User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-7484840-20130719072932/@comment-5529393-20130719082126

Unfortunately, I don't know what you mean by I(a_1,a_2,...,(a_k),0...0) = b |--> I(a_1,a_2,...a_k,b,0...0). The key point is that weakly inaccesibles and hyper-weakly inaccessibles are "sufficiently large and far apart", so that it is impossible to reach one from below using weaker operations (same as with regular cardinals). So, for example, it is impossible to reach I(0) using any hierarchy based on a -> Omega_a, and it is impossible to reach I(1, 0) using any hierarchy based on a -> I(a). Note that all of these hierarchies go higher and higher as we extend the notation, as adding weakly Mahlos, hyper-weakly Mahlos and the like allow use to define higher and higher functions of I(0) and I(1, 0), and therefore stronger diagonalizations of a -> Omega_a and a -> I(a). It is this interplay between the various levels that give the notations their strength. So we can't define I(0) as the limit of the Veblen function based on a -> Omega_a, or as the SVO or LVO or BHO version of a -> Omega_a; the hierarchy keeps going up forever, so we need I(0) to always be larger no matter how high the hierarchy goes. (And I(1) must be larger than the next hierarchy, etc.)