The Lowenheim-Skolem theorem for Godel logic

We prove the following Lowenheim-Skolem theorems for first-order Godel logic:(1) For the Godel logic G[0,1], a sentence phi has models of every infinite cardinality if and only if it has a model of cardinality n omega(= sup{aleph 0, 2 aleph 0, ...}).(2) For an arbitrary Godel logic GT, a sentence phi has models of every infinite cardinality if and only if it has a model of cardinality n omega 1. Moreover, (1) becomes false if n omega is replaced by a smaller cardinality, and (2) becomes false if n omega 1 is replaced by a smaller cardinality.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).