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Denis Maksudov

aka Denis

  • I live in Russian Federation, Republic of Bashkortostan, Ufa
  • I was born on January 15
  • I am Male
  • Denis Maksudov

    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 FS-system 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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  • Denis Maksudov

    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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  • Denis Maksudov

    ===Up to .

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  • Denis Maksudov

    I-notation

    May 20, 2017 by Denis Maksudov

    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

    Fast-growing hierarchy

    a|b corresponds to \(f_b(a)\) where \(f_b\) is a function of fast-growing 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 b-th 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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  • Denis Maksudov

    The Buchholz's psi-functions are a hierarchy of single-argument 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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