Transitive Closure and the Mechanization of Mathematics

We argue that the concept of transitive closure is the key for understanding finitary inductive definitions and reasoning, and we provide evidence for the thesis that logics which are based on it (in which induction is a logical rule) are the right logical framework for the formalization and mechanization of mathematics. We investigate the expressive power of languages with the most basic transitive closure operation TC. We show that with TC one can define all recursive predicates and functions from 0, the successor function and addition, yet with TC alone addition is not definable from 0 and the successor function. However, in the presence of a pairing function, TC does suffice for having all types of finitary inductive definitions of relations and functions. This result is used for presenting a simple version of Feferman’s framework FS 0, demonstrating that TC-logics provide in general an excellent framework for mechanizing formal systems. An interesting side effect of these results is a simple characterization of recursive enumerability and a new, concise version of the Church thesis. We end with a use of TC for a formalization of set theory which is based on purely syntactical considerations, and reflects real mathematical practice.