We define the concept of a logic frame , which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive logic frame is called complete , if every finite consistent theory has a model. We show that for logic frames built from the cardinality quantifiers “there exists at least λ ” completeness always implies .0-compactness. On (...) the other hand we show that a recursively compact logic frame need not be ℵ0-compact. (shrink)

We give an exposition of the compactness of L(QcfC), for any set C of regular cardinals.

The local homogeneity property is defined as in [Mak]. We show thatL ωω(Q1) and some related logics do not have the local homogeneity property, whereas cofinality logicL ωω(Q cfω) has the homogeneity property. Both proofs use forcing and absoluteness arguments.