Delta is eliminable propositionally in Goedel logics via restricted semantics yielding disjunctions of chain formulas, but fails in general first-order due to recursion and topology constraints though recoverable under witnessed semantics.
Linear Kripke frames and Gödel logics.Journal of Symbolic Logic, 72(1):26–44, 2007
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Goedel Logics: On the Elimination of The Absoluteness Operator
Delta is eliminable propositionally in Goedel logics via restricted semantics yielding disjunctions of chain formulas, but fails in general first-order due to recursion and topology constraints though recoverable under witnessed semantics.