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

TL;DR

This paper demonstrates that, assuming Schanuel's Conjecture, the prime model of real exponentiation can be embedded into any o-minimal exponential field, advancing understanding of their model-theoretic relationship.

Contribution

It establishes a conditional embedding of the prime model into o-minimal exponential fields, linking conjectural number theory to model theory.

Findings

Conditional embeddability under Schanuel's Conjecture

Unconditional embeddability result via K ext{"o}nig's Lemma

Progress towards the Transfer Conjecture for 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 K\H{o}nig's Lemma.