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The notion of countable well order admits an alternative definition in terms of embeddings between initial segments. We use the framework of reverse mathematics to investigate the logical strength of this definition and its connection with Fraïssé’s conjecture, which has been proved by Laver. We also fill a small gap in Shore’s proof that Fraïssé’s conjecture implies arithmetic transfinite recursion over 
$\mathbf {RCA}_0$, by giving a new proof of 
$\Sigma ^0_2$-induction.
