Seyed Mohammad Amin Khatami - Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran
Massoud Pourmahdian - Department of Mathematics and Computer Science, Amirk- abir University of Technology, Tehran, Iran

We study three kinds of compactness in some variants of G"{o}del logic: compactness,entailment compactness, and approximate entailment compactness.For countable first-order underlying language we use the Henkinconstruction to prove the compactness property of extensions offirst-order g logic enriched by nullary connective or the Baaz'sprojection connective. In the case of uncountable first-order languagewe use the ultraproduct method to derive the compactness theorem.

G {o}del logic, Compactness theorem