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# Denis Maksudov

## aka Denis

My favorite wikis
• I live in Russian Federation, Republic of Bashkortostan, Ufa
• I was born on January 15
• I am Male
• ## FS for Hypcos's notation (with weakly inaccessibles cardinals) up to Î¨(M^2)

August 17, 2017 by 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,\â€¦ Read more > • ## Canonical fundamental sequences for Hypcos's notation with the first weakly inaccessible cardinal. August 14, 2017 by 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 Read more > • ## Comparison of notations May 28, 2017 by Denis Maksudov ===Up to . Read more > • ## 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 wâ€¦

• ## The extension of Buchholz's function

April 15, 2017 by 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}\righâ€¦