User blog comment:Bubby3/Walkthrough of BMS./@comment-35470197-20190216042718/@comment-35470197-20190220134728

Thank you. But your conversion to standard Buchholz seems to be incorrect. Concerning the follow equality

> (0,0)(1,1)(2,2)(3,3)(2,2) = ψ_0(Ω_3+ψ_1(Ω_3+ψ_2(Ω_3)+Ω_2))

we have

G_0(Ω_3+ψ_1(Ω_3+ψ_2(Ω_3)+Ω_2)) = {0,Ω_3+ψ_2(Ω_3)+Ω_2,Ω_3}

and hence Ω_3+ψ_1(Ω_3+ψ_2(Ω_3)+Ω_2) does ont belong to C_0(Ω_3+ψ_1(Ω_3+ψ_2(Ω_3)+Ω_2)). (Here G_0 is the primitive recursive relation introduced by Buchholz in his original paper.) Therefore it is a non-standard expression. Although you might intend "standard OCF" but not "standard expression", I think that the result itself is incorrect. I think that it corresponds to ψ_0(Ω_3+Ω_2), which is strictly greater than your result. (I am sorry that I do not have a way to explain the reason without proof.)