# Incompleteness of an infinite-valued first-order Gödel logic and of some temporal logics of programs

Baaz, Matthias, Alexander Leitsch, and Richard Zach. 1996. “Incompleteness of an Infinite-Valued First-Order Gödel Logic and of Some Temporal Logics of Programs.” In Computer Science Logic. CSL 1995. Selected Papers, edited by E. Börger, 1–15. LNCS 1092. Berlin: Springer. DOI: 10.1007/3-540-61377-3_28

It is shown that the infinite-valued first-order Gödel logic G0 based on the set of truth values $$\{0\} \cup \{1/k : k = 1, 2, 3, \dots\}$$ is not r.e. The logic G0 is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger’s temporal logic of programs (even of the fragment without the nexttime operator) and of the authors’ temporal logic of linear discrete time with gaps follows.