Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The reals as a subset of an ultraproduct of finite fields

Roee Sinai

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.

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.
Paper Structure (9 sections, 62 theorems, 110 equations)

This paper contains 9 sections, 62 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Some almost internal sub­fields of $\mathbb{R}'$
  3. $\sigma$-sets and $\delta$-sets
  4. The complicated situation of $\mathbb R'$
  5. Characterizations of superfields of $\mathbb R$
  6. Other algebraic structures included in $\mathbb R^\delta$
  7. The case where the field contains a square root of -1
  8. A more elaborate discussion of cuts
  9. The proof of Lemma \ref{['alg-integer']}

Key Result

Theorem 5

No copy of $\mathbb R$ inside $\mathbb F_{\hat{p}}$ can be a $\sigma$-set and neither can one be a $\delta$-set.

Theorems & Definitions (143)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Proposition 1.1
  • proof
  • ...and 133 more