User blog comment:PsiCubed2/Help Me Understand Ordinals Beyond the BHO/@comment-30118230-20170220203230

Well,.....

theta_1 works just like theta but on \(\Omega\).

If we take for example \(\varepsilon_0\) which is also representable as \(\vartheta(\Omega)\),then what is \(\vartheta_1(\Omega_2)\)?Well,it's \(\verepsilon_{\Omega+1}\).\(\vertheta_1\) works like \(\vartheta\) but using \(\Omega+1\) at the end,instead of 0.Another example would be using \(\Gamma_0\),which us the same thing as \(\vartheta(\Omega^2)\).What is \(\vartheta_1(\Omega^2_2)\)?It's \(\Gamma_{\Omega +1}\) and so on.It basically the same thing as \(\vartheta\) but with the primary definition changed so that \(\vartheta(\alpha) = \omega^\alpha\) and \(\vartheta_1(\alpha) = \Omega^\alpha\).Then we need a cardinal bigger than anything constructable with \(\Omega\).We call that \(\Omega_2\) and \(\Omega_2\) diagonalizes over \(\vartheta_1\) the same way \(\Omega\) diagonalises over \(\vartheta\).Then,after things like \(\varepsilon_{\Omega_2 +1}\) and \(\Gamma_{\Omega_2 +1}\) and so on we need to diagonalise over \(\Omega_2\).For that we use \(\vartheta_2\),which works like \(\vartheta_1\) but on \(\Omega_2\) instead of \(\Omega\).And now we use \(\Omega_3\) to diagonalize over it and so on.Notice that \(\vartheta_k\) works on \(\Omega_k\) and it uses \(\Omega_{k+1}\) as a diagonalizer and that \(\vartheta(\Omega_{k+1})\) is shorthand for \(\vartheta(\vartheta_1(\vartheta_2(.....\vartheta_k(\Omega_{k+1})....)))\).Then,if \(\alpha\) is a limit ordinal,then \(\Omega_\alpha\) is the supremum of \(\Omega_\beta\) for all ordinals \(\beta < \alpha\).