User blog comment:DrCeasium/Hyperfactorial array notation: Analysis part 3/@comment-1605058-20130529124938/@comment-5150073-20130529131729

I know that canonical notations for theta function can go as far as the first ordinal such that theta(Omega_alpha) = alpha, but it is not a big problem to define some ordinals beyond. For example, let's define:

theta(alpha_0) = lim(Omega,Omega_Omega,Omega_Omega_Omega,...)

theta(alpha_(beta+1)) = lim(Omega_(alpha_beta),Omega_Omega_(alpha_beta),Omega_Omega_Omega(alpha_beta),...)

theta(superalpha_0) = lim(alpha,alpha_alpha,alpha_alpha_alpha,...)

theta(superalpha_(beta+1)) = lim(alpha_(superalpha_beta),alpha_alpha_(superalpha_beta),...)

Etc, we can continue with duperalpha_0, truperalpha_0 (a bit unserious names, but you see the logic). We can note that we have fixed points here, and make the function theta_1. I already proposed such name in one of my sub-pages, but now I decided to adopt it for recursive ordinals. Okay, let's theta_1(0) = Omega, theta_1(1) = alpha_0, theta_1(2) = superalpha_0, etc. It otherwords, each increment here means next indexing fixed point in theta(alpha). All other rules can be mirrored from theta(alpha).

We can define higher order theta functions, for example, let each increment in theta_2(alpha) means next indexing fixed point in theta_1(alpha) and generally:

In theta_(beta+1)(alpha) each increment means next indexing fixed point in theta_(beta)(alpha). So, just imagine how large the limit ordinal of all this can be: it is the first one such that theta_alpha(alpha) = alpha. But largeness of ordinals isn't graspable in googological way, so to help googology, I can just define the number f_alpha(100).

Anyone can compare f_alpha(100) to Loader's number?