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

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