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-orde...

Abstract We can measure the complexity of a logical formula by counting number alternations between existential and universal quantifiers. Suppose that an elementary first-order $\varphi $ (in $\mathcal {L}_{\omega ,\omega }$ ) is equivalent to infinitary language {L}_{\infty with n prove finitary Thus using logic does not allow us express in simpler way.