggregation of weak interpolants: for each $\varepsilon=2^{-n}$ one builds $\rho_k$ and combines them via a continuous function to obtain a single $V\cap W$-sentence $\theta$ with $\models\varphi\ge\theta$ and $\models\theta\ge\psi-\varepsilon$, for all $\varepsilon>0$. These results connect to linear continuous logic and enable uniform interpolation constructions and Beth-type results in the metric setting, utilizing special/saturated models and compactness techniques to ensure coherence across $[0,1]$-valued structures.

Abstract

We prove analogues of the Craig interpolation theorem for the continuous model theory of metric structures.

Figures (2)

• Figure 1: Interpolation when $\models \varphi\ge\psi.$
• Figure 2: Graphs of $\varphi^{\mathcal{M}}, \psi^{\mathcal{M}}$, and $\rho_k^{\mathcal{M}}$

Theorems & Definitions (29)

• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Corollary 1.6
• proof
• Corollary 2.2
• proof
• Theorem 2.7
• proof