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Abstract Propositional temporal logic over the real number time flow is finitely axiomatisable, but its first-order counterpart is not recursively axiomatisable. We study the logic that combines the propositional axiomatisation with the usual axioms for first-order logic with identity, and develop an alternative “admissible” semantics for it, showing that it is strongly complete for admissible models over the reals. By contrast there is no recursive axiomatisation of the first-order temporal logic of admissible models whose time flow is the integers, or any scattered linear ordering.
Robert Goldblatt (Wed,) studied this question.