An Axiomatic Theory of Ordinal Numbers

Abstract

In this paper, a formal theory of ordinal numbers is developed on an axiomatic basis whose details are described in § 1. Our primitive notions are set, class, collection, a binary relation ∈, and collection formation {!}. Sets and classes in our theory play similar roles as sets and classes respectively in Gödel [1] except the difference that an element of a class is a class but not necessarily a set. A new notion, introduced into our theory is that of collections. A collection relates to a class, just as a class relates to a set in von Neumann’s theory. That is; a set is a class and a class is a collection but the converses are not generally the case. For example, all the natural numbers, all the real numbers etc. constitute sets, the ordinal numbers which are sets constitute a proper class, and the totality of ordinal numbers as well as that of all classes are proper collections. These relations are described by axiom group (A).