User blog comment:B1mb0w/Fundamental Sequences/@comment-1605058-20151112165051/@comment-5529393-20151114151804

There aren't any "generally accepted" fundamental sequences past $$\varepsilon_0$$, and arguably not even before. If we were to to define our fundamental sequences based on Schutte's Klammersymbolen, then the most natural value of {(1,w)}[2] would be {(1,2)}, i.e. we would have SVO[2] = Gamma_0. Now suppose we use a notation up to the Bachmann-Howard ordinal;  using the $$\psi$$ notation from my blog, we would most naturally have $$\psi(\Omega^{\Omega^\omega})[2] = \psi(\Omega^{\Omega^2})$$ which would mean SVO[2] = phi(1,0,0,0) (the Ackermann ordinal). Using the $$\vartheta$$ notation, we would most naturally have $$\vartheta(\Omega^\omega) = \vartheta(\Omega^2)$$ which would mean SVO[2] = Gamma_0. Finally, if we use the $$\theta$$ notation we would most naturally have $$\theta(\Omega^\omega,0)[2] = \theta(\Omega^2,0)$$, which would mean SVO[2] = phi(1,0,0,0).

So, some notations suggest SVO[2] = Gamma_0, and some suggest that SVO[2] = phi(1,0,0,0); those are the two most likely candidates.