Infinite Rules in Finite Systems

Abstract

Publisher Summary This chapter focuses on the infinite rules in finite systems. The idea of making use of rules of inference with infinitely many premises in the discussion of finite formulae is certainly not new. Some of the paradoxes of antiquitity were paradoxes because of (implicit) uses of such infinite rules, for example, the paradox of Achilles and the Tortoise. The chapter discusses the proof-theory, first-order classical number theory, recursively restricted ω-rule, combinatorial condition, preliminaries from recursive function theory, formal mathematical theories, languages of many formulae (or sequents), Carnap theory, infinitary theory, applications of theorems, canonical spreads, canonical derivations for the theorems, an unfaithful axiomatization of intuitionism, and various lemmas.