The Craig Interpolation Property in First-order G\"odel Logic

TL;DR

This paper proves that first-order G"odel logic and its extension possess the Craig interpolation property using a model-theoretic approach, addressing a previously open question in the field.

Contribution

It introduces a model-theoretic proof that first-order G"odel logic and its extension have the Craig interpolation property, advancing understanding of their logical characteristics.

Findings

First-order G"odel logic has the Craig interpolation property.

Extension G^ also enjoys the Craig interpolation property.

Provides partial answers to open questions in G"odel logic research.

Abstract

In this article, a model-theoretic approach is proposed to prove that the first-order G\"odel logic, $\mathbf{G}$, as well as its extension $\mathbf{G}^\Delta$ associated with first-order relational languages enjoy the Craig interpolation property. These results partially provide an affirmative answer to a question posed in [Aguilera, Baaz, 2017, Ten problems in G\"odel logic].