Arithmetical Set Theory

Abstract

It is well known that number theory can be interpreted in the usual set theories, e.g. ZF, NF and their extensions. The problem I posed for myself was to see if, conversely, a reasonably strong set theory could be interpreted in number theory. The reason I am interested in this problem is, simply, that number theory is more basic or more concrete than set theory, and hence a more concrete foundation for mathematics. A partial solution to the problem was accomplished by WTN in [2], where it was shown that a predicative set theory could be interpreted in a natural extension of pure number theory, PN, (i.e. classical first-order Peano Arithmetic). In this paper, we go a step further by showing that a reasonably strong fragment of predicative set theory can be interpreted in PN itself. We then make an attempt to show how to develop predicative fragments of mathematics in PN. If one wishes to know what is meant by “reasonably strong” and “fragment” please read on.