User blog comment:Ubersketch/A proposal for a standard/@comment-11227630-20190813030043

OCFs by Rathjen or Stegert are unnecessarily complicated in some way. They aimed for ordinal analysis, so their structure might match some structure in proofs. But they are not suitable for googology, because some of those structures are unnecessary.

Take Rathjen's OCF for \(\Pi_3\)-reflection for example. (\(\mathcal K\) is the weakly compact cardinal)



OCFs after simplification can be used in googology. The simplification takes much work for complicated OCF, though.
 * 1) The \(\xi<\mathcal K\) limitation in \((\xi\mapsto\Omega_\xi)_{\xi<\mathcal K}\) is unnecessary. Removal of the limitation will improve the strength.
 * 2) After 1, the \(\varphi\) is unnecessary and can be removed without affecting the strength. (But without 1, the removal will weaken the notation below \(\mathcal K\cdot\omega\))
 * 3) The \(\Xi\) is unnecessary and can be removed without affecting the strength.
 * 4) \(M^0\) and \(M^\alpha\) are defined separately. "Stationary set" operation starts with regular cardinals so \(M^0\) must be defined separately. However, "\(\Pi^1_0\)-indescribable on" operation yields similar hierarchy without defining \(M^0\) separately. (In this hierarchy, \(M^1\) is inaccessible cardinals instead of regular cardinals; higher \(M^\alpha\) are Mahlo cardinals instead of weakly Mahlo cardinals.) Anyway, this change does not affect the syntax.
 * 5) Finally and the most sophisticated, removing the "\(\land\pi,\alpha\in C(\alpha,\rho)\)" in the definition of \(\Psi_\pi^\xi(\alpha)\) and the "\(\land\alpha\in C(\alpha,\rho)\)" in the definition of \(M^\alpha\) does not affect the strength. In the original definition, \(\Psi^0_\Omega(0)=\varphi(1,0,0)\), \(\Psi^0_\Omega(1)=\varphi(1,0,1)\), \(\Psi^0_\Omega(\omega)=\varphi(1,0,\omega)\), etc. until \(\Psi^0_\Omega(\varphi(1,1,0))=\varphi(1,0,\varphi(1,1,0)+1)\), which skip over the \(\varphi(1,1,0)\), and \(\Psi^0_\Omega(\Omega)=\varphi(1,1,0)\) fill the "hole". Original \(\Psi_\pi^\xi(\alpha)\) is not monotonic in \(\alpha\), and it works in the \(\vartheta\)-way. After the removal, \(\Psi_\pi^\xi(\alpha)\) becomes monotonic in \(\alpha\), and it generate values in \(\psi\)-way. For example, \(\Psi^0_\Omega(0)=\varphi(1,0,0)\), \(\Psi^0_\Omega(1)=\varphi(1,0,1)\), \(\Psi^0_\Omega(\omega)=\varphi(1,0,\omega)\), etc. until \(\Psi^0_\Omega(\varphi(1,1,0))=\varphi(1,1,0)\), and \(\Psi^0_\Omega(\alpha)=\varphi(1,1,0)\) for \(\varphi(1,1,0)\le\alpha\le\Omega\), and then \(\Psi^0_\Omega(\Omega+1)=\varphi(1,0,\varphi(1,1,0)+1)\).