  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
