User blog:Simply Beautiful Art/A hierarchy of Greatly Mahlo OCFs

Recently I've been trying to make a greatly Mahlo OCF, and here's what I have:

Fix a natural \(k\).

Define the following propositions:

\(G_0(A)\Leftrightarrow\forall f:\sup A\mapsto\sup A,\exists\alpha\in A,\forall\eta\in\alpha,f(\eta)\in\alpha\)

\(G_{n+1}(A)\Leftrightarrow\exists F\subsetneq\mathcal P(\sup A),\bigwedge_{k\in5}G_{n+1}^k(A,F)\)

\(G_{n+1}^0(A,F)\Leftrightarrow\emptyset\notin F\land A\in F\)

\(G_{n+1}^1(A,F)\Leftrightarrow\forall B\in F,\forall C\subseteq\sup A,B\subseteq C\Rightarrow C\in F\)

\(G_{n+1}^2(A,F)\Leftrightarrow\forall B\subseteq F,\operatorname{card}(B)\in\sup A\Rightarrow\bigcap B\in F\)

\(G_{n+1}^3(A,F)\Leftrightarrow\forall f:\sup A\mapsto F,\bigg\{\alpha\in\sup A~|~\alpha\in\bigcap_{\beta\in\alpha}f(\beta)\bigg\}\)

\(G_{n+1}^4(A,F)\Leftrightarrow\forall B\in F,\{\alpha\in B~|~\omega\in\operatorname{cof}(\alpha)\land G_n(\alpha\cap B)\}\in F\)

We can then define the following \(k+1\) OCFs for \(n\in k+1\), \(K\) as some sufficiently large ordinal, and \(\Xi\) either as some sufficiently large ordinal greater than \(K\) or the class of all ordinals:

\(B_n(\alpha,\kappa)_0=\kappa\cup\{0,K\}\)

\(B_n(\alpha,\kappa)_{m+1}=\{\gamma+\delta~|~\gamma,\delta\in B_n(\alpha,\kappa)_m\}\)

\({}\cup\{\theta_\eta^{\pi_0,\dots,\pi_{k-n-1}}(\mu)~|~\forall i\in k-n,\pi_i,\mu,\eta\in B_n(\alpha,\kappa)_m\land\eta\in\alpha\}\)

\(B_n(\alpha,\kappa)=\bigcup_{m\in\mathbb N}B_n(\alpha,\kappa)_m\)

\(\Xi(\pi_0,\dots,\pi_{k-n-1};\alpha)=\bigg\{\kappa\in\Xi(\pi_0,\dots;\pi_{k-n-1})~\bigg|~\kappa\notin B_n(\alpha,\kappa)\land{}\)

\(\alpha\in\operatorname{cl}(B_n(\alpha,\kappa))\land0\in\alpha\Rightarrow G^n\bigg(\kappa\cap\bigcap_{\eta\in\alpha\cap B_n(\alpha,\kappa)}(\pi_0,\dots,\pi_{k-n-1};\eta)\bigg)\bigg\}\)

\(\theta_\alpha^{\pi_0,\dots,\pi_{k-n-1}}=\operatorname{enum}(\Xi(\pi_0,\dots,\pi_{k-n-1};\alpha))\)

We can then define our lowest OCF as follows:

\(C(\alpha,\kappa)_0=\kappa\cup\{0,K\}\)

\(C(\alpha,\kappa)_{n+1}=\{\gamma+\delta~|~\gamma,\delta\in C(\alpha,\kappa)_n\}\)

\({}\cup\{\psi_\mu^{\pi_0,\dots,\pi_k}(\eta)~|~\forall i\in k+1,\pi_i,\mu,\eta\in C(\alpha,\kappa)_n\land\eta\in\alpha\}\)

\(C(\alpha,\kappa)=\bigcup_{n\in\mathbb N}C(\alpha,\kappa)_n\)

\(\psi_\mu^{\pi_0,\dots,\pi_k}(\alpha)=\min\{\kappa\in\Xi(\pi_0,\dots,\pi_{k-1};\pi_k)~|~\kappa=\theta_{\pi_k+1}^{\pi_0,\dots,\pi_{k-1}}(\mu)\cap C(\alpha,\kappa)\}\)

This is assuming my understanding of Greatly Mahlos is accurate. Some basic values:

\(\Xi(0,\dots,0;0)\) are the additive principals.

\(\Xi(0,\dots,0;1)\) are the uncountable regular ordinals.

\(\Xi(0,\dots,0;2)\) are the Mahlo ordinals.

\(\Xi(0,\dots,1;0)\) are the Greatly Mahlo ordinals.

\(\psi_0^{0,\dots,1,0}(0)\) being the first Greatly Mahlo ordinal.

\(\Xi(0,\dots,0;\psi_0^{0,\dots,1,0}(0))\) being the hyper-Mahlo ordinals less than \(\psi_0^{0,\dots,1,0}(0)\) and the \(\psi_0^{0,\dots,1,0}(0)\)-Mahlo ordinals.

\(\Xi(0,\dots,1;1)\) are the points where the Greatly Mahlo ordinals are stationary.

\(\Xi(0,\dots,1;2)\) are the points where \(\Xi(0,\dots,1;1)\) is stationary.

\(\Xi(0,\dots,2;0)\) are the points that cannot be reached by repeatedly taking stationary points and diagonalizing over the Greatly Mahlo ordinals.

\(\Xi(0,\dots,1,0;0)\) are the points that cannot be reached by repeatedly taking stationary points and diagonalizing over the set/class of ordinals of the form \(\Xi(0,\dots,0,\kappa;0)\).

And so on.