Semantics of the infinitistic rules of proof

This paper is devoted to the study of the infinitistic rules of proof i.e. those which admit an infinite number of premises. The best known of these rules is the ω-rule. Some properties of the ω-rule and its connection with the ω-models on the basis of the ω-completeness theorem gave impulse to the development of the theory of models for admissible fragments of the language . On the other hand the study of representability in second order arithmetic with the ω-rule added revealed for the first time an analogy between the notions of re-cursivity and hyperarithmeticity which had an important influence on the further development of generalized recursion theory. The consideration of the subject of infinitistic rules in complete generality seems to be reasonable for several reasons. It is not completely clear which properties of the ω-rule were essential for the development of the above-mentioned topics. It is also worthwhile to examine the proof power of infinitistic rules of proof and what distinguishes them from finitistic rules of proof. What seemed to us the appropriate point of view on this problem was the examination of the connection between the semantics and the syntax of the first order language equipped with an additional rule of proof.