INTERPOLATION, COMPACTNESS AND JEP IN SOFT MODEL THEORY

Let[Figure not available: see fulltext.] be the following statement: "for any infinite regular κ, for any uniform ultrafilter D on κ, D is λ-descendingly incomplete for all infinite λ".[Figure not available: see fulltext.] is weaker than {bottom left crop}0#. Assuming[Figure not available: see fulltext.] we prove the following: let L be a logic in which the class of sentences of type τ is a set if so is τ; then: (I)L is compact iff L has JEP; (II)L satisfies Robinson Consistency Theorem iff L is compact and satisfies Craig Interpolation theorem; (III) if, in addition, L is single-sorted, then L satisfies Robinson Consistency Theorem iff L has JEP#. JEP (resp. JEP#) are the natural generalizations for logic L of the familiar Joint Embedding Property of elementary (resp. complete) embeddings in first order logic. As a corollary, we characterize first order logic as the only logic having Löwenheim number equal to ω together with JEP.