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Recursive logic frames. (English) Zbl 1093.03024
Summary: 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 (recursively compact, $$\aleph_0$$-compact), if every finite (respectively: recursive, countable) consistent theory has a model. We show that for logic frames built from the cardinality quantifiers “there exists at least $$\lambda$$” completeness always implies $$\aleph_0$$-compactness. On the other hand we show that a recursively compact logic frame need not be $$\aleph_0$$-compact.

##### MSC:
 03C95 Abstract model theory 03C80 Logic with extra quantifiers and operators 03C55 Set-theoretic model theory
##### Keywords:
Generalized quantifiers; identities; compact logics
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