Thirty Five Years of Automating Mathematics
Volume 28 of the series Applied Logic Series pp 149171
Transitive Closure and the Mechanization of Mathematics
 Arnon Avron
Abstract
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 TClogics 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.
 Title
 Transitive Closure and the Mechanization of Mathematics
 Book Title
 Thirty Five Years of Automating Mathematics
 Pages
 pp 149171
 Copyright
 2003
 DOI
 10.1007/9789401702539_7
 Print ISBN
 9789048164400
 Online ISBN
 9789401702539
 Series Title
 Applied Logic Series
 Series Volume
 28
 Series ISSN
 13862790
 Publisher
 Springer Netherlands
 Copyright Holder
 Springer Science+Business Media Dordrecht
 Additional Links
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 Editors

 Fairouz D. Kamareddine ^{(2)}
 Editor Affiliations

 2. School of Mathematical and Computer Sciences, HeriotWatt University
 Authors
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