Abstract
Classical results concerning the categoricity of Euclidean geometry can be viewed upon as proofs of finite axiomatizability of the theory of the Euclidean plane in the strong second-order logic. The chapter conceives the problem of axiomatizability: (1) a formal language L , (2) the notion of consequence in L , that is, a function Cn that correlates with every set of closed formulas of L another such set, and (3) a mathematical structure, for example, the set of integers together with the arithmetical operations or the field of real numbers or the like. Let V be the set of formulas of L that are true in the considered structure. The problem is whether there exists a finite (or a recursively enumerable) set X of closed formulas, such that V = Cn ( X ). If such a set exists, the theory of the considered mathematical structure is finitely axiomatizable (or simply axiomatizable) in the logic determined by L and Cn .
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