User blog comment:Ubersketch/What is the most powerful OCF?/@comment-11227630-20181104033916

The strongest OCF defined using Skolem Hull is an OCF by Michael Rathjen, which uses \(<\Gamma_0\) many reducible cardinals and the hierarchy of \(\mathcal A\)-\(\eta\)-\(\mathcal F\)-reducible cardinals (where \(\mathcal A\) is a class of ordinals, \(\eta\) is an ordinal, \(\mathcal F\) is a class of formulas). It might go further than \(\Delta^1_3\text{-CA}\).

Taranovsky's ordinal notation might be not so strong as he claimed because it missed some structure such as somewhere between "x is C(C(Ω2+C(Ω2,C(Ω2,x)),x),C(Ω2,x))-stable" and "x is C(Ω2,C(Ω2,x))-stable", somewhere between "x is C(C(Ω2+C(Ω2,C(Ω2,C(Ω2,x))),x),C(Ω2,C(Ω2,x)))-stable" and "x is C(Ω2,C(Ω2,C(Ω2,x)))-stable", and so on. More details are shown here.

"A large cardinal an OCF can diagonalize over or collapse" can be arbitrarily large, but the resulting notation might be weak. For example, in the OCF \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0,\Omega\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\omega^\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{\psi(\gamma)|\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \psi(\alpha) &=& \min\{\beta<\Omega|C(\alpha,\beta)\cap\Omega\subseteq\beta\} \end{eqnarray*} the \(\Omega\) can be as small as the BHO itself, or as large as the admissible ordinal, or the uncountable cardinal, or something even larger. However large \(\Omega\) is, its strength is always BHO. What really matters, is not the absolute size of the ordinal for collapsing, but is the relative complexity of the hierarchy of large ordinals. Using the concept "\(\alpha\) is \(\Pi^1_n\)-indescribable on set \(A\)" a stronger OCF can be built than only using the concept "\(\alpha\) is \(\Pi^1_n\)-indescribable", although their sizes are comparable. Another example of complex hierarchy is the hierarchy for stable ordinals. More discussions are shown here.