Abstract
We continue to develop a new first-order combinatorial approach presenting a conceptual framework for investigations concerning expressive power of first-order logic. In this work, we consider the case of infinitary first-order combinatorics. Based on the universal construction of finitely axiomatizable theories, we introduce some common scheme yielding finitely axiomatizable theories with pre-assigned sets of model-theoretic properties. At an initial stage, a maximum common Turing’s computation is performed (one can say, computable Brute Force). Starting from an input block of parameters, the computation yields a computably axiomatizable theory T . Finally, by applying an available version of the universal construction, the theory T is transformed into a finitely axiomatizable theory F that inherits model-theoretic properties of T within the infinitary semantic layer. We also give three demonstrations showing possibilities of this method.