The Use of Ordinals in the Constructive Foundations of Mathematics

Abstract

Publisher Summary The chapter presents a method of analyzing formal theories by the constructive use of ordinals. Constructive foundations and reductive proof theory of mathematics are discussed in the chapter. It has become customary to encode infinite derivation trees by natural numbers in a manner similar to the encoding of the Church–Kleene constructive ordinals. It makes the metamathematics more elementary because now the domain of individuals in the metamathematics consists of natural numbers. The main emphasis, however, has been in the use of such an encoding as a technical tool. The use of encodings has led to an awareness of the continuity of the syntactical transformations employed in the process of cut-elimination. Because the mappings of the codes are defined by Kleene's recursion theorem, they are defined on all trees rather than merely on well-founded trees. Moreover these transformations can be extended to all recursive functions by employing an extensional effective operation, which maps an arbitrary recursive function into a function that defines a tree.