# Why do we define the proof-theoretic ordinal of a theory the way we do when there are unnatural well-orderings out there?

The proof-theoretic ordinal of first-order arithmetic ($$\mathsf{PA}$$) is $$\varepsilon_{0}$$. However, in pages 3 and 4 of Andreas Weiermann's Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results, he mentions it is possible to define a primitive recursive well-ordering $$\prec$$ whose order type is $$\omega$$ but such that $$\mathsf{PA}$$ cannot prove transfinite induction along $$\omega$$. No reference is provided other than the name Kreisel.

First, does anyone know how to construct $$\prec$$, or perhaps may provide a modern reference where this is accomplished?

Second, if such a pathological well-ordering $$\prec$$ exists, could not we use it to prove the consistency of $$\mathsf{PA}$$ by using finitary methods plus transfinite induction along $$\prec$$? If this is the case, why do we define the proof-theoretic ordinal of a theory $$T$$ the way we do, and not simply as

The least ordinal $$\alpha$$ such that transfinite induction along $$\alpha$$ (plus finitary methods) proves the consistency of $$T$$.

I guess this would depend on the ordinal notation used; and since the definition of proof-theoretic ordinal does not depend on a notation, it is then more robust. Is that it, or there is more?

EDIT: I just found out that my question is also answered in Math Overflow. There are very informative and useful answers provided there.

Re: your first question, this is disappointingly easy: writing "$$Con(\mathsf{PA},x)$$" for "There is no proof of a contradiction in $$\mathsf{PA}$$ of length $$," consider the relation given by $$a\prec b\quad \equiv \quad (a Intuitively, this describes the following process: we start listing the natural numbers in their usual order, and if we ever see a contradiction in $$\mathsf{PA}$$ we shift tactics (so to speak) and build a descending chain. For example, if there were a shortest proof of a contradiction in $$\mathsf{PA}$$ of length $$5$$, the $$\prec$$-order would look like $$0,1,2,3,4, ... ,8,7,6,5$$ and have ordertype $$5+\omega^*$$ (which is not a well-order, to put it mildly). This is a primitive recursive process, though, and so we have a p.r. "copy" of $$\omega$$ which $$\mathsf{PA}$$ can't appropriately analyze.

Note that this trick works with any ("reasonable") theory in place of $$\mathsf{PA}$$. In particular, your proposal would lead to every proof-theoretic ordinal being just $$\omega$$.