Recursive axiomatizations from separation properties
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We define a fragment of monadic infinitary second-order logic corresponding to a kind of abstract separation property. We use this to define certain subclasses of elementary classes as separation subclasses. We use model theoretic techniques and games to show that separation subclasses which are, in a sense, recursively enumerable in our second-order fragment can also be recursively axiomatized in their original first-order language. We pin down the expressive power of this formalism with respect to first-order logic, and investigate some questions relating to decidability and computational complexity. As applications, we use simple characterizations as separation subclasses to obtain axiomatizability results related to graph colourings and partial algebras.
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