The reals as a subset of an ultraproduct of finite fields

TL;DR

The paper explores constructing subsets of nonstandard models of arithmetic, demonstrating conditions under which real numbers or related fields can be embedded or constructed within ultraproducts of finite fields.

Contribution

It introduces new methods for constructing external subsets in nonstandard models and characterizes when real numbers or their algebraic closures can be embedded in ultraproducts of finite fields.

Findings

If an ultraproduct of prime finite fields contains a copy of the real numbers, then either the real numbers or their algebraic closure can be constructed using the new methods.

No copy of the real numbers can be constructed in these ways, but hyperreal fields or large algebraically closed fields can be.

The results clarify the limitations and possibilities of embedding real number structures in ultraproducts 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.