A formalization of the theory of ordinal numbers

Abstract

Although Peano's arithmetic can be developed in set theories, it can also be developed independently. This is also true for the theory of ordinal numbers. The author formalized the theory of ordinal numbers in logical systems GLC (in [2]) and FLC (in [3]). These logical systems which contain the concept of ‘arbitrary predicates’ or ‘arbitrary functions’ are of higher order than the first order predicate calculus with equality. In this paper we shall develop the theory of ordinal numbers in the first order predicate calculus with equality as an extension of Peano's arithmetic. This theory will prove to be easy to manage and fairly powerful in the following sense: If A is a sentence of the theory of ordinal numbers, then A is a theorem of our system if and only if the natural translation of A in set theory is a theorem of Zermelo-Fraenkel set theory. It will be treated as a natural extension of Peano's arithmetic. The latter consists of axiom schemata of primitive recursive functions and mathematical induction, while the theory of ordinal numbers consists of axiom schemata of primitive recursive functions of ordinal numbers (cf. [5]), of transfinite induction, of replacement and of cardinals. The latter three axiom schemata can be considered as extensions of mathematical induction.In the theory of ordinal numbers thus developed, we shall construct a model of Zermelo-Fraenkel's set theory by following Gödel's construction in [1]. Our intention is as follows: We shall define a relation α ∈ β as a primitive recursive predicate, which corresponds to F′ α ε F′ β in [1]; Gödel defined the constructible model Δ using the primitive notion ε in the universe or, in other words, using the whole set theory.