Difference between revisions of "User blog:Zetapology/Extended Madore's Psi"

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== The First Impredictive Ordinal With this Definition ==
 
== 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 [[admissible|$\omega_1^{ck}$]]. This ordinal is so large that it dwarfs the supremum of $\{\varepsilon_0,BHO,\psi_1(\varepsilon_{\Omega+1})...\}$, which is larger than any known countable ordinal that can be proven to exist in ZFC (for an example of an ordinal which is countable if it exists but cannot be proven in ZFC is the $\vartheta$ collapse of a Mahlo cardinal.)
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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 [[admissible|$\omega_1^{ck}$]]. This ordinal is so large that it dwarfs the supremum of $\{\varepsilon_0,BHO,\psi_1(\varepsilon_{\Omega+1})...\}$, which itself is incredibly large.

Latest revision as of 08:14, 9 October 2017


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.