I define this "function" for use uncountable ordinal in ordinal hierarchies.
Definition[]
Denoted as \(\alpha[a,b]\).
- If cofinality of \(\alpha\) equal of smaller than \(\omega\): \(\alpha[a,b] = \alpha[a]\)
- Otherwise (cofinality of \(\alpha\) larger than \(\omega\)): \(\alpha[a,b] = \alpha[g_{\beta}(a)\Omega(b)]\) where \(\beta\) is the largest possible ordinal smaller than cofinality of \(\alpha\) satisfies \(g_{\beta}(b)\Omega(b) = a\)
I define Uncountable SGH as:
- \(g_{\alpha[a,b]}(a)\Omega(b) = g_\alpha(a)\Omega(b)\)