User blog:Simply Beautiful Art/Uncountably Nested Wfp OCF

Recently I came up with an OCF which may very well simulate the strength of $$M_{M_{M_{\ddots}}}$$ using only $$\Omega_\alpha$$ and a lot of fixed points. It never reaches the first inaccessible, and here goes for the definition:

$$D(\alpha,\varphi)= \{0\}$$ $$\cup\{\gamma+\delta,\Omega_\gamma^\delta,\chi_\pi(\eta)~|~\zeta\in\varphi\land\gamma,\delta,\pi,\eta\in D(\alpha,\zeta)\land\eta\in\alpha\}$$ $$\cup\{\sup B~|~\xi\in\varphi\land B\subset D(\alpha,\xi)\}$$

$$\chi_\pi(0)=\sup D(0,\Omega_\pi)$$

$$\chi_\pi(\alpha)=\sup(D(\alpha,\Omega_\pi)\cap\chi_{\pi+1}(0))$$

$$C(\alpha,0)=\{0\}$$

$$C(\alpha,n+1)=\{\gamma+\delta,\Omega_\gamma^\delta,\chi_\gamma(\delta),\psi_\gamma(\eta)~|~\gamma,\delta,\eta\in C(\alpha,n)\land\eta\in\alpha\}$$

$$C(\alpha)=\bigcup_{n\in\omega}C(\alpha,n)$$

$$\psi_\beta(\alpha)=\sup(C(\alpha)\cap\Omega_\beta)$$

Some values of $$\chi$$:

$$\chi_0(0)=\Omega_{\Omega_{\Omega_{\ddots}}}$$

$$\chi_0(1)=\Omega_{\Omega_{\Omega_{\ddots_{\Omega_{\chi_0(0)+1}}}}}$$

$$\chi_0(\omega)=\sup\{\chi_0(0),\chi_0(1),\chi_0(2),\dots\}$$

$$\chi_0(\Omega)=\chi_0(\varepsilon_0)$$

$$\chi_1(0)=\Omega$$th fixed point of $$\beta\mapsto\Omega_\beta$$

$$\chi_1(1)=\Omega2$$th fixed point of $$\beta\mapsto\Omega_\beta$$

$$\chi_1(2)=\Omega3$$th fixed point of $$\beta\mapsto\Omega_\beta$$

$$\chi_1(\Omega)=\Omega^2$$th fixed point of $$\beta\mapsto\Omega_\beta$$

$$\chi_1(\Omega_2)=\chi_1(\varepsilon_{\Omega2})$$

$$\chi_1(\chi_2(0))=\Omega$$th fixed point of $$\beta\mapsto\chi_1(\beta)$$

And it continues further from there. In any case, I speculate that $$\chi_1(0)$$ behaves like the first inaccessible, $$\chi_1(1)$$ behaves like the second inaccessible, etc., $$\chi_2(0)$$ behaves like the first weakly Mahlo, though I'm a little lost on what higher values may correspond to.