User:Wythagoras/Ordinals/Extensions of non-recursive ordinals

We'll start with \(\omega_1^CK\). It is the first ordinal which cannot be obtained using \(\omega\) and recursion of the type that is used in the compact ordinals and beyond. That recursion we denote with \(R_1\).

\(\omega_2^CK\) is the second ordinal which cannot be obtained using \(\omega\) and \(R_1\). Lets call it \(R_1(2)\) We can also have the \(\omega\)th ordinal which cannot be obtained using \(\omega\) and \(R_1\), \(R_1(\omega)\), and the \(\alpha\)th ordinal, \(R_1(\alpha)\). The first fixed point of the \(R_1(n)\) function is \(R_1(\Omega)\), which is equal to \(\varepsilon^CK_0\). We can have \(\Omega_2\) inside the \(R_1\) function, or even compact ordinals inside the \(R_1\) function.

The first ordinal which cannot be obtained using the \(R_1\) function and \(R_1\) itself is \(R_2(1)\), or \(\omega_{1,2}^CK\) in Ikosarakt's system. Then we have \(\omega_{2,2}^CK\), which is \(R_1(R_2(1)+1)\), and \(\omega_{1,3}^CK\), which cannot be obtained using the \(R_1\) function, \(R_1\) itself and \(R_2(1)\).

We can continue with \(R_2(\omega)=\omega_{1,\omega}^CK\) and so on. The first ordinal which cannot be obtained using the \(R_2\) function and \(R_1\) is \(\omega_{1,1,2}^CK\), but it isn't the limit of the \(R_2\) function, which is \(\omega_{1\text{|}2}^CK\) in Ikosarakt's system and \(R_3(1)\) in my system.

Next we have \(R_4(1)\) which is \(\omega_{1\text{|}_22}^CK\) and \(R_\omega(1)\) which is \(\omega_{1\text{|}_\omega2}^CK\)