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We consider the Gödel bi-modal logic determined by fuzzy Kripke models where both the propositions and the accessibility relation are infinitely valued over the standard Gödel algebra [0,1], and prove strong completeness of the Fischer Servi intuitionistic modal logic  IK  plus the prelinearity axiom with respect to this semantics. We axiomatize also the bi-modal analogues of classical  T ,  S 4 and  S 5, obtained by restricting to models over frames satisfying the [0,1]-valued versions of the structural properties which characterize these logics. As an application of the completeness theorems we obtain a representation theorem for bi-modal Gödel algebras.

Bi-modal Gödel logic over [0,1]-valued Kripke frames