Extended Madore's $\psi$

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Definition of $\psi_\alpha$

First let $Cl(F,S)$ for a set $S$ and a set of functions $F$ to be the smallest (up to $\subseteq$) set $X$ such that $S\subseteq X$ and $X$ is closed under every function in $F$.

Then we let $\psi_\beta(\alpha)=\min\{\gamma:\gamma\not\in C_\beta(\alpha)\}$ where $C_\beta(\alpha)=Cl(\{\psi_\beta\upharpoonright\alpha,+,\cdot,\wedge\}\cup\{\psi_\gamma:\gamma<\beta\},\{0,1,\omega,\Omega\})$ where $f\upharpoonright\alpha$ is $f$ restricted to the domain $\alpha$.

Clearly by definition of Madore's $\psi$, $\psi(\alpha)=\psi_0(\alpha)$.

Values of $\psi_1$

The following is known about $\psi_1$:

  • $\psi_1(0)=\psi_0(\varepsilon_{\Omega+1})=BHO$ where $BHO$ is the Bachmann-Howard Ordinal
  • $\psi_1(\alpha)=\varepsilon_{BHO+\alpha}$ whenever $\alpha\leq\zeta_{BHO+1}$
  • $\psi_1(\alpha)=\zeta_{BHO+1}$ whenever $\zeta_{BHO+1}\leq\alpha\leq\Omega$
  • $\psi_1(\Omega+\alpha)=\varepsilon_{\zeta_{BHO+1}+\alpha}$ whenever $\alpha\leq\zeta_{BHO+2}$
  • $\psi_1(\Omega\cdot 2)=\varphi_3(BHO+1)$ where $\varphi$ is the Veblen function
  • $\psi_1(\Omega\cdot 3)=\varphi_3(BHO+2)$
  • $\psi_1(\Omega^2)=\varphi_4(BHO+1)$
  • $\psi_1(\Omega^2\cdot 2)=\varphi_3(\varphi_4(BHO+1)+1)$
  • $\psi_1(\Omega^3)=\varphi_5(BHO+1)$

At this point we must introduce a new function $\Theta$ similar to the Extended Veblen function. $\Theta_\alpha(\beta)$ is the $\alpha$-th fixed point of $\varphi_\gamma(\beta)$ (with constant $\beta$). Then $\Theta_{\gamma,0}(\alpha)$ is the $\gamma$-th fixed point of $\Theta_{\beta}(\alpha)$ (with constant $\alpha$). The $\beta$-th fixed point of $\Theta_{\gamma,0}(\alpha)$ is $\Theta_{\beta,1}(\alpha)$. The $\gamma$-th fixed point of $\Theta_{\beta,1}(\alpha)$ is $\Theta_{\gamma,2}(\alpha)$. This continues, until the $\gamma$-th fixed point of $\Theta_{\zeta,\beta}(\alpha)$ (with constants $\zeta$ and $\alpha$) which is $\Theta_{\gamma,\zeta,0}(\alpha)$.

  • $\psi_1(\Omega^\Omega)=\Theta_0(BHO+1)$
  • $\psi_1(\Omega^\Omega + \Omega)=\varphi_3(\Theta_0(BHO+1)+1)$
  • $\psi_1(\Omega^\Omega + \Omega\cdot 2)=\varphi_3(\Theta_0(BHO+1)+2)$
  • $\psi_1(\Omega^\Omega+\Omega^2)=\varphi_4(\Theta_0(BHO+1)+1)$
  • $\psi_1(\Omega^\Omega\cdot 2)=\varphi_{\psi_1(\Omega^\Omega\cdot 2)}(\Theta_0(BHO+1))=\Theta_1(BHO+1)$
  • $\psi_1(\Omega^{\Omega+1})=\Theta_{\psi_1(\Omega^{\Omega+1})}(BHO+1)=\Theta_{0,0}(BHO+1)$
  • $\psi_1(\Omega^{\Omega+1}+\Omega^2)=\varphi_4(\Theta_{0,0}(BHO+1)+1)$
  • $\psi_1(\Omega^{\Omega+1}+\Omega^\Omega)=\varphi_{\psi_1(\Omega^{\Omega+1}+\Omega^\Omega)}(\Theta_{0,0}(BHO+1))=\Theta_0(\Theta_{0,0}(BHO+1))$
  • $\psi_1(\Omega^{\Omega+1}+\Omega^\Omega\cdot 2)=\varphi_{\psi_1(\Omega^{\Omega+1}+\Omega^\Omega)}(\Theta_{0,0}(BHO+1))=\Theta_1(\Theta_{0,0}(BHO+1))$
  • $\psi_1(\Omega^{\Omega+1}\cdot 2)=\Theta_{\psi_1(\Omega^{\Omega+1}\cdot 2)}(\Theta_{0,0}(BHO+1))=\Theta_{0,0}(\Theta_{0,0}(BHO+1))=\Theta_{1,0}(BHO+1)$
  • $\psi_1(\Omega^{\Omega+2})=\Theta_{\psi_1(\Omega^{\Omega+2}),0}(BHO+1)=\Theta_{0,1}(BHO+1)$
  • $\psi_1(\Omega^{\Omega\cdot 2})=\Theta_{0,0,0}(BHO+1)$
  • $\psi_1(\Omega^{\Omega^2})=\Theta_{0,0,0,0,0,...}(BHO+1)$
  • $\psi_1(\Omega^{\Omega^\omega})=\Theta_{0,0,0,0...(\psi_1(\Omega^{\Omega^\omega})\;\mathrm{zeros})}(BHO+1)$
  • $\psi_1(\varepsilon_{\Omega+1})=\psi_1(\alpha)$ whenever $\alpha\geq\varepsilon_{\Omega+1}$

The First Impredictive Ordinal With this Definition

The first impredictive ordinal with this definition is the first ordinal $\alpha$ such that $\psi_\alpha(0)=\alpha$. Let it's name be $\mu_0$. It is actually not known just how large this ordinal is, but it is known to be smaller than $\omega_1^{ck}$. This ordinal is so large that it dwarfs the supremum of $\{\varepsilon_0,BHO,\psi_1(\varepsilon_{\Omega+1})...\}$, which itself is incredibly large.