User blog comment:P進大好きbot/What is the greatest ordinal notation now?/@comment-1605058-20180623095923/@comment-1605058-20180623162012

What Gentzen's proof only establishes that if we have transfinite induction along \(\varepsilon_0\), then we can prove PA consistent, but it doesn't tell us that this is the least such ordinal. Indeed, it's not difficult to construct a shorter ordering, of length \(\omega\), which has the property that if induction holds along it, then PA is consistent. You can find the details here, page 3. The advantage of Gentzen's proof is that it uses a certain "natural" ordering of length \(\varepsilon_0\), but there is no way to make the notion of "natural" formal, which would be necessary to make sense of ordinal analysis at all.