My most recent OCF

P-bot has decided that this is likely welldefined: (Note : the whole of the LaTeX may not be visible)

\begin{eqnarray*}C_{0,m}(\alpha) = \{0,\Omega\} \\ C_{,m}(\alpha)=\bigcup_{n<\omega}(C_{n,m}(\alpha))\\ C_{n+1,m+1}(\alpha)=\{\gamma+\delta,\gamma\delta,\gamma^\delta,\phi_\gamma(\delta),(\psi_{0,m+1}(η))| \gamma,\delta,\eta\in(C_{n,m+1}(\alpha));\eta<\alpha\}\bigcup\\\{(\psi_{1,o}(\gamma)),(\psi_{0,o}(\gamma)),\Omega_\gamma| \gamma\in(C_{n,m+1}(\alpha));o<(m+1)\}\\ C_{n+1,0}(\alpha)=\{\gamma+\delta,\gamma\delta,\gamma^\delta,\phi_\gamma(\delta),(\psi_{0,0}(\eta)),\Omega_\gamma|\gamma,\delta,\eta\in(C_{n,0}(\alpha));\eta<\alpha\}\\ \psi_{0,m}(\alpha)=min\{\beta\in\Omega|\beta\notin(C_{,m}(\alpha))\}\\ \psi_{1,m}(\alpha)=sup((C_{,m}(\alpha)))\\ \\\end{eqnarray*}

Just posting this so that I can stop cluttering my old OCF post.

In plain text:

C_0,0(α)={0,Ω}

C_n+1,0(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,0(η)),w_γ|γ,δ,η∈(C_n,0(α));η<α}

C,0(α)=⋃(n<ω)C_n,0(α)

ψ_0,0(α)=min{β∈Ω|β∉C,0(α)}

ψ_1,0(α)=sup(C,0(α))

C_0,m(α)={0,Ω}

C_n+1,m+1(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,m+1(η)),(ψ_1,o(γ)),(ψ_0,o(γ)),w_γ| γ,δ,η∈(C_n,m+1(α));η<α,o<(m+1)}

C,m(α)=⋃(n<ω)(C_n,m(α))

ψ_0,m(α)=min{β∈Ω|β∉(C,m(α))}

ψ_1,m(α)=sup((C,m(α)))

Basically, it's a system of OCFs where each one can access the prior ones. This is why I thought it was welldefined. (And it is).

EDIT: another related one: \begin{eqnarray*}C_{0,m}(\alpha) = \{0,\Omega\} \\ C_{,m}(\alpha)=\bigcup_{n<\omega}(C_{n,m}(\alpha))\\ C_{n+1,m+1}(\alpha)=\{\gamma+\delta,\gamma\delta,\gamma^\delta,\phi_\gamma(\delta),(\psi_{0,m+1}(η))| \gamma,\delta,\eta\in(C_{n,m+1}(\alpha));\eta<\alpha\}\bigcup\\\{(\psi_{2,o}(\gamma)),(\psi_{1,o}(\gamma)),(\psi_{0,o}(\gamma)),\Omega_\gamma| \gamma\in(C_{n,m+1}(\alpha));o<(m+1)\}\\ C_{n+1,0}(\alpha)=\{\gamma+\delta,\gamma\delta,\gamma^\delta,\phi_\gamma(\delta),(\psi_{0,0}(\eta)),\Omega_\gamma|\gamma,\delta,\eta\in(C_{n,0}(\alpha));\eta<\alpha\}\\ \psi_{0,m}(\alpha)=min\{\beta\in\Omega|\beta\notin(C_{,m}(\alpha))\}\\ \psi_{1,m}(\alpha)=min(\beta>\Omega|\beta\notin(C_{,m}(\alpha)))\\ \psi_{2,m}(\alpha)=sup((C_{,m}(\alpha)))\\ \\\end{eqnarray*}

This one has a third function which finds the minimum ordinal \(\beta\) that is larger than \(\Omega\) and not in the set. This makes it somewhat stronger than the previous.