User blog comment:P進大好きbot/Relation between an OCF and an Ordinal Notation/@comment-30754445-20180810102658/@comment-30754445-20180810125741

I should have phrased that last part more carefully:

"And this cannot be done with any current notation confirmed to reaches PTO(ZFC) or even PTO(Z2)".

The limitation here is not in the notations themselves, but in our analyzing power. Nobody (yet) knows how the step-by-step process up to PTO(Z2) looks like. To achieve that goal, a strong OCF is not enough. We also need a way to track our progress, and this is something that TON doesn't provide (it is actually known to be an insanely difficult notation to analyze).

So the bottom line is: We don't know how strong TON is. It might(*) reach PTO(Z2), and it might not. Either way, at present time, it does not provide us with a step-by-step roadmap up to PTO(Z2).

On the other hand, we do have notations that are easily proven to reach PTO(Z2) and beyond. I don't really have any idea how they work, but I know that they do exist. I also know that by their very nature, they defy any attempt at doing a step-by-step analysis.

And that's the distinction I wanted to make in my previous comment.

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(*) It should be noted that the math pros on this wiki are quite skeptical of the claim that TON gets anywhere near PTO(Z2). The consensus among those-in-the-know seems to be that TON's strength is - at most - an ordinal called PTO(Π12−CA) (which can be seen as the second ordinal in the fundamental sequence of PTO(Z2)).