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Recursive logic frames
Author(s) -
Shelah Saharon,
Väänänen Jouko
Publication year - 2006
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200410058
Subject(s) - mathematics , intermediate logic , cardinality (data modeling) , axiom , frame (networking) , second order logic , discrete mathematics , higher order logic , compact space , algebra over a field , algorithm , theoretical computer science , computer science , description logic , pure mathematics , telecommunications , geometry , data mining
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 , ℵ 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 λ ” completeness always implies .0‐compactness. On the other hand we show that a recursively compact logic frame need not be ℵ 0 ‐compact. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)