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This is the continuation of my previous post. I added few new rules in fundamental sequences system to extend up to \(\psi(I_{I_{I...}})=\psi(M^2)\). That allow to define FS for Hypcos's notation with weakly inaccessibles. I publish post to take into account possible critical remarks and then to add given FSsystem in the article List of systems of fundamental sequences.
\(\rho\) and \(\pi\) are always regular cardinals written as \(\Omega_{\nu+1}\) or \(I_{\mu+1}\) i.e. \(\text{cof}(\rho)=\rho\) and \(\text{cof}(\pi)=\pi\).
\(\Omega_\alpha\) with \(\alpha>0\) is the \(\alpha\)th uncountable cardinal, \(I_\alpha\) with \(\alpha>0\) is the \(\alpha\)th weakly inaccessible cardinal and for this notation \(I_0=\Omega_0=0\).
Then,
\(C_0(\alpha,\…
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This is the system of fundamental sequences which I propose to use as canonical for Hypcos's notation with the first weakly inaccessible cardinal.
Defenition:
\(\rho\) and \(\pi\) are always regular cardinals i.e. \(\rho,\pi\in\{\Omega_{\nu+1}\}\cup\{ I\}\) i.e. \(\text{cof}(\rho)=\rho\) and \(\text{cof}(\pi)=\pi\).
\( I\) is the first weakly inaccessible cardinal, \(\Omega_\alpha\) with \(\alpha>0\) is the \(\alpha\)th uncountable cardinal and for this notation \(\Omega_0=0\).
Then,
\(C_0(\alpha,\beta) = \beta\cup\{0,I\} \)
\(C_{n+1}(\alpha,\beta) = \{\gamma+\delta\gamma,\delta\in C_n(\alpha,\beta)\} \cup \{\Omega_\gamma\gamma\in C_n(\alpha,\beta)\} \cup \{\psi_\pi(\gamma)\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma
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===Up to .
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Expression written in notation is exactly equal to output of function of fast growing hierarchy indexed by ordinal number generated by Buchholz's function.
What inspired me:
Chronolegends's Egg Notation
Deedlit's notation
Buchholz's function
Fastgrowing hierarchy
ab corresponds to \(f_b(a)\) where \(f_b\) is a function of fastgrowing hierarchy
To the right of the sign "" :
1) () corresponds to 1, (()) corresponds to \(\omega\) and (...) always corresponds to a countable ordinal number,
2) \(()_b\) corresponds to \(\Omega_b\) where \(\Omega_b=\aleph_b=\psi_b(0)\) denotes bth uncountable ordinal,
3) \((...)_b\) corresponds to \(\psi_b(...)\) where \(\psi_b\) denotes Buchholz's function.
notation allows to obtain ultimatively short ruleset for w…
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The Buchholz's psifunctions are a hierarchy of singleargument ordinal functions \(\psi_\nu(\alpha)\) introduced by Wilfried Buchholz in 1986. These functions are a simplified version of the \(\theta\)functions, but nevertheless have the same strength as those.
Definition
\(C_\nu^0(\alpha) = \Omega_\nu\),
\(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma  P(\gamma) \subseteq C_\nu^n(\alpha)\} \cup \{\psi_\mu(\xi)  \xi \in \alpha \cap C_\nu^n(\alpha) \wedge \xi \in C_\mu(\xi) \wedge \mu \leq \omega\}\),
\(C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)\),
\( \psi_\nu(\alpha) = \min\{\gamma  \gamma \not\in C_\nu(\alpha)\}\),
where
\(\Omega_\nu=\left\{\begin{array}{lcr} 1\text{ if }\nu=0\\ \aleph_\nu\text{ if }\nu>0\\ \end{array}\righ…
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