AbstractChapter IV contains a discussion of the general properties of the system R. In this chapter, the general properties of the system will be used for a detailed development of the theory of numbers in the system R.
Section 41 contains a detailed discussion of natural numbers and functions of natural numbers. The discussion begins by introducing the concept of an R-formula I-representing a natural number. Speaking informally, this is saying that certain R-formulas serve as names for natural numbers. This definition, together with the primitive rules for W are used to develop, in detail, the properties of addition,multiplication, exponentiation and proper subtraction in the system R.
Sections 42 and 43 are devoted to developing more general ways of dealing with functions of natural numbers in the system R. The basic idea is to introduce certain classes of functions of natural numbers and show that each function has an I-representation in system R. The particular classes of functions which are considered the primitive recursive functions and the partial recursive functions. Section 44 contains a discussion of general methods which can be used to show that relations among natural numbers have I-representations in the system R. There is also a discussion of limitations on relations which can be represented in the system R. In section 45, the theory of natural numbers in the system R is used to continue the development of the theory of combinators which was begun in section 31. Using natural numbers, it is possible to prove many additional properties of combinators as well as to define combinators which have interesting and useful properties. Finally, in section 46, the concept of an ordinal number is introduced. Then it is shown that a substantial fragment of several versions of the theory of ordinal numbers can be developed within the system R.
In summary, the objective of this chapter is to develop the theory of natural numbers and ordinal numbers in the system R.
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