I am not asking which order types PA proves are well ordered. That would be all up to $\epsilon\_0$. Rather I mean, assuming a stronger ambient theory such as Zermelo set theory, which ordinals have the order type of some relation on $\mathbb{N}$ that is defined by a formula of PA (not requiring that PA prove the relation is a well ordering).



What ordinals are definable relations in Peano Arithmetic?
These are the recursive ordinals. The same well-order-types can be realized by recursive relations as by hyperarithmetical relations. PA-definable, i.e., arithmetical, falls nicely between these two. (I think you can go considerably lower, say to PTime-computable relations, and still have the same order-types.)


