User blog comment:Boboris02/Ordinal Analysis of Theories/@comment-11227630-20180213035846/@comment-30118230-20180214092214

Let me just clarify something.

A notation can reach past the PTO of ZFC and still be provable within ZFC.

The definition of a PTO is not the least ordinal not provable in a theory T.

Instead, a PTO of a theory T is the least ordinal \(\alpha\) such that for all well orders \(\mathbb{X},R\)  of length \(\alpha\), T cannot prove R recursive.

The reason why the first definition doesn't work is because most theories can't talk about ordinals directly either because they are too weak, or in the case of arithmetical theories, because they only talk about finite numbers and functions on finite numbers, while other thoeries like ZFC can talk about ordinals directly and are strong enough to talk about ordinals directly. ZFC can prove well-orderings for all ordinals, so by the first "definition", ZFC has not PTO. Also, according to the first definition, all arithmetical theories have a PTO of \(\omega\), which is just nonsense.

Again, ZFC can prove well-orderings for any ordinal: recursive, unrecursive, uncounable, inaccessible, you name it. ZFC can use powerset operators to prove successor oridnals, so there is no real limit to that use. This means it can prove well-orderings for it's own PTO easily.