User:Ikosarakt1/Extension of traditional notations for non-recursive ordinals

First time, I invented \(\theta_1(\alpha)\) function to notate some non-recursive ordinals in traditional notation more compactly. We define \(\theta_1(0,\alpha) = \omega_{\alpha}^\text{CK}\) and keep all other rules from \(\theta(\alpha)\) function the same (adopted, of course, to \(\theta_1(\alpha)\) function).

\(\theta_1(0,1) = \omega_1^\text{CK}\)

\(\theta_1(0,2) = \omega_2^\text{CK}\)

\(\theta_1(0,m) = \omega_m^\text{CK}\)

\(\theta_1(0,\omega) = \omega_{\omega}^\text{CK}\)

\(\theta_1(0,\theta(0,1)) = \omega_{\omega_1^\text{CK}}^\text{CK}\)

\(\theta_1(1,0) = \alpha \) (Goucher's ordinal)

\(\theta_1(0,\theta_1(1,0)+1) = \omega_{\alpha+1}^\text{CK}\)

\(\theta_1(0,\theta_1(0,\theta_1(1,0)+1)) = \omega_{\omega_{\alpha+1}^\text{CK}}^\text{CK}\)

\(\theta_1(1,1) = \beta \) (Let allow me to call it second Goucher's ordinal)

\(\theta_1(1,m)\) (m-1th Goucher's ordinal, also m-1th fixed point of function that Little Peng mentioned here)

\(\theta_1(2,0) = \theta_1(2)\) (guessed order type of \(\Xi_2(n)\), Xi function with oracle combinator)

\(\theta_1(m)\) (guessed order type of \(\Xi_m(n)\), Xi function with m-1 oracle combinators)

\(\theta_1(\omega)\)

\(\theta_1(\theta(\Omega))\)

\(\theta_1(\theta_1(0,1))\)

\(\theta_1(\Omega)\)