User blog comment:Googology Noob/Ordinal FGH, with an actual definition!/@comment-26428969-20151228152713/@comment-27173506-20151230163547

f_0(alpha)=alpha+1

f_1(alpha)={alpha+1,alpha+2,alpha+3...}[alpha]=alpha+alpha=alpha*2

f_2(alpha)={alpha*2,alpha*2*2,alpha*2*2*2...}[alpha]={alpha*2,alpha*4,alpha*8...}[alpha]=alpha*2^alpha=alpha*alpha=alpha^2

f_3(alpha)={alpha^2,(alpha^2)^2,((alpha^2)^2)^2...}[alpha]={alpha^2,alpha^4,alpha^8...}[alpha]=alpha^2^alpha=alpha^alpha

All good up to here, right?

f_4(alpha)={alpha^alpha,(alpha^alpha)^(alpha^alpha),((alpha^alpha)^(alpha^alpha))^((alpha^alpha)^(alpha^alpha))...}[alpha]~{alpha^alpha,alpha^alpha^alpha,alpha^alpha^alpha^alpha}[alpha]=alpha^^alpha~e_alpha (or e_alpha*2 for fixed points, but that's why we use approximations)

f_5(alpha)={f_4(alpha),f_4(f_4(alpha)),f_4(f_4(f_4(alpha)))...}[alpha]~{e_alpha,e_e_alpha,e_e_e_alpha...}[alpha]~zeta_alpha (again, or zeta_alpha*2 for fixed points)

You may have noticed that these comparisions do not work for all ordinals (for example, plain old omega). This is because they hold only for quite large ordinals, but they are quite accurate eventually.

From here, each increase of the subscript by one will produce an (infinite!) recursion of the previous function, just like the 2 argument Veblen hierarchy, so f_m+3(alpha)~phi(m,alpha). Therefore:

f_w(alpha)~phi(alpha,0)

f_w+1(alpha)={f_w(alpha),f_w(f_w(alpha)),f_w(f_w(f_w(alpha)))...}[alpha]~{phi(alpha,0),phi(phi(alpha,0),0),phi(phi(phi(alpha,0),0),0)...}[alpha]~phi(1,0,alpha) (for sufficiently large alpha).

f_w+2(alpha)~{phi(1,0,alpha),phi(1,0,phi(1,0,alpha)),phi(1,0,phi(1,0,phi(1,0,alpha)))...}[alpha]~phi(1,alpha,0) (likewise)

f_w+3(alpha)~{phi(1,alpha,0),phi(1,alpha,phi(1,alpha,0)),phi(1,alpha,phi(1,alpha,phi(1,alpha,0)))...}[alpha]~phi(1,alpha*2,0) (for sufficiently large alpha)

Do I have to go over all of the things I did up to f_w*2(alpha), which are pretty much in the blog post?

f_w2(alpha)~phi(1,alpha*alpha,0)

f_w2+1(alpha)={f_w2(alpha),f_w2(f_w2(alpha)),f_w2(f_w2(f_w2(alpha)))...}[alpha]~{phi(1,alpha*alpha,0),phi(1,(phi(1,alpha*alpha,0),0),phi(1,phi(1,phi(1,alpha*alpha,0),0),0)...}[alpha]~phi(alpha,0,0) (for sufficiently large alpha. With omega it merely produces the limit of finite nestings, which is equal to phi(2,0,0)

<p style="font-weight:normal;">Is this good? Is there something that I did wrong?

<p style="font-weight:normal;">I realize that it is not the most formal definition, but I can't think of any better. If I do find (or if anyone can give me advice on how to do it) rest be assured that I'll post here ASAP.