Finite Models of Elementary Recursive Nonstandard Analysis

This paper provides a new proof of the consistency of a formal system similar to the one presented by Chuaqui and Suppes in 2, 9]. First, a simpler, yet in some respects stronger, system, called Elementary Re-cursive Nonstandard Analysis (ERNA) will be provided. Indeed, it will be shown that ERNA proves the main axioms of the Chuaqui and Sup-pes system. Then a nitary consistency proof of ERNA will be given; in particular, we will show that PRA, the system of primitive recursive arithmetic, which is generally recognized as capturing Hilbert's notion of nitary, proves the consistency of ERNA. From the consistency proof we can extract a constructive method for obtaining nite approximations of models of nonstandard analysis. We present an isomorphism theorem for models that are nite substructures of innnite models.