Compactness in first-order Godel logics

摘要

Our aim in this article is twofold. First, for an arbitrary closed subset V of [0,1] containing both 0 and 1, the compactness theorem for the Goedel logic Gv is investigated. Next, following Cintula and Navara (2004, Fuzzy sets and systems, 143, 59-73) and Tavana et al. (2012, Logic J. IGPL, 20, 254-265), for any subset K of [0,1], the notions of K-satisfiability and k-compactness are introduced for the standard first-order Goedel logic G_r. It is shown that whenever K is closed, the K-compactness fails for Gr if and only if K is infinitely countable and 1 ￠ K.