The reals as a subset of an ultraproduct of finite fields

Abstract

In this paper we present new ways to construct external subsets of nonstandard models of arithmetic using mostly internal sets, and show that if an ultraproduct of prime finite fields includes a copy of the algebraic real numbers then either this copy or its algebraic closure can be constructed in some of these ways. We also show that no copy of the field of real numbers inside such an ultraproduct can ever be constructed in any of these ways, but there is either a hyperreal field or an algebraically closed field of cardinality larger or equal to the continuum that can be.