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Completeness/intuitionistic predicate calculus/Berka: the completeness with regard to the semantics of Kripke and Lorenzen has been proved several times, but always with classical means.
An intuitionist completeness proof has not yet been found (1983). On the contrary. Kreisel (1962) proved that the intuitionist predicate calculus follows intuitionistically from the intuitionist Church thesis.
Constructive Philosophy Cambridge 1987