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Embedding the prime model of real exponentiation into o-minimal exponential fields

Lothar Sebastian Krapp

Abstract

Motivated by the decidability question for the theory of real exponentiation and by the Transfer Conjecture for o-minimal exponential fields, we show that, under the assumption of Schanuel's Conjecture, the prime model of real exponentiation is embeddable into any o-minimal exponential field, where the embedding is not necessarily elementary. This is a consequence of an unconditional model theoretic embeddability result that we obtain by applying Kőnig's Lemma.

Embedding the prime model of real exponentiation into o-minimal exponential fields

Abstract

Motivated by the decidability question for the theory of real exponentiation and by the Transfer Conjecture for o-minimal exponential fields, we show that, under the assumption of Schanuel's Conjecture, the prime model of real exponentiation is embeddable into any o-minimal exponential field, where the embedding is not necessarily elementary. This is a consequence of an unconditional model theoretic embeddability result that we obtain by applying Kőnig's Lemma.
Paper Structure (5 sections, 5 theorems, 1 equation)

This paper contains 5 sections, 5 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Embedding substructures
  4. Exponential algebraic closures
  5. Prime model of real exponentiation

Key Result

Theorem 2.1

Let $\mathcal{L}$ be a countable language, let $\mathcal{A}$ and $\mathcal{B}$ be two $\mathcal{L}$-structures and let $\mathcal{A}'$ be a countably infinite substructure of $\mathcal{A}$. Suppose that the following hold: Then $\mathcal{A}'$ can be embedded into $\mathcal{B}$ as an $\mathcal{L}$-substructure.

Theorems & Definitions (9)

  • Theorem 2.1
  • proof
  • Definition 3.1
  • proof
  • Proposition 3.5
  • Theorem 3.6
  • proof
  • Corollary 4.1
  • Theorem 4.3