User blog comment:Edwin Shade/Ordinal Questions/@comment-1605058-20171129160105

6. The key here is to figure out how to express "an ordinal is well-ordered" in the language of arithmetic. This is done indirectly - firstly, we don't deal with ordinals themselves, but rather with definable orderings - formulas \(R(a,b)\) which express \(a\prec b\) in some ordering.

Now we have to somehow express what it means that this ordering is a well-order. It's easy to state that it is a linear order, so I omit it. The last condition we express using something akin to the induction axiom scheme. Precisely, we define the transfinite induction scheme for \(\prec\), denoted \(\mathsf{TI}_\prec\), as the following collection of formulas: for each \(\varphi(n)\) a formula in the language of arithmetic, \[(\forall n:(\forall m:m\prec n\implies\varphi(m))\implies\varphi(n))\implies\forall n:\varphi(n).\] We now say that PA proves \(\prec\) is a well-order if it proves it's a linear order and it proves every axiom of \(\mathsf{TI}_\prec\). Finally, we say PA proves an ordinal \(\alpha\) is well-ordered if there is a recursive ordering \(\prec\) (i.e. the relation \(a\prec b\) is decidable) of order type \(\alpha\) which PA proves to be a well-order. Lastly, the proof-theoretic ordinal of PA is the supremum of ordinals which PA proves well-ordered.