On the compactness property of extensions of first-order G\"odel logic

TL;DR

This paper investigates various forms of compactness in extensions of first-order G"odel logic, proving compactness theorems for countable and uncountable languages using Henkin and ultraproduct methods.

Contribution

It establishes the compactness property for certain extensions of first-order G"odel logic, including those with nullary and Baaz's projection connectives, using novel proof techniques.

Findings

Proves compactness for countable languages with specific connectives.

Establishes compactness for uncountable languages via ultraproducts.

Clarifies the role of different connectives in G"odel logic extensions.

Abstract

We study three kinds of compactness in some variants of G\"odel logic: compactness, entailment compactness, and approximate entailment compactness. For countable first-order underlying language we use the Henkin construction to prove the compactness property of extensions of first-order G\"odel logic enriched by nullary connective or the Baaz's projection connective. In the case of uncountable first-order language we use the ultraproduct method to derive the compactness theorem