ИСТИНА

In definability theory, the main question is: Which classes of structures are definable by (sets of) formulas of a given language? For many languages and classes of structures, the criteria (necessary and sufficient conditions) are found for a class of structures to be finitely axiomatizable (definable by a single formula), axiomatizable (definable by some set of formulas), or to be the union of (finitely) axiomatizable classes (these four options are exhaustive in some sense). At the same time, one can introduce dual notions for sets of formulas: the set of formulas that are true in some model is called a basic theory; a theory is the set of formulas that are true in some class of models (thus it is the intersection of some basic theories); a co-theory is the union of some basic theories; a quasi-theory is the union of some theories (again, these four options are exhaustive). We present the criteria (necessary and sufficient conditions) for a set of formulas to be a quasi-theory. Notably, like in the criteria for models, the proof of the result does not involve the axiom of choice and gives an explicit representation of a quasi-theory as the union of theories. We also discuss possible criteria for theories, co-theories, and basic theories.