Abstract
Let be the following statement: “for any infinite regularκ, for any uniform ultrafilterD onκ,D isλ-descendingly incomplete for all infiniteλ”. is weaker than ⌍0#. Assuming we prove the following: letL be a logic in which the class of sentences of typeτ is a set if so isτ; then: (I)L is compact iffL has JEP; (II)L satisfies Robinson Consistency Theorem iffL is compact and satisfies Craig Interpolation theorem; (III) if, in addition,L is single-sorted, thenL satisfies Robinson Consistency Theorem iffL has JEP#. JEP (resp. JEP#) are the natural generalizations for logicL 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 Lowenheim number equal toω together with JEP.
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