Compactness in first order Łukasiewicz logic
There are different non-equivalent ways to approach first-order Łukasiewicz logic. In this paper the authors focus on a first-order logic in which the formulas are interpreted in the interval \([0,1]\) and the equality axioms are interpreted by a similarity relation (with respect to Łukasiewicz t-norm). This logic is not complete, there are true formulas that are not provable; see [\textit{B. Scarpellini}, ``Die Nichtaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Łukasiewicz'', J. Symb. Log. 27, 159--170 (1963; Zbl 0112.24503)]. The authors prove, using the ultraproduct construction, that for this logic the compactness theorem holds: if a set of formulas is such that each of its finite subsets is \(K\)-satisfiable (there is an evaluation assigning to each formula a truth value in \(K\)) then the set itself is \(K\)-satisfiable. A comment on the relationship of this result with the incompleteness of the logic would have been appropriate.
scientific article; zbMATH DE number 6062916