My system of number names (Fast Growing Series, FGS)

This is a naming system for numbers that are defined using the fast-growing hierarchy.

The numbers were grouped into several series also called fast-growing series or FGS. Aims of this system of naming for numbers:

1. number names must be connected with the fast-growing hierarchy,

2. number names must allow to easily restore mathematical expressions, which define those numbers.

Section I. The fast-growing hierarchy

The fast-growing hierarchy is defined as follows:

• \(f_0(n) = n+1\)
• \(f_\alpha^{m+1}(n) = f_\alpha(f_\alpha^m(n))\)
• \(f_\alpha^0(n) = n\)
• \(f_{\alpha+1}(n) = f_\alpha^n(n)\)
• \(f_\alpha(n) = f_{\alpha[n]}(n)\) if \(\alpha\) is a countable limit ordinal

where \(n\), \(m\) are non-negative integers and \(\alpha[n]\) is the \(n\)-th element of the fundamental sequence assigned to the limit ordinal \(\alpha\).

Section II. Fundamental sequences for limit ordinals written in Cantor normal form

Every nonzero ordinal \(\alpha<\varepsilon_0\) can be represented in a unique Cantor normal form \(\alpha=\omega^{\beta_{1}}+ \omega^{\beta_{2}}+\cdots+\omega^{\beta_{k-1}}+\omega^{\beta_{k}}\) where \(\alpha>\beta_1\geq\beta_2\geq\cdots\geq\beta_{k-1}\geq\beta_k\). If \(\beta_k>0\) then \(\alpha\) is a limit and we can assign to it a fundamental sequence as follows

\(\alpha[n]=\omega^{\beta_{1}}+ \omega^{\beta_{2}}+\cdots+\omega^{\beta_{k-1}}+\left\{\begin{array}{lcr} \omega^\gamma n \text{ if } \beta_k=\gamma+1\\ \omega^{\beta_k[n]} \text{ if } \beta_k \text{ is a limit}\\ \end{array}\right.\)

Note: \(\omega^0=1\)

If \(\alpha=\varepsilon_0\) then \(\alpha[0]=0\) and \(\alpha[n+1]=\omega^{\alpha[n]}\)

Section III. Fundamental sequences for the Veblen function

Let \(z\) be an empty string or a string consisting of one or more comma-separated zeros \(0,0,...,0\) and \(s\) be an empty string or a string consisting of one or more comma-separated ordinals \(\alpha _{1},\alpha _{2},...,\alpha _{n}\) with \(\alpha _{1}>0\). The binary function \(\varphi (\beta ,\gamma )\) can be written as \(\varphi (s,\beta ,z,\gamma )\) where both \(s\) and \(z\) are empty strings. The finitary Veblen functions are defined as follows:

• \(\varphi (\gamma )=\omega ^{\gamma }\)
• \(\varphi (z,s,\gamma )=\varphi (s,\gamma )\)
• if \(\beta >0\), then \(\varphi (s,\beta ,z,\gamma )\) denotes the \((1+\gamma )\)-th common fixed point of the functions \(\xi \mapsto \varphi (s,\delta ,\xi ,z)\) for each \(\delta <\beta\)

Every non-zero ordinal \(\alpha\) less than the small Veblen ordinal (SVO) can be uniquely written in normal form for the finitary Veblen function:

\(\alpha =\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k})\)

where

• \(k\) is a positive integer
• \(\varphi (s_{1})\geq \varphi (s_{2})\geq \cdots \geq \varphi (s_{k})\)
• \(s_{m}\) is a string consisting of one or more comma-separated ordinals \(\alpha _{m,1},\alpha _{m,2},...,\alpha _{m,n_{m}}\) where \(\alpha _{m,1}>0\) and each \(\alpha _{m,i}<\varphi (s_{m})\)

Fundamental sequences for limit ordinals of finitary Veblen function

Any countable limit ordinal has cofinality \(\omega\). The fundamental sequence for an ordinal number \(\alpha\) with cofinality \(\text{cof}(\alpha )=\omega\) is a strictly increasing sequence \((\alpha [n])_{n<\omega }\) with length \(\omega\) and with limit \(\alpha\), where \(\alpha [n]\) is the \(n\)-th element of this sequence.

For limit ordinals \(\alpha<SVO\), written in normal form for the finitary Veblen function:

• \((\varphi(s_1)+\varphi(s_2)+\cdots+\varphi(s_k))[n]=\varphi(s_1)+\varphi(s_2)+\cdots+\varphi(s_k)[n]\),
• \(\varphi(\gamma)[n]=\left\{\begin{array}{lcr} n \quad \text{if} \quad \gamma=1\\ \varphi(\gamma-1)\cdot n \quad \text{if} \quad \gamma \quad \text{is a successor ordinal}\\ \varphi(\gamma[n]) \quad \text{if} \quad \gamma \quad \text{is a limit ordinal}\\ \end{array}\right.\),
• \(\varphi(s,\beta,z,\gamma)[0]=0\) and \(\varphi(s,\beta,z,\gamma)[n+1]=\varphi(s,\beta-1,\varphi(s,\beta,z,\gamma)[n],z)\) if \(\gamma=0\) and \(\beta\) is a successor ordinal,
• \(\varphi(s,\beta,z,\gamma)[0]=\varphi(s,\beta,z,\gamma-1)+1\) and \(\varphi(s,\beta,z,\gamma)[n+1]=\varphi(s,\beta-1,\varphi(s,\beta,z,\gamma)[n],z)\) if \(\gamma\) and \(\beta\) are successor ordinals,
• \(\varphi(s,\beta,z,\gamma)[n]=\varphi(s,\beta,z,\gamma[n])\) if \(\gamma\) is a limit ordinal,
• \(\varphi(s,\beta,z,\gamma)[n]=\varphi(s,\beta[n],z,\gamma)\) if \(\gamma=0\) and \(\beta\) is a limit ordinal,
• \(\varphi(s,\beta,z,\gamma)[n]=\varphi(s,\beta[n],\varphi(s,\beta,z,\gamma-1)+1,z)\) if \(\gamma\) is a successor ordinal and \(\beta\) is a limit ordinal.

Section IV. The extended Wilfried Buchholz's functions

Definition of Buchholz's family of ordinal functions

Buchholz defined his functions as follows:

• \(C_\nu^0(\alpha) = \Omega_\nu\)
• \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma | P(\gamma) \subseteq C_\nu^n(\alpha)\}\cup \{\psi_\mu(\xi) | \xi \in \alpha \cap C_\nu^n(\alpha) \wedge \xi \in C_\mu(\xi) \wedge \mu \leq \omega\}\)
• \(C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha) \)
• \(\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\} \)

where

\(\Omega_\nu=\left\{\begin{array}{lcr} 1\text{ if }\nu=0\\ \aleph_\nu\text{ if }\nu>0\\ \end{array}\right. \)

and \(P(\gamma)=\{\gamma_1,...,\gamma_k\}\) is the set of additive principal numbers in form \( \omega^\xi\)

\(P=\{\alpha\in On: 0<\alpha \wedge \forall \xi, \eta < \alpha (\xi+\eta < \alpha)\}=\{\omega^\xi: \xi \in On\}\)

the sum of which gives this ordinal \(\gamma\):

\(\gamma=\alpha_1+\alpha_2+\cdots+\alpha_k\) where \(\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_k\) and \(\alpha_1,\alpha_2,...,\alpha_k \in P(\gamma) \)

Note: \(On \) denotes the class of all ordinals.

The limit of this notation is Takeuti-Feferman-Buchholz ordinal.

Properties

Buchholz showed following properties of those functions:

• \(\psi_\nu(0)=\Omega_\nu\)
• \(\psi_\nu(\alpha)\in P\)
• \(\psi_\nu(\alpha+1)=\text{min}\{\gamma\in P: \psi_\nu(\alpha)<\gamma\}\text{ if }\alpha\in C_\nu(\alpha) \)
• \(\Omega_\nu\le\psi_\nu(\alpha)<\Omega_{\nu+1} \)
• \(\alpha\le\beta\Rightarrow\psi_\nu(\alpha)\le\psi_\nu(\beta) \)
• \(\psi_0(\alpha)=\omega^\alpha \text{ if }\alpha<\varepsilon_0\)
• \(\psi_\nu(\alpha)=\omega^{\Omega_\nu+\alpha} \text{ if }\alpha<\varepsilon_{\Omega_\nu+1} \text{ and } \nu\neq 0\)
• \(\theta(\varepsilon_{\Omega_\nu+1},0)=\psi_0(\varepsilon_{\Omega_\nu+1})\) for \(0<\nu\le\omega\)

The extended Wilfried Buchholz's functions

We rewrite Buchholz's definition as follows:

• \(C_\nu^0(\alpha) = \{\beta|\beta<\Omega_\nu\}\)
• \(C_\nu^{n+1}(\alpha) = \{\beta+\gamma,\psi_\mu(\eta)|\mu,\beta, \gamma,\eta\in C_{\nu}^n(\alpha)\wedge\eta<\alpha\}\)
• \(C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)\)
• \(\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}\)

where

\(\Omega_\nu=\left\{\begin{array}{lcr} 1\text{ if }\nu=0\\ \text{smallest ordinal with cardinality }\aleph_\nu \text{ if }\nu>0\\ \end{array}\right.\)

and \(\omega\) is the smallest infinite ordinal.

There is only one little detail difference with original Buchholz definition: ordinal \(\mu\) is not limited by \(\omega\), now ordinal \(\mu\) belongs to previous set \(C_n\). Limit of this notation must be omega fixed point \(\psi_0(\Omega_{\Omega_{\Omega_{...}}})=\psi_0(\psi_{\psi_{...}(0)}(0))\)

Normal form for the extended Wilfried Buchholz's functions

The normal form for 0 is 0. If \(\alpha\) is a nonzero ordinal number \(\alpha<\Lambda=\text{min}\{\beta|\psi_\beta(0)=\beta\}\) then the normal form for \(\alpha\) is \(\alpha=\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)\) where \(k\) is a positive integer and \(\psi_{\nu_1}(\beta_1)\geq\psi_{\nu_2}(\beta_2)\geq\cdots\geq\psi_{\nu_k}(\beta_k)\) and each \(\nu_i, \beta_i\) are also written in normal form.

Fundamental sequences for the extended Wilfried Buchholz's functions

The fundamental sequence for an ordinal number \(\alpha\) with cofinality \(\text{cof}(\alpha)=\beta\) is a strictly increasing sequence \((\alpha[\eta])_{\eta<\beta}\) with length \(\beta\) and with limit \(\alpha\), where \(\alpha[\eta]\) is the \(\eta\)-th element of this sequence. If \(\alpha\) is a successor ordinal then \(\text{cof}(\alpha)=1\) and the fundamental sequence has only one element \(\alpha[0]=\alpha-1\). If \(\alpha\) is a limit ordinal then \(\text{cof}(\alpha)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}\)

For nonzero ordinals \(\alpha<\Lambda\), written in normal form, fundamental sequences are defined as follows:

• If \(\alpha=\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)\) where \(k\geq2\) then \(\text{cof}(\alpha)=\text{cof}(\psi_{\nu_k}(\beta_k))\) and \(\alpha[\eta]=\psi_{\nu_1}(\beta_1)+\cdots+\psi_{\nu_{k-1}}(\beta_{k-1})+(\psi_{\nu_k}(\beta_k)[\eta])\)
• If \(\alpha=\psi_{0}(0)=1\), then \(\text{cof}(\alpha)=1\) and \(\alpha[0]=0\)
• If \(\alpha=\psi_{\nu+1}(0)\), then \(\text{cof}(\alpha)=\Omega_{\nu+1}\) and \(\alpha[\eta]=\Omega_{\nu+1}[\eta]=\eta\)
• If \(\alpha=\psi_{\nu}(0)\) and \(\text{cof}(\nu)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}\), then \(\text{cof}(\alpha)=\text{cof}(\nu)\) and \(\alpha[\eta]=\psi_{\nu[\eta]}(0)=\Omega_{\nu[\eta]}\)
• If \(\alpha=\psi_{\nu}(\beta+1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_{\nu}(\beta)\cdot \eta\) (and note: \(\psi_\nu(0)=\Omega_\nu\))
• If \(\alpha=\psi_{\nu}(\beta)\) and \(\text{cof}(\beta)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu<\nu\}\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\psi_{\nu}(\beta[\eta])\)
• If \(\alpha=\psi_{\nu}(\beta)\) and \(\text{cof}(\beta)\in\{\Omega_{\mu+1}|\mu\geq\nu\}\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_{\nu}(\beta[\gamma[\eta]])\) where \(\left\{\begin{array}{lcr} \gamma[0]=\Omega_\mu \\ \gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])\\ \end{array}\right.\)

If \(\alpha=\Lambda\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=0\) and \(\alpha[\eta+1]=\psi_{\alpha[\eta]}(0)=\Omega_{\alpha[\eta]}\)

Section V. Fundamental sequences for the Hypcos's functions

Definition of the Hypcos's functions

\(\Omega_\alpha\) 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 \(I_0=\Omega_0=0\)

In this section the variables \(\rho, \pi\) are reserved for uncountable regular cardinals of the form \(\Omega_{\nu+1}\) or \(I_{\mu+1}\)

Then,[1]

\(C_0(\alpha,\beta) = \beta\cup\{0\}\)

\(C_{n+1}(\alpha,\beta) = \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\}\)

\(\cup \{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\}\)

\(\cup \{I_\gamma|\gamma\in C_n(\alpha,\beta)\}\)

\(\cup \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\}\)

\(C(\alpha,\beta) = \bigcup_{n<\omega} C_n(\alpha,\beta)\)

\(\psi_\pi(\alpha) = \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}\)

Properties

• \(\psi_{\pi}(0)=1\)
• \(\psi_{\Omega_1}(\alpha)=\omega^\alpha\) for \(\alpha<\varepsilon_0\)
• \(\psi_{\Omega_{\nu+1}}(\alpha)=\omega^{\Omega_\nu+\alpha}\) for \(1\le\alpha<\varepsilon_{\Omega_\nu+1}\) and \(\nu>0\)

Standard form for ordinals \(\alpha<\beta=\text{min}\{\xi|I_\xi=\xi\}\)

• The standard form for 0 is 0
• If \(\alpha\) is of the form \(\Omega_\beta\), then the standard form for \(\alpha\) is \(\alpha= \Omega_\beta\) where \(\beta<\alpha\) and \(\beta\) is expressed in standard form
• If \(\alpha\) is of the form \(I_\beta\), then the standard form for \(\alpha\) is \(\alpha= I_\beta\) where \(\beta<\alpha\) and \(\beta\) is expressed in standard form
• If \(\alpha\) is not additively principal and \(\alpha>0\), then the standard form for \(\alpha\) is \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\), where the \(\alpha_i\) are principal ordinals with \(\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n\), and the \(\alpha_i\) are expressed in standard form
• If \(\alpha\) is an additively principal ordinal but not of the form \(\Omega_\beta\) or \(I_\gamma\), then \(\alpha\) is expressible in the form \(\psi_\pi(\delta)\). Then the standard form for \(\alpha\) is \(\alpha=\psi_\pi(\delta)\) where \(\pi\) and \(\delta\) are expressed in standard form

Fundamental sequences

The fundamental sequence for an ordinal number \(\alpha\) with cofinality \(\text{cof}(\alpha)=\beta\) is a strictly increasing sequence \((\alpha[\eta])_{\eta<\beta}\) with length \(\beta\) and with limit \(\alpha\), where \(\alpha[\eta]\) is the \(\eta\)-th element of this sequence.

Let \(S=\{\alpha|\text{cof}(\alpha)=1\}\) and \(L=\{\alpha|\text{cof}(\alpha)\geq\omega\}\) where \(S\) denotes the set of successor ordinals and \(L\) denotes the set of limit ordinals.

For non-zero ordinals written in standard form fundamental sequences are defined as follows:

• If \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\) with \(n\geq 2\) then \(\text{cof}(\alpha)=\text{cof}(\alpha_n)\) and \(\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])\)
• If \(\alpha=\psi_{\pi}(0)\) then \(\alpha=\text{cof}(\alpha)=1\) and \(\alpha[0]=0\)
• If \(\alpha=\psi_{\Omega_{\nu+1}}(1)\) then \(\text{cof}(\alpha)=\omega\) and \(\left\{\begin{array}{lcr} \alpha[\eta]=\Omega_{\nu}\cdot\eta \text{ if }\nu>0 \\ \alpha[\eta]=\eta \text{ if }\nu=0\\ \end{array}\right.\)
• If \(\alpha=\psi_{\Omega_{\nu+1}}(\beta+1)\) and \(\beta\geq 1\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_{\Omega_{\nu+1}}(\beta)\cdot\eta\)
• If \(\alpha=\psi_{ I_{\nu+1}}(1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=I_\nu+1\) and \(\alpha[\eta+1]=\Omega_{\alpha[\eta]}\)
• If \(\alpha=\psi_{ I_{\nu+1}}(\beta+1)\) and \(\beta\geq 1\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=\psi_{ I_{\nu+1}}(\beta)+1\) and \(\alpha[\eta+1]=\Omega_{\alpha[\eta]}\)
• If \(\alpha=\pi\) then \(\text{cof}(\alpha)=\pi\) and \(\alpha[\eta]=\eta\)
• If \(\alpha=\Omega_\nu\) and \(\nu\in L\) then \(\text{cof}(\alpha)=\text{cof}(\nu)\) and \(\alpha[\eta]=\Omega_{\nu[\eta]}\)
• If \(\alpha=I_\nu\) and \(\nu\in L\) then \(\text{cof}(\alpha)=\text{cof}(\nu)\) and \(\alpha[\eta]=I_{\nu[\eta]}\)
• If \(\alpha=\psi_\pi(\beta)\) and \(\omega\le\text{cof}(\beta)<\pi\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\psi_\pi(\beta[\eta])\)
• If \(\alpha=\psi_\pi(\beta)\) and \(\text{cof}(\beta)=\rho\geq\pi\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_\pi(\beta[\gamma[\eta]])\) with \(\gamma[0]=1\) and \(\gamma[\eta+1]=\psi_{\rho}(\beta[\gamma[\eta]])\)

Limit of this notation is \(\lambda\). If \(\alpha=\lambda\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=1\) and \(\alpha[\eta+1]=I_{\alpha[\eta]}\)

Section VI. Fundamental sequences for the functions collapsing \(\alpha\)-weakly inaccessible cardinals

Definition of the functions collapsing \(\alpha\)-weakly inaccessible cardinals

An ordinal is \(\alpha\)-weakly inaccessible if it's an uncountable regular cardinal and it's a limit of \(\gamma\)-weakly inaccessible cardinals for all \(\gamma<\alpha\)

Let \(I(\alpha, 0)\) be the first \(\alpha\)-weakly inaccessible cardinal, \(I(\alpha, \beta+1)\) be the next \(\alpha\)-weakly inaccessible cardinal after \(I(\alpha,\beta)\), and \(I(\alpha,\beta)=\sup\{I(\alpha,\gamma)|\gamma<\beta\}\) for limit ordinal \(\beta\)

In this section the variables \(\rho\), \(\pi\) are reserved for uncountable regular cardinals of the form \(I(\alpha,0)\) or \(I(\alpha,\beta+1)\)

Then,

\(C_0(\alpha,\beta) = \beta\cup\{0\}\)

\(C_{n+1}(\alpha,\beta) = \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\}\)

\(\cup \{I(\gamma,\delta)|\gamma,\delta\in C_n(\alpha,\beta)\}\)

\(\cup \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\}\)

\(C(\alpha,\beta) = \bigcup_{n<\omega} C_n(\alpha,\beta)\)

\(\psi_\pi(\alpha) = \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}\)

Properties

• \(I(0,\alpha)=\Omega_{1+\alpha}=\aleph_{1+\alpha}\)
• \(I(1,\alpha)=I_{1+\alpha}\)
• \(\psi_{I(0,0)}(\alpha)=\omega^\alpha\) for \(\alpha<\varepsilon_0\)
• \(\psi_{I(0,\alpha+1)}(\beta)=\omega^{I(0,\alpha)+1+\beta}\) for \(\beta<\varepsilon_{I(0,\alpha)+1}\)

Standard form

1) \(\alpha=_{NF}\alpha _1+\cdots +\alpha _n:\Leftrightarrow \alpha =\alpha _1+\cdots +\alpha _n\wedge \alpha>\alpha _1\geq \cdots \geq \alpha _n\wedge \alpha _1,\cdots ,\alpha _n\in P\) where \(P\) is the class of additive principal numbers

2) \(\alpha=_{NF}I(\beta,\gamma):\Leftrightarrow \alpha =I(\beta,\gamma)\wedge \beta,\gamma<\alpha \)

3) \(\alpha=_{NF}\psi_\pi(\beta):\Leftrightarrow\alpha=\psi_\pi(\beta) \wedge\beta\in C(\beta, \psi_\pi(\beta))\)

Definition of the set \(T\) of ordinals expressible using symbols \(0,+,I, \psi \)

1) \(0 \in T\)

2) \(\alpha=_{NF}\alpha_1+\alpha_2+\cdots+\alpha_n\wedge\alpha_1,\alpha_2,...,\alpha_n\in T\Rightarrow\alpha\in T\)

3) \(\alpha=_{NF}I(\beta,\gamma)\wedge\beta,\gamma\in T\Rightarrow\alpha\in T\)

4) \(\alpha=_{NF}\psi_\pi(\beta)\wedge\pi, \beta \in T\Rightarrow\alpha\in T\)

Fundamental sequences

The fundamental sequence for an ordinal number \(\alpha\) with cofinality \(\text{cof}(\alpha)=\beta\) is a strictly increasing sequence \((\alpha[\eta])_{\eta<\beta}\) with length \(\beta\) and with limit \(\alpha\), where \(\alpha[\eta]\) is the \(\eta\)-th element of this sequence.

Let \(S=\{\alpha|\text{cof}(\alpha)=1\}\) and \(L=\{\alpha|\text{cof}(\alpha)\geq\omega\}\) where \(S\) denotes the set of successor ordinals and \(L\) denotes the set of limit ordinals.

Definition of fundamental sequences for non-zero ordinals \(\alpha\in T\):

• If \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\) with \(n\geq 2\) then \(\text{cof}(\alpha)=\text{cof}(\alpha_n)\) and \(\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])\)
• If \(\alpha=\psi_{I(0,0)}(0)\) then \(\alpha=\text{cof}(\alpha)=1\) and \(\alpha[0]=0\)
• If \(\alpha=\psi_{I(0,\beta+1)}(0)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=I(0,\beta)\cdot\eta\)
• If \(\alpha=\psi_{I(0,\beta)}(\gamma+1)\) and \(\beta\in\{0\}\cup S\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_{I(0,\beta)}(\gamma)\cdot\eta\)
• If \(\alpha=\psi_{I(\beta+1,0)}(0)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=0\) and \(\alpha[\eta+1]=I(\beta,\alpha[\eta])\)
• If \(\alpha=\psi_{I(\beta+1,\gamma+1)}(0)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=I(\beta+1,\gamma)+1\) and \(\alpha[\eta+1]=I(\beta,\alpha[\eta])\)
• If \(\alpha=\psi_{I(\beta+1,\gamma)}(\delta+1)\) and \(\gamma\in\{0\}\cup S\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=\psi_{I(\beta+1,\gamma)}(\delta)+1\) and \(\alpha[\eta+1]=I(\beta,\alpha[\eta])\)
• If \(\alpha=\psi_{I(\beta,0)}(0)\) and \(\beta\in L\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=I(\beta[\eta],0)\)
• If \(\alpha=\psi_{I(\beta,\gamma+1)}(0)\) and \(\beta\in L\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=I(\beta[\eta],I(\beta,\gamma)+1)\)
• If \(\alpha=\psi_{I(\beta,\gamma)}(\delta+1)\) and \(\beta\in L\) and \(\gamma\in \{0\}\cup S\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=I(\beta[\eta],\psi_{I(\beta,\gamma)}(\delta)+1)\)
• If \(\alpha=\pi\) then \(\text{cof}(\alpha)=\pi\) and \(\alpha[\eta]=\eta\)
• If \(\alpha=I(\beta,\gamma)\) and \(\gamma\in L\) then \(\text{cof}(\alpha)=\text{cof}(\gamma)\) and \(\alpha[\eta]=I(\beta,\gamma[\eta])\)
• If \(\alpha=\psi_\pi(\beta)\) and \(\omega\le\text{cof}(\beta)<\pi\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\psi_\pi(\beta[\eta])\)
• If \(\alpha=\psi_\pi(\beta)\) and \(\text{cof}(\beta)=\rho\geq\pi\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_\pi(\beta[\gamma[\eta]])\) with \(\gamma[0]=1\) and \(\gamma[\eta+1]=\psi_{\rho}(\beta[\gamma[\eta]])\)

Limit of this notation \(\psi_{I(1,0,0)}(0)\). If \(\alpha=\psi_{I(1,0,0)}(0)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=0\) and \(\alpha[\eta+1]=I(\alpha[\eta],0)\)

Section VII. The function collapsing weakly Mahlo cardinals

Definition

An ordinal is weakly Mahlo if it's an uncountable regular cardinal, and regular cardinals in it (in another word, less than it) are stationary. Let \(M_{0}=0\), \(M_{\alpha +1}\) be the next weakly Mahlo cardinal after \(M_{\alpha }\), and \(M_{\alpha }=\sup\{M_{\beta }|\beta <\alpha \}\) for limit ordinal \(\alpha\).

Then,[2]

\(C_{0}(\alpha ,\beta )=\beta \cup \{0\}\)

\(C_{n+1}(\alpha ,\beta )=\{\gamma +\delta |\gamma ,\delta \in C_{n}(\alpha ,\beta )\}\)

\(\cup \{M_{\gamma }|\gamma \in C_{n}(\alpha ,\beta )\}\)

\(\cup \{\chi _{\pi }(\gamma )|\pi ,\gamma \in C_{n}(\alpha ,\beta )\wedge \gamma <\alpha \wedge \pi {\text{ is weakly Mahlo}}\}\)

\(\cup \{\psi _{\pi }(\gamma )|\pi ,\gamma \in C_{n}(\alpha ,\beta )\wedge \gamma <\alpha \wedge \pi {\text{ is uncountable regular}}\}\)

\(C(\alpha ,\beta )=\bigcup _{n<\omega }C_{n}(\alpha ,\beta )\)

\(\chi _{\pi }(\alpha )=\min\{\beta <\pi |C(\alpha ,\beta )\cap \pi \subseteq \beta \wedge \beta {\text{ is uncountable regular}}\}\)

\(\psi _{\pi }(\alpha )=\min\{\beta <\pi |C(\alpha ,\beta )\cap \pi \subseteq \beta \}\)

Fundamental sequences for the function collapsing weakly Mahlo cardinals

For the function collapsing weakly Mahlo cardinals fundamental sequences defined as follows:

• \(C_{0}=\{0,1\}\)
• \(C_{n+1}=\{\alpha +\beta ,M_{\gamma },\chi _{\delta }(\epsilon ),\psi _{\zeta }(\eta )|\alpha ,\beta ,\gamma ,\delta ,\epsilon ,\zeta ,\eta \in C_{n}\wedge \delta \in W\wedge \zeta \in R\}\)
• \(L(\alpha )=\text{min}\{n<\omega |\alpha \in C_{n}\}\)
• \(\alpha [n]=\text{max}\{\beta <\alpha |L(\beta )\leq L(\alpha )+n\}\)

where \(R\) denotes set of all uncountable regular cardinals and \(W\) denotes set of all weakly Mahlo cardinals.

Section VIII. Fundamental sequences for the function collapsing \(M(\alpha; \beta)\)

Definition of \(M(\alpha ;\beta )\)

If \(\beta =0\) or \(\beta =\beta '+1\) then \(M(\alpha ;\beta )\) is an \(\alpha\)-weakly Mahlo cardinal.

Let in this section \(W(\alpha)\) be the set of all \(\alpha\)-weakly Mahlo cardinals, \(R\) be the set of all uncountable regular cardinals less than \(\text{min}\{\xi |\xi =M(\xi ;0)\}\) and the variable \(\pi\) be reserved for uncountable regular cardinals of the forms \(M(\alpha ;\beta +1)\) or \(M(\alpha ;0)\).

• If \(\alpha =0\) then \(W(\alpha )=R\)
• If \(\alpha >0\) then \(W(\alpha )=\{\gamma \in R|\forall \delta <\alpha :W(\delta )\cap \gamma \text{ is stationary in }\gamma \}\)
• If \(\beta =0\) then \(M(\alpha ;\beta )=\text{min}W(\alpha )\)
• If \(\beta =\beta '+1\) then \(M(\alpha ;\beta )=\text{min}\{\gamma \in W(\alpha )|\gamma >M(\alpha ;\beta ')\}\)
• If \(\beta\) is a limit ordinal then \(M(\alpha ;\beta )=\sup\{M(\alpha ;\gamma )|\gamma <\beta \}\)

Definition of \(\psi _{\pi }(\alpha)\)

\(C_{0}(\alpha ,\beta )=\beta \cup \{0\}\)

\(C_{n+1}(\alpha ,\beta )=\{\gamma +\delta |\gamma ,\delta \in C_{n}(\alpha ,\beta )\}\)

\(\cup \{M(\gamma ;\delta )|\gamma ,\delta \in C_{n}(\alpha ,\beta )\}\)

\(\cup \{\chi _{\pi }^{\gamma }(\delta )|\pi ,\gamma ,\delta \in C_{n}(\alpha ,\beta )\wedge \delta <\alpha \}\)

\(\cup \{\psi _{\pi }(\gamma )|\pi ,\gamma \in C_{n}(\alpha ,\beta )\wedge \gamma <\alpha \}\)

\(C(\alpha ,\beta )=\bigcup _{n<\omega }C_{n}(\alpha ,\beta )\)

\(\chi _{\pi }^{\gamma }(\alpha )=\min(\{\beta <\pi |C(\alpha ,\beta )\cap \pi \subseteq \beta \wedge \beta \in W(\gamma )\}\cup \{\pi \})\)

\(\psi _{\pi }(\alpha )=\min(\{\beta <\pi |C(\alpha ,\beta )\cap \pi \subseteq \beta \}\cup \{\pi \})\)

Fundamental sequences

The fundamental sequence for a countable limit ordinal \(\alpha\) is defined as follows:

\(C_{0}=\{0\}\)

\(C_{n+1}=\{\gamma +\delta |\gamma ,\delta \in C_{n}\}\)

\(\cup \{M(\gamma ;\delta )|\gamma ,\delta \in C_{n}\}\)

\(\cup \{\chi _{\pi }^{\gamma }(\delta )|\pi ,\gamma ,\delta \in C_{n}\}\)

\(\cup \{\psi _{\pi }(\gamma )|\pi ,\gamma \in C_{n}\}\)

\(L(\alpha )=\text{min}\{n<\omega |\alpha \in C_{n}\}\)

\(\alpha [n]=\text{max}\{\beta <\alpha |L(\beta )\leq L(\alpha )+n\}\)

Section IX. My system of number names (FGS)

Codes of operations (group 1)

codes	operations
add	Addition \(+\)
ult	Multiplication \(\times\)
ex	Exponentiation \(\uparrow\)
tetr	Tetration \(\uparrow^2\)
pent	Pentation \(\uparrow^3\)
hex	Hexation \(\uparrow^4\)
hept	Heptation \(\uparrow^5\)

and so on

Codes of some natural numbers (group 2)

codes	natural numbers
zer(o)	\(0\)
un(i)	\(1\)
b(i)	\(2\)
tr(i)	\(3\)
quadr(i)	\(4\)
quint(i)	\(5\)
sext(i)	\(6\)
sept(i)	\(7\)
oct(i)	\(8\)
non(i)	\(9\)
dek(o)	\(10\)
hekt(o)	\(10^2\)
kil(o)	\(10^3\)
meg(o)	\(10^6\)
gig(o)	\(10^9\)
ter(o)	\(10^{12}\)
pet(o)	\(10^{15}\)
ex(o)	\(10^{18}\)
zett(o)	\(10^{21}\)
yott(o)	\(10^{24}\)

Note 1: do not write the letter in parentheses if this code is followed by any vowel letter

Note 2: codes of two-digit numbers from 11 to 99 can be obtained at the reading of those two-digit numbers from right to left by replacing the digits in them with codes of corresponding one-digit numbers (for example: trib(i) for 23)

Codes of ordinals and functions (group 3)

codes	ordinals and functions
alum	a finite ordinal
om	the first transfinite ordinal \(\omega\)
ep	ordinal \(\varepsilon_0\)
zet	ordinal \(\zeta_0\)
et	ordinal \(\eta_0\)
phi	the Veblen function \(\varphi\) of two variables
gam	ordinal \(\Gamma_0\)
omm	the first uncountable ordinal \(\Omega\)
thet	Feferman's theta-function \(\theta\)
wil	the extended Wilfried Buchholz's function \(\psi_0\)
ot	the first weakly inaccessible cardinal \(I_1\)
os	the Hypcos's function \(\psi_{\Omega_1}\)
ah	the function collapsing \(\alpha\)-weakly inaccessible cardinals \(\psi_{I(0,0)}\)
im	the first 0-weakly inaccessible cardinal \(I(0,0)\)
em	the first weakly Mahlo cardinal \(M_1\)
ar	the function collapsing weakly Mahlo cardinals \(\psi_{\alpha}\) where \(\alpha=\chi_{M_1}(0)\)
am	the first 0-weakly Mahlo cardinal \(M(0;0)\)
us	the function collapsing \(M(\alpha;\beta)\) into countable ordinals \(\psi_{M(0;0)}\)

In general case use the following rules for generation of number names:

rule 1.

If a number \(n\) is defined using a function of the fast-growing hierarchy, then read ordinal index of the function from right to left using codes to get the name of the number \(n\) and in case if argument of the function is not equal to ten also write code of the argument in the beginning of the name, adding "argum-". Default argument is equal to ten.

Example 1. \(f_{\omega^3+\omega.2+3}(10)\) is traddbultomaddtrexom and \(f_{\omega^\omega+3}(10)\) is traddomexom.

Example 2. \(f_{\omega+1}(10)\) is unaddom, but \(f_{\omega+1}(3)\) is trargum-unaddom.

Thus, if name of a number \(n\) contains at least one code from group 3, then \(n\) is defined using a function of the fast-growing hierarchy where the ordinal index of the function at the reading from right to left corresponds to sequence of codes inside name of the number \(n\).

rule 2.

If name of a number \(n\) does not contain codes from group 3, then \(n=a\uparrow^b c\) where \(a\uparrow^b c\) at the reading from right to left corresponds to sequence of codes from groups 1-2 inside name of the number \(n\). Default \(a=10\)

For example: quadritetr is \(10\uparrow^2 4\), quintipent is \(10\uparrow^3 5\), nonihex is \(10\uparrow^4 9\)

Special operations

1) in-operation (code: in)

If name of a number \(n\) contains "in" between code of a natural number \(b\) and code of \(\alpha\), then this case corresponds to repeating \(b\) times of insertion of \(\alpha\) inside ordinal index of function of the fast-growing hierarchy, which used for definition of the number \(n\):

\(\underbrace{\alpha(\alpha(...(\alpha(i))...))}_{b+1\quad \alpha 's}\) where \(i=\left\{\begin{array}{lcr} 0\text{ if }\alpha(0)>0\\ 1\text{ if }\alpha(0)=0\\ \end{array}\right.\)

example 1. "Trinep" is equal to \(f_{\varepsilon(\varepsilon(\varepsilon(\varepsilon(0))))}(10)=f_{\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}}(10)\)

example 2. "Trinphi" is equal to \(f_{\varphi(\varphi(\varphi(\varphi(0,0),0),0),0)}(10)\), where \(\varphi(0,0)=\omega^0=1\)

2) mix -operation (code: mix)

If name of a number \(n\) contains "mix" between code of a natural number \(b\) and codes of \(\alpha\) and \(\beta\), then this case corresponds to repeating \(b\) times of the alternation of \(\alpha\) and \(\beta\) inside ordinal index of function of the fast-growing hierarchy, which used for definition of the number \(n\):

\(\underbrace{\beta(\alpha(\beta(\alpha...(\beta(\alpha(1))...)))}_{b \quad\alpha's}\)

example 1. Trimixommthet

\(f_{\theta(\Omega(\theta(\Omega(\theta(\Omega(1))))))}(10)=\)

\(=f_{\theta(\Omega_{\theta(\Omega_{\theta(\Omega)})})}(10)\)

example 2. trimixphithet

\(f_{\theta(\varphi(\theta(\varphi(\theta(\varphi(1,0)),0)),0))}(10)\)

Default

If name of a number \(n\) contains directly following one another codes of \(\alpha\) and \(\beta\) without "mix" before them, and code of \(\beta\) belongs to group 3, then this case corresponds to \(\beta(\alpha)\) inside ordinal index of function of the fast-growing hierarchy, which used for definition of the number \(n\).

example 1. "triphi" is equal to \(f_{\varphi(3,0)}(10)\) where \(\varphi\) is the Veblen function of two variables

example 2. "ommthet" is equal to \(f_{\theta(\Omega)}(10)\)

example 3. "trommthet" is equal to \(f_{\theta(\Omega(3))}(10)=f_{\theta(\Omega_3)}(10)\)

example 4. "Trinommwil" is equal to \(f_{\psi_0(\Omega(\Omega(\Omega(\Omega(0)))))}(10)=f_{\psi_0(\Omega(\Omega(\Omega(1))))}(10)=f_{\psi_0(\Omega_{\Omega_{\Omega}})}(10)\) since for the extended Wilfried Buchholz's functions \(\Omega_0=1\) but "Trinommos" is equal to \(f_{\psi_{\Omega_1}(\Omega(\Omega(\Omega(\Omega(1)))))}(10)=f_{\psi_{\Omega_1}(\Omega_{\Omega_{\Omega_{\Omega}}})}(10)\) since for the Hypcos's functions \(\Omega_0=0\)

Note: zeromm is \(\Omega(0)=\Omega_0\), omm=unomm is \(\Omega(1)=\Omega_1=\Omega\) - first uncountable ordinal (i.e. the smallest ordinal that has cardinality \(\aleph_1\)), bomm is \(\Omega(2)=\Omega_2\) - the smallest ordinal that has cardinality \(\aleph_2\), tromm is \(\Omega(3)=\Omega_3\) and so on; ep is \(\varepsilon_0\) and gamm is \(\Gamma_0\) but unep is \(\varepsilon_1\) and unigam is \(\Gamma_1\) and so on.

if code of number is not written before code of operation, then number is 10 default

addom is \(f_{\omega+10}(10)\)=dekaddom

ultom is \(f_{\omega.10}(10)\)=dekultom

exom is \(f_{\omega^{10}}(10)\)=dekexom

tetrom is \(f_{\omega\uparrow^2 10}(10)\)=dekotetrom

inphi is \(f_{\underbrace{\varphi(\varphi(...(\varphi(0,0),0)...),0)}_{11 \quad \varphi's}}(10)\)=dekinphi

mixommthet is \(f_{\underbrace{\theta(\Omega_{\theta(\Omega_{..._{\theta(\Omega)}})})}_{10 \Omega 's}}(10)\)= dekomixommthet

inommthet is \(f_{\underbrace{\theta(\Omega_{\Omega_{..._{\Omega}}})}_{10 \quad\Omega 's}}(10)\)=dekinommthet

addultexom is \(f_{\omega^{10}.10+10}(10)\)

Appendix 1: huge units of measurement

l-\(\bullet\) is the distance \(\bullet\) meters,

c-\(\bullet\) is the \(\bullet\)-dimensional hypercube with side length \(\bullet\) meters,

t-\(\bullet\) is the time interval \(\bullet\) seconds,

m-\(\bullet\) is the mass \(\bullet\) kg.

Example: l-inommthet = inommthet meters.

Appendix 2: googological regions

Let's define: Ra\(\bullet\) is spherical region of the universe bounded by imaginary sphere with center at the Earth's center and with radius of \(\bullet\) meters, where \(\bullet\) is a name of number, which was created according this system of names (if name of number begins with a vowel then don't write "a" in "Ra").

Note 1: this definition is made on the assumption that our universe is infinite.

Note 2: a sphere with center \(c\) and with radius \(r\) is the set of all points that are at distance \(r\) from \(c\).

Example 1. Ratrimah is spherical region of the universe bounded by imaginary sphere with center at the Earth's center and with radius of trimah meters. Trimah is equal to \(f_{\psi_{I(0,0)}(I(3,0))}(10)\) using the fast-growing hierarchy with fundamental sequences for the functions collapsing \(\alpha\)-weakly inaccessible cardinals.

Example 2. Roctinemar is spherical region of the universe bounded by imaginary sphere with center at the Earth's center and with radius of octinemar meters. Octinemar is equal to \(f_{\psi _{\alpha }(\beta [9])}(10)\) using the fast-growing hierarchy with fundamental sequences for the function collapsing weakly Mahlo cardinals, where \( \alpha =\chi _{M_{1}}(0)\) and \( \beta [0]=1\) and \( \beta [n+1]=M_{\beta [n]}\) for all integers \(n\geq 0\)

Other examples: Runaddom, Runaddep, Runingam, Roctinommwil, Ratrinotos, Ratrinimah, Roctinamus, Ratarintar.

About unaddom, unaddep, uningam, octinommwil, trinotos, trinimah, octinamus, tarintar you can read below.

Section X. List of my numbers

All that you can see below is a list of my numbers defined using the fast-growing hierarchy.

1) Series with finite ordinals

Series 1.1 (Alpha series)

Names of all numbers of series 1.1 contain code: alum.

See also the fast-growing hierarchy.

Zeralum \(f_0(10)=10+1=11\)

Unalum \(f_1(10)=f_0^{10} (10)= f_0(f_0 (f_0(f_0(f_0( f_0( f_0( f_0( f_0( f_0( 10))))))))))=20\)

Balum \(f_2(10)=f_1^{10} (10)= f_1(f_1 (f_1(f_1(f_1( f_1( f_1( f_1( f_1( f_1( 10))))))))))=2^{10}\times 10=10240\)

Tralum \(f_3(10)=f_2^{10} (10)= f_2(f_2(f_2(f_2(f_2(f_2(f_2(f_2(f_2(f_2(10))))))))))\approx\)

\(\approx 10^{10^{10^{10^{10^{10^{10^{10^{1,0865890600\times 10^{3086}}}}}}}}}\)

Quadralum \(f_4(10)=f_3^{10}(10)\)

Quintalum \(f_5(10)=f_4^{10}(10)\)

Sextalum \(f_6(10)=f_5^{10}(10)\)

Septalum \(f_7(10)=f_6^{10}(10)\)

Octalum \(f_8(10)=f_7^{10}(10)\)

Nonalum \(f_9(10)=f_8^{10}(10)\)

Dekalum \(f_{10} (10)=f_9^{10}(10)\)

Hektalum \(f_{100} (10)\)

Kilalum \(f_{1000} (10)\)

Megalum \(f_{10^{6}} (10)\)

Gigalum \(f_{10^{9}} (10)\)

Teralum \(f_{10^{12}} (10)\)

Petalum \(f_{10^{15}} (10)\)

Exalum \(f_{10^{18}} (10)\)

Zettalum \(f_{10^{21}} (10)\)

Yottalum \(f_{10^{24}}(10)\)

Full list of the names of the numbers in this series, see here: https://sites.google.com/site/travelingtotheinfinity/my-system-of-number-names/generator-of-googologisms/before-w

2) Omega series

These four series 2.1-2.4 consist of my numbers defined using the fast-growing hierarchy with fundamental sequences for limit ordinals written in Cantor normal form.

Series 2.1 (Omega-addition series)

Names of all numbers of series 2.1 contain following codes: add, om.

Zeraddom \(f_{\omega} (10)=f_{10} (10)\)

Unaddom \(f_{\omega+1} (10)=f_{\omega } ^{10} (10)\)

Baddom \(f_{\omega+2 }(10) = f_{\omega+1}^{10} (10)\)

Traddom \(f_{\omega+3 } (10)\)

Quadraddom \(f_{\omega+4 } (10)\)

Quintaddom \(f_{\omega+5 } (10)\)

Sextaddom \(f_{\omega+6 } (10)\)

Septaddom \(f_{\omega+7 } (10)\)

Octaddom \(f_{\omega+8 } (10)\)

Nonaddom \(f_{\omega+9 } (10)\)

Dekaddom \(f_{\omega+10 } (10)\)

Hektaddom \(f_{\omega+100 } (10)\)

Kiladdom \(f_{\omega+10^{3} } (10)\)

Megaddom \(f_{\omega+10^{6} } (10)\)

Gigaddom \(f_{\omega+10^{9} } (10)\)

Teraddom \(f_{\omega+10^{12} } (10)\)

Petaddom \(f_{\omega+10^{15} } (10)\)

Exaddom \(f_{\omega+10^{18} } (10)\)

Zettaddom \(f_{\omega+10^{21} } (10)\)

Yottaddom \(f_{\omega+10^{24} } (10)\)

Series 2.2 (Omega- multiplication series)

Names of all numbers of series 2.2 contain following codes: ult, om.

Bultom \(f_{\omega.2 } (10)= f_{\omega+10} (10)\)

Trultom \(f_{\omega.3 } (10)= f_{\omega.2+10} (10)\)

Quadrultom \(f_{\omega.4 } (10)= f_{\omega.3+10} (10)\)

Quintultom \(f_{\omega.5 } (10)= f_{\omega.4+10} (10)\)

Sextultom \(f_{\omega.6 } (10)\)

Septultom \(f_{\omega.7 } (10)\)

Octultom \(f_{\omega.8 } (10)\)

Nonultom \(f_{\omega.9 } (10)\)

Dekultom \(f_{\omega.10 } (10)\)

Hektultom \(f_{\omega.100 } (10)\)

Kilultom \(f_{\omega.1000 } (10)\)

Megultom \(f_{\omega.10^{6} } (10)\)

Gigultom \(f_{\omega.10^{9} } (10)\)

Terultom \(f_{\omega.10^{12} } (10)\)

Petultom \(f_{\omega.10^{15} } (10)\)

Exultom \(f_{\omega.10^{18} } (10)\)

Zettultom \(f_{\omega.10^{21} } (10)\)

Yottultom \(f_{\omega.10^{24} } (10)\)

Series 2.3 (Omega- exponentiation series)

Names of all numbers of series 2.3 contain following codes: ex, om.

Bexom \(f_{\omega^2} (10)=f_{\omega.10 } (10)\)

Trexom \(f_{\omega^{3}} (10)\)

Quadrexom \(f_{\omega^{4}} (10)\)

Quintexom \(f_{\omega^{5}} (10)\)

Sextexom \(f_{\omega^{6}} (10)\)

Septexom \(f_{\omega^{7}} (10)\)

Octexom \(f_{\omega^{8}} (10)\)

Nonexom \(f_{\omega^{9}} (10)\)

Dekexom \(f_{\omega^{10}} (10)\)

Hektexom \(f_{\omega^{100}} (10)\)

Kilexom \(f_{\omega^{10^{3}}} (10)\)

Megexom \(f_{\omega^{10^{6}}} (10)\)

Gigexom \(f_{\omega^{10^{9}}} (10)\)

Terexom \(f_{\omega^{10^{12}}} (10)\)

Petexom \(f_{\omega^{10^{15}}} (10)\)

Exexom \(f_{\omega^{10^{18}}} (10)\)

Zettexom \(f_{\omega^{10^{21}}} (10)\)

Yottexom \(f_{\omega^{10^{24}}} (10)\)

Series 2.4 (Omega- tetration series)

Names of all numbers of series 2.4 contain following codes: tetr, om.

Bitetrom \(f_{\omega\uparrow\uparrow 2 } (10)= f_{\omega^{\omega }} (10)=f_{\omega^{10}} (10)\)

Tritetrom \(f_{\omega\uparrow\uparrow 3 } (10)= f_{\omega^{\omega^{\omega }}} (10)\)

Quadritetrom \(f_{\omega\uparrow\uparrow 4 } (10)\)

Quintitetrom \(f_{\omega\uparrow\uparrow 5 } (10)\)

Sextitetrom \(f_{\omega\uparrow\uparrow 6 } (10)\)

Septitetrom \(f_{\omega\uparrow\uparrow 7 } (10)\)

Octitetrom \(f_{\omega\uparrow\uparrow 8 } (10)\)

Nonitetrom \(f_{\omega\uparrow\uparrow 9 } (10)\)

Dekotetrom \(f_{\omega\uparrow\uparrow 10 } (10)\)

Hektotetrom \(f_{\omega\uparrow\uparrow 100 } (10)\)

Kilotetrom \(f_{\omega\uparrow\uparrow 1000 } (10)\)

Megotetrom \(f_{\omega\uparrow\uparrow 10^{6} } (10)\)

Gigotetrom \(f_{\omega\uparrow\uparrow 10^{9} } (10)\)

Terotetrom \(f_{\omega\uparrow\uparrow 10^{12} } (10)\)

Petotetrom \(f_{\omega\uparrow\uparrow 10^{15} } (10)\)

Exotetrom \(f_{\omega\uparrow\uparrow 10^{18} } (10)\)

Zettotetrom \(f_{\omega\uparrow\uparrow 10^{21} } (10)\)

Yottotetrom \(f_{\omega\uparrow\uparrow 10^{24} } (10)\)

3) Epsilon series

3.1) Epsilon(0) series

Series 3.1.1 (Epsilon(0)-addition series)

In series 3.1.1 \(f_{\varepsilon(0)+n}(10)=f_{\varphi(1,0)+n}(10)\) using the fast-growing hierarchy with fundamental sequences for the Veblen function, where:

• \(n\) is a non-negative integer
• \(\varphi\) is the Veblen function of two variables.

Names of all numbers of series 3.1.1 contain following codes: add, ep.

Zeraddep \(f_{\varepsilon(0)} (10)\)

Unaddep \(f_{\varepsilon(0)+1} (10)\)

Baddep \(f_{\varepsilon(0)+2 }(10)\)

Traddep \(f_{\varepsilon(0)+3 } (10)\)

Quadraddep \(f_{\varepsilon(0)+4 } (10)\)

Quintaddep \(f_{\varepsilon(0)+5 } (10)\)

Sextaddep \(f_{\varepsilon(0)+6 } (10)\)

Septaddep \(f_{\varepsilon(0)+7 } (10)\)

Octaddep \(f_{\varepsilon(0)+8 } (10)\)

Nonaddep \(f_{\varepsilon(0)+9 } (10)\)

Dekaddep \(f_{\varepsilon(0)+10 } (10)\)

Hektaddep \(f_{\varepsilon(0)+100 } (10)\)

Kiladdep \(f_{\varepsilon(0)+10^{3} } (10)\)

Megaddep \(f_{\varepsilon(0)+10^{6} } (10)\)

Gigaddep \(f_{\varepsilon(0)+10^{9} } (10)\)

Teraddep \(f_{\varepsilon(0)+10^{12} } (10)\)

Petaddep \(f_{\varepsilon(0)+10^{15} } (10)\)

Exaddep \(f_{\varepsilon(0)+10^{18} } (10)\)

Zettaddep \(f_{\varepsilon(0)+10^{21} } (10)\)

Yottaddep \(f_{\varepsilon(0)+10^{24} } (10)\)

Series 3.1.2 (Epsilon(0)-multiplication series)

In series 3.1.2 \(f_{\varepsilon(0).k}(10)=f_{\varphi(1,0)\cdot k}(10)\) using the fast-growing hierarchy with fundamental sequences for the Veblen function, where:

• \(k\) is a positive integer
• \(\varphi\) is the Veblen function of two variables.

Names of all numbers of series 3.1.2 contain following codes: ult, ep.

Bultep \(f_{\varepsilon(0).2 } (10)\)

Trultep \(f_{\varepsilon(0).3 } (10)\)

Quadrultep \(f_{\varepsilon(0).4 } (10)\)

Quintultep \(f_{\varepsilon(0).5 } (10)\)

Sextultep \(f_{\varepsilon(0).6 } (10)\)

Septultep \(f_{\varepsilon(0).7 } (10)\)

Octultep \(f_{\varepsilon(0).8 } (10)\)

Nonultep \(f_{\varepsilon(0).9 } (10)\)

Dekultep \(f_{\varepsilon(0).10 } (10)\)

Hektultep \(f_{\varepsilon(0).100 } (10)\)

Kilultep \(f_{\varepsilon(0).1000 } (10)\)

Megultep \(f_{\varepsilon(0).10^{6} } (10)\)

Gigultep \(f_{\varepsilon(0).10^{9} } (10)\)

Terultep \(f_{\varepsilon(0).10^{12} } (10)\)

Petultep \(f_{\varepsilon(0).10^{15} } (10)\)

Exultep \(f_{\varepsilon(0).10^{18} } (10)\)

Zettultep \(f_{\varepsilon(0).10^{21} } (10)\)

Yottultep \(f_{\varepsilon(0).10^{24} } (10)\)

Series 3.1.3 (Epsilon(0)-exponentiation series)

In series 3.1.3 \(f_{\varepsilon(0)^k}(10)=f_{\varphi(0,\varphi(1,0)\cdot k)}(10)\) using the fast-growing hierarchy with fundamental sequences for the Veblen function, where:

• \(k>1\) is an integer
• \(\varphi\) is the Veblen function of two variables.

Names of all numbers of series 3.1.3 contain following codes: ex, ep.

Bexep \(f_{\varepsilon(0)^2}(10)\)

Trexep \(f_{\varepsilon(0)^{3}} (10)\)

Quadrexep \(f_{\varepsilon(0)^{4}} (10)\)

Quintexep \(f_{\varepsilon(0)^{5}} (10)\)

Sextexep \(f_{\varepsilon(0)^{6}} (10)\)

Septexep \(f_{\varepsilon(0)^{7}} (10)\)

Octexep \(f_{\varepsilon(0)^{8}} (10)\)

Nonexep \(f_{\varepsilon(0)^{9}} (10)\)

Dekexep \(f_{\varepsilon(0)^{10}} (10)\)

Hektexep \(f_{\varepsilon(0)^{100}} (10)\)

Kilexep \(f_{\varepsilon(0)^{10^{3}}} (10)\)

Megexep \(f_{\varepsilon(0)^{10^{6}}} (10)\)

Gigexep \(f_{\varepsilon(0)^{10^{9}}} (10)\)

Terexep \(f_{\varepsilon(0)^{10^{12}}} (10)\)

Petexep \(f_{\varepsilon(0)^{10^{15}}} (10)\)

Exexep \(f_{\varepsilon(0)^{10^{18}}} (10)\)

Zettexep \(f_{\varepsilon(0)^{10^{21}}} (10)\)

Yottexep \(f_{\varepsilon(0)^{10^{24}}} (10)\)

Series 3.1.4 (Epsilon(0)-tetration series)

In series 3.1.4 \(f_{\varepsilon(0)\uparrow\uparrow k}(10)=f_{\alpha[k]}(10)\) using the fast-growing hierarchy with fundamental sequences for the Veblen function, where:

• \(k>1\) is an integer
• \(\alpha[0]=\varphi(1,0)+\varphi(1,0)\) and \(\alpha[n+1]=\varphi(0,\alpha[n])\) where \(n\geq0\) is an integer and \(\varphi\) is the Veblen function of two variables.

Names of all numbers of series 3.1.4 contain following codes: tetr, ep.

Bitetrep \(f_{\varepsilon(0)\uparrow\uparrow 2 } (10)\)

Tritetrep \(f_{\varepsilon(0)\uparrow\uparrow 3 } (10)\)

Quadritetrep \(f_{\varepsilon(0)\uparrow\uparrow 4 } (10)\)

Quintitetrep \(f_{\varepsilon(0)\uparrow\uparrow 5 } (10)\)

Sextitetrep \(f_{\varepsilon(0)\uparrow\uparrow 6 } (10)\)

Septitetrep \(f_{\varepsilon(0)\uparrow\uparrow 7 } (10)\)

Octitetrep \(f_{\varepsilon(0)\uparrow\uparrow 8 } (10)\)

Nonitetrep \(f_{\varepsilon(0)\uparrow\uparrow 9 } (10)\)

Dekotetrep \(f_{\varepsilon(0)\uparrow\uparrow 10 } (10)\)

Hektotetrep \(f_{\varepsilon(0)\uparrow\uparrow 100 } (10)\)

Kilotetrep \(f_{\varepsilon(0)\uparrow\uparrow 1000 } (10)\)

Megotetrep \(f_{\varepsilon(0)\uparrow\uparrow 10^{6} } (10)\)

Gigotetrep \(f_{\varepsilon(0)\uparrow\uparrow 10^{9} } (10)\)

Terotetrep \(f_{\varepsilon(0)\uparrow\uparrow 10^{12} } (10)\)

Petotetrep \(f_{\varepsilon(0)\uparrow\uparrow 10^{15} } (10)\)

Exotetrep \(f_{\varepsilon(0)\uparrow\uparrow 10^{18} } (10)\)

Zettotetrep \(f_{\varepsilon(0)\uparrow\uparrow 10^{21} } (10)\)

Yottotetrep \(f_{\varepsilon(0)\uparrow\uparrow 10^{24} } (10)\)

3.2) Epsilon(1) series

Series 3.2.1 (Epsilon(1)-addition series)

In series 3.2.1 \(f_{\varepsilon(1)+n}(10)=f_{\varphi(1,1)+n}(10)\) using the fast-growing hierarchy with fundamental sequences for the Veblen function, where:

• \(n\) is a non-negative integer
• \(\varphi\) is the Veblen function of two variables.

Names of all numbers of series 3.2.1 contain following codes: add, un, ep.

Unaddunep \(f_{\varepsilon(1)+1} (10)\)

Baddunep \(f_{\varepsilon(1)+2 }(10)\)

Traddunep \(f_{\varepsilon(1)+3 } (10)\)

Quadraddunep \(f_{\varepsilon(1)+4 } (10)\)

Quintaddunep \(f_{\varepsilon(1)+5 } (10)\)

Sextaddunep \(f_{\varepsilon(1)+6 } (10)\)

Septaddunep \(f_{\varepsilon(1)+7 } (10)\)

Octaddunep \(f_{\varepsilon(1)+8 } (10)\)

Nonaddep \(f_{\varepsilon(1)+9 } (10)\)

Dekaddunep \(f_{\varepsilon(1)+10 } (10)\)

Hektaddunep \(f_{\varepsilon(1)+100 } (10)\)

Kiladdunep \(f_{\varepsilon(1)+10^{3} } (10)\)

Megaddunep \(f_{\varepsilon(1)+10^{6} } (10)\)

Gigaddunep \(f_{\varepsilon(1)+10^{9} } (10)\)

Teraddunep \(f_{\varepsilon(1)+10^{12} } (10)\)

Petaddunep \(f_{\varepsilon(1)+10^{15} } (10)\)

Exaddunep \(f_{\varepsilon(1)+10^{18} } (10)\)

Zettaddunep \(f_{\varepsilon(1)+10^{21} } (10)\)

Yottaddunep \(f_{\varepsilon(1)+10^{24} } (10)\)

Series 3.3 (Inserted epsilon series)

In series 3.3 \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{k\quad\varepsilon's}} (10)=f_{\varphi(2,0)[k]}(10)\) using the fast-growing hierarchy with fundamental sequences for the Veblen function, where:

• \(k\) is a positive integer
• \(\varphi\) is the Veblen function of two variables.

Names of all numbers of series 3.3 contain following codes: in, ep.

Uninep \(f_{\varepsilon_{\varepsilon_{0}}} (10)\)

Binep \(f_{\varepsilon_{\varepsilon_{\varepsilon_{0}}}} (10)\)

Trinep \(f_{\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_{0}}}}} (10)\)

Quadrinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{5\quad\varepsilon's}} (10)\)

Quintinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{6\quad\varepsilon's}} (10)\)

Sextinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{7\quad\varepsilon's}} (10)\)

Septinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{8\quad\varepsilon's}} (10)\)

Octinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{9\quad\varepsilon's}} (10)\)

Noninep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{10\quad\varepsilon's}} (10)\)

Dekinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{11\quad\varepsilon's}} (10)\)

Hektinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{101\varepsilon's}} (10)\)

Kilinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{10^{3}+1\varepsilon's}} (10)\)

Meginep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{10^{6}+1\varepsilon's}} (10)\)

Giginep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{10^{9}+1\varepsilon's}} (10)\)

Terinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{10^{12}+1\varepsilon's}} (10)\)

Petinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{10^{15}+1\varepsilon's}} (10)\)

Exinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{10^{18}+1\varepsilon's}} (10)\)

Zettinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{10^{21}+1\varepsilon's}} (10)\)

Yottinep \(f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{10^{24}+1\varepsilon's}} (10)\)

4) Zeta series

Series 4.1 (Inserted zeta series)

In series 4.1 \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{k\quad\zeta's}} (10)=f_{\varphi(3,0)[k]}(10)\) using the fast-growing hierarchy with fundamental sequences for the Veblen function, where:

• \(k\) is a positive integer
• \(\varphi\) is the Veblen function of two variables.

Names of all numbers of series 4.1 contain following codes: in, zet.

Uninzet \(f_{\zeta _{\zeta_{0}}} (10)\)

Binzet \(f_{\zeta_{\zeta_{\zeta_{0}}}} (10)\)

Trinzet \(f_{\zeta_{\zeta_{\zeta_{\zeta_{0}}}}} (10)\)

Quadrinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{5\quad\zeta's}} (10)\)

Quintinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{6\quad\zeta's}} (10)\)

Sextinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{7\quad\zeta's}} (10)\)

Septinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{8\quad\zeta's}} (10)\)

Octinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{9\quad\zeta's}} (10)\)

Noninzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{10\quad\zeta's}} (10)\)

Dekinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{11\quad\zeta's}} (10)\)

Hektinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{101\zeta's}} (10)\)

Kilinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{10^{3}+1\zeta's}} (10)\)

Meginzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{10^{6}+1\zeta's}} (10)\)

Giginzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{10^{9}+1\zeta's}} (10)\)

Terinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{10^{12}+1\zeta's}} (10)\)

Petinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{10^{15}+1\zeta's}} (10)\)

Exinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{10^{18}+1\zeta's}} (10)\)

Zettinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{10^{21}+1\zeta's}} (10)\)

Yottinzet \(f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{10^{24}+1\zeta's}} (10)\)

5) Eta series

Series 5.1 (Inserted eta series)

In series 5.1 \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{k\quad\eta's}} (10)=f_{\varphi(4,0)[k]}(10)\) using the fast-growing hierarchy with fundamental sequences for the Veblen function, where:

• \(k\) is a positive integer
• \(\varphi\) is the Veblen function of two variables.

Names of all numbers of series 5.1 contain following codes: in, et.

Uninet \(f_{\eta _{\eta_{0}}} (10)\)

Binet \(f_{\eta_{\eta_{\eta_{0}}}} (10)\)

Trinet \(f_{\eta_{\eta_{\eta_{\eta_{0}}}}} (10)\)

Quadrinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{5\quad\eta's}} (10)\)

Quintinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{6\quad\eta's}} (10)\)

Sextinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{7\quad\eta's}} (10)\)

Septinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{8\quad\eta's}} (10)\)

Octinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{9\quad\eta's}} (10)\)

Noninet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{10\quad\eta's}} (10)\)

Dekinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{11\quad\eta's}} (10)\)

Hektinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{101\eta's}} (10)\)

Kilinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{10^{3}+1\eta's}} (10)\)

Meginet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{10^{6}+1\eta's}} (10)\)

Giginet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{10^{9}+1\eta's}} (10)\)

Terinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{10^{12}+1\eta's}} (10)\)

Petinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{10^{15}+1\eta's}} (10)\)

Exinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{10^{18}+1\eta's}} (10)\)

Zettinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{10^{21}+1\eta's}} (10)\)

Yottinet \(f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{10^{24}+1\eta's}} (10)\)

6) Phi-series

For series 6.1 and 6.2 the Veblen function \(\varphi\) of two variables is used. These two series consist of my numbers defined using the fast-growing hierarchy with fundamental sequences for the Veblen function.

Series 6.1

Names of all numbers of series 6.1 contain code: phi.

Uniphi \(f_{\varphi(1,0)}(10)=f_{\varepsilon_0}(10)\)

Biphi \(f_{\varphi(2,0)}(10)=f_{\zeta_0}(10)\)

Triphi \(f_{\varphi(3,0)}(10)=f_{\eta_0}(10)\)

Quadriphi \(f_{\varphi(4,0)}(10)\)

Quintiphi \(f_{\varphi(5,0)}(10)\)

Sextiphi \(f_{\varphi(6,0)}(10)\)

Septiphi \(f_{\varphi(7,0)}(10)\)

Octiphi \(f_{\varphi(8,0)}(10)\)

Noniphi \(f_{\varphi(9,0)}(10)\)

Dekophi \(f_{\varphi(10,0)}(10)\)

Hektophi \(f_{\varphi(100,0)}(10)\)

Kilophi \(f_{\varphi(1000,0)}(10)\)

Megophi \(f_{\varphi(10^{6},0)}(10)\)

Gigophi \(f_{\varphi(10^{9},0)}(10)\)

Terophip \(f_{\varphi(10^{12},0)}(10)\)

Petophi \(f_{\varphi(10^{15},0)}(10)\)

Exophi \(f_{\varphi(10^{18},0)}(10)\)

Zettophi \(f_{\varphi(10^{21},0)}(10)\)

Yottophi \(f_{\varphi(10^{24},0)}(10)\)

Series 6.2 (Inserted phi series)

Names of all numbers of series 6.2 contain following codes: in, phi.

Uninphi \(f_{\varphi(\varphi(0,0),0)} (10)\)

Binphi \(f_{\varphi(\varphi(\varphi(0,0),0),0)} (10)\)

Trinphi \(f_{\varphi(\varphi(\varphi(\varphi(0,0),0),0),0)} (10)\)

Quadrinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{5 \quad \varphi's}} (10)\)

Quintinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{6 \quad \varphi's}} (10)\)

Sextinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0),0)...,0)}_{7 \quad \varphi's}} (10)\)

Septinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{8 \quad \varphi's}} (10)\)

Octinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{9 \quad \varphi's}} (10)\)

Noninphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0),...0),0)}_{10 \quad \varphi's}} (10)\)

Dekinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{11 \quad \varphi's}} (10)\)

Hektinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{101 \quad \varphi's}} (10)\)

Kilinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{10^{3}+1 \quad \varphi's}} (10)\)

Meginphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{10^{6}+1 \quad \varphi's}} (10)\)

Giginphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{10^{9}+1 \quad \varphi's}} (10)\)

Terinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{10^{12}+1 \quad \varphi's}} (10)\)

Petinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{10^{15}+1 \quad \varphi's}} (10)\)

Exinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{10^{18}+1 \quad \varphi's}} (10)\)

Zettinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{10^{21}+1 \quad \varphi's}} (10)\)

Yottinphi \(f_{\underbrace{\varphi(\varphi(\cdots\varphi(\varphi(0,0),0)...,0),0)}_{10^{24}+1 \quad \varphi's}} (10)\)

7) Gamma series

Series 7.1 (Inserted gamma series)

In series 7.1 \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{k\quad\Gamma's}} (10)=f_{\varphi(1,1,0)[k]}(10)\) using the fast-growing hierarchy with fundamental sequences for the Veblen function, where:

• \(k\) is a positive integer
• \(\varphi\) is the Veblen function of three variables.

Names of all numbers of series 7.1 contain following codes: in, gam.

Uningam \(f_{\Gamma _{\Gamma_{0}}} (10)\)

Bingam \(f_{\Gamma_{\Gamma_{\Gamma_{0}}}} (10)\)

Tringam \(f_{\Gamma_{\Gamma_{\Gamma_{\Gamma_{0}}}}} (10)\)

Quadringam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{5\quad\Gamma's}} (10)\)

Quintingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{6\quad\Gamma's}} (10)\)

Sextingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{7\quad\Gamma's}} (10)\)

Septingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{8\quad\Gamma's}} (10)\)

Octingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{9\quad\Gamma's}} (10)\)

Noningam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{10\quad\Gamma's}} (10)\)

Dekingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{11\quad\Gamma's}} (10)\)

Hektingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{101\Gamma's}} (10)\)

Kilingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{10^{3}+1\Gamma's}} (10)\)

Megingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{10^{6}+1\Gamma's}} (10)\)

Gigingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{10^{9}+1\Gamma's}} (10)\)

Teringam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{10^{12}+1\Gamma's}} (10)\)

Petingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{10^{15}+1\Gamma's}} (10)\)

Exingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{10^{18}+1\Gamma's}} (10)\)

Zettingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{10^{21}+1\Gamma's}} (10)\)

Yottingam \(f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{10^{24}+1\Gamma's}} (10)\)

8) Theta-series

In these series 8.1-8.3: \(\theta(\alpha)=\theta_{\alpha}(0)\) where \(\theta_{\alpha}\) is a Feferman's theta-function.

Series 8.1 (Theta- exponentiation series)

In series 8.1 \(f_{\theta (\Omega ^{k})}(10)=f_{\varphi (1,\underbrace {0,0,...,0} _{k+1\;\,0's})}(10)\) using the fast-growing hierarchy with fundamental sequences for the Veblen function, where:

• \(k\) is a positive integer
• \(\varphi\) is the Veblen function of \(k+2\) variables.

Names of all numbers of series 8.1 contain following codes: ex, omm, thet.

Unexommthet \(f_{\theta(\Omega)}(10)=f_{\varphi(1,0,0)}(10)=\)

\(=f_{\Gamma_0}(10)=f_{\varphi(\varphi(\varphi\cdots(\varphi(1,0),0)\cdots),0)}(10)\)

Bexommthet \(f_{\theta(\Omega^2)}(10)=f_{\varphi(1,0,0,0)}(10)=\)

\(=f_{\varphi(\varphi(\varphi\cdots(\varphi(1,0,0),0,0)\cdots),0,0)}(10)\)

Trexommthet \(f_{\theta(\Omega^{3})}(10)=f_{\varphi(1,0,0,0,0)}(10)\)

Quadrexommthet \(f_{\theta(\Omega^{4})}(10)\)

Quintexommthet \(f_{\theta(\Omega^{5})}(10)\)

Sextexommthet \(f_{\theta(\Omega^{6})}(10)\)

Septexommthet \(f_{\theta(\Omega^{7})}(10)\)

Octexommthet \(f_{\theta(\Omega^{8})}(10)\)

Nonexommthet \(f_{\theta(\Omega^{9})}(10)\)

Dekexommthet \(f_{\theta(\Omega^{10})}(10)\)

Small Veblen ordinal level

Hektexommthet \(f_{\theta(\Omega^{100})}(10)\)

Kilexommthet \(f_{\theta(\Omega^{10^{3}})}(10)\)

Megexommthet \(f_{\theta(\Omega^{10^{6}})}(10)\)

Gigexommthet \(f_{\theta(\Omega^{10^{9}})}(10)\)

Terexommthet \(f_{\theta(\Omega^{10^{12}})}(10)\)

Petexommthet \(f_{\theta(\Omega^{10^{15}})}(10)\)

Exexommthet \(f_{\theta(\Omega^{10^{18}})}(10)\)

Zettexommthet \(f_{\theta(\Omega^{10^{21}})}(10)\)

Yottexommthet \(f_{\theta(\Omega^{10^{24}})}(10)\)

Series 8.2 (Theta- tetration series)

In series 8.2 \(f_{\theta (\Omega \uparrow \uparrow k)}(10)=f_{\psi _{0}(\Omega \uparrow \uparrow (k+1))}(10)=f_{\psi _{0}(\psi_1^{k+2}(0))}(10)\) using the fast-growing hierarchy with fundamental sequences for the extended Wilfried Buchholz's functions, where:

• \(\psi_1^{k+2}(0)=\underbrace{\psi_1(\psi_1(\cdots(\psi_1}_{k+2\quad \psi's}(0))\cdots))\)
• \(k\) is a positive integer
• \(\psi_0, \psi_1\) are extended Wilfried Buchholz's functions.

Names of all numbers of series 8.2 contain following codes: tetr, omm, thet.

Bitetrommthet \(f_{\theta(\Omega\uparrow\uparrow 2)}(10)= f_{\theta(\Omega^\Omega)}(10)\)

Large Veblen ordinal level

Tritetrommthet \(f_{\theta(\Omega\uparrow\uparrow 3)}(10)\)

Quadritetrommthet \(f_{\theta(\Omega\uparrow\uparrow 4)}(10)\)

Quintitetrommthet \(f_{\theta(\Omega\uparrow\uparrow 5)}(10)\)

Sextitetrommthet \(f_{\theta(\Omega\uparrow\uparrow 6)}(10)\)

Septitetrommthet \(f_{\theta(\Omega\uparrow\uparrow 7)}(10)\)

Octitetrommthet \(f_{\theta(\Omega\uparrow\uparrow 8)}(10)\)

Nonitetrommthet \(f_{\theta(\Omega\uparrow\uparrow 9)}(10)\)

Dekotetrommthet \(f_{\theta(\Omega\uparrow\uparrow 10)}(10)\)

Hektotetrommthet \(f_{\theta(\Omega\uparrow\uparrow 100)}(10)\)

Kilotetrommthet \(f_{\theta(\Omega\uparrow\uparrow 10^{3})}(10)\)

Megotetrommthet \(f_{\theta(\Omega\uparrow\uparrow 10^{6})}(10)\)

Gigotetrommthet \(f_{\theta(\Omega\uparrow\uparrow 10^{9})}(10)\)

Terotetrommthet \(f_{\theta(\Omega\uparrow\uparrow 10^{12})}(10)\)

Petotetrommthet \(f_{\theta(\Omega\uparrow\uparrow 10^{15})}(10)\)

Exotetrommthet \(f_{\theta(\Omega\uparrow\uparrow 10^{18})}(10)\)

Zettotetrommthet \(f_{\theta(\Omega\uparrow\uparrow 10^{21})}(10)\)

Yottotetrommthet \(f_{\theta(\Omega\uparrow\uparrow 10^{24})}(10)\)

Series 8.3 (Omega subscript series)

In series 8.3 \(f_{\theta (\Omega _{i})}(10)=f_{\psi _{0}(\Omega _{i}^{\Omega _{i}})}(10)=f_{\psi _{0}(\psi_i^3(0))}(10)\) using the fast-growing hierarchy with fundamental sequences for the extended Wilfried Buchholz's functions, where:

• \(\psi_i^3(0)=\psi_i(\psi_i(\psi_i(0)))\)
• \(i\) is a positive integer
• \(\psi_0, \psi_i\) are extended Wilfried Buchholz's functions.

Names of all numbers of series 8.3 contain following codes: omm, thet.

Bommthet \(f_{\theta(\Omega_2)}(10)\)

Trommthet \(f_{\theta(\Omega_3)}(10)\)

Quadrommthet \(f_{\theta(\Omega_4)}(10)\)

Quintommthet \(f_{\theta(\Omega_5)}(10)\)

Sextommthet \(f_{\theta(\Omega_6)}(10)\)

Septommthet \(f_{\theta(\Omega_7)}(10)\)

Octommthet \(f_{\theta(\Omega_8)}(10)\)

Nonommthet \(f_{\theta(\Omega_9)}(10)\)

Dekommthet \(f_{\theta(\Omega_{10})}(10)\)

Hektommthet \(f_{\theta(\Omega_{100})}(10)\)

Kilommthet \(f_{\theta(\Omega_{10^{3}})}(10)\)

Megommthet \(f_{\theta(\Omega_{10^{6}})}(10)\)

Gigommthet \(f_{\theta(\Omega_{10^{9}})}(10)\)

Terommthet \(f_{\theta(\Omega_{10^{12}})}(10)\)

Petommthet \(f_{\theta(\Omega_{10^{15}})}(10)\)

Exommthet \(f_{\theta(\Omega_{10^{18}})}(10)\)

Zettommthet \(f_{\theta(\Omega_{10^{21}})}(10)\)

Yottommthet \(f_{\theta(\Omega_{10^{24}})}(10)\)

9) Psi-series

For series 9.1 and 9.2 the extended Wilfried Buchholz's function \(\psi_0\) is used.

Series 9.1

In series 9.1 \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{k\quad\Omega's}}(10)=f_{\alpha[k]}(10)\) using the fast-growing hierarchy with fundamental sequences for the extended Wilfried Buchholz's functions, where:

• \(k\) is a positive integer
• \(\alpha[0]=1\)
• \(\alpha[n+1]=\psi_0(\Omega_{\alpha[n]})\) where \(n\geq0\) is an integer and \(\psi_0\) is the extended Wilfried Buchholz's function.

Names of all numbers of series 9.1 contain following codes: mix, omm, wil.

Bimixommwil \(f_{\psi(\Omega_{\psi(\Omega) })}(10)\)

Trimixommwil \(f_{\psi(\Omega_{\psi(\Omega_{\psi(\Omega) }) })}(10)\)

Quadrimixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{4\quad\Omega's}}(10)\)

Quintimixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{5\quad\Omega's}}(10)\)

Sextimixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{6\quad\Omega's}}(10)\)

Septimixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{7\quad\Omega's}}(10)\)

Octimixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{8\quad\Omega's}}(10)\)

Nonimixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{9\quad\Omega's}}(10)\)

Dekomixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{10\quad\Omega's}}(10)\)

Hektomixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{100\quad\Omega's}}(10)\)

Kilomixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{10^{3}\quad\Omega's}}(10)\)

Megomixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{10^{6}\quad\Omega's}}(10)\)

Gigomixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{10^{9}\quad\Omega's}}(10)\)

Teromixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{10^{12}\quad\Omega's}}(10)\)

Petomixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{10^{15}\quad\Omega's}}(10)\)

Exomixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{10^{18}\quad\Omega's}}(10)\)

Zettomixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{10^{21}\quad\Omega's}}(10)\)

Yottomixommwil \(f_{\underbrace{\psi(\Omega_{\psi(\Omega_{\cdots_{\psi(\Omega) }\cdots})})}_{10^{24}\quad\Omega's}}(10)\)

Series 9.2

In series 9.2 \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{k\quad\Omega's}}(10)=f_{\psi_0(\alpha[k])}(10)\) using the fast-growing hierarchy with fundamental sequences for the extended Wilfried Buchholz's functions, where:

• \(k\) is a positive integer
• \(\alpha[0]=1\)
• \(\alpha[n+1]=\Omega_{\alpha[n]}\) for all integers \(n\geq0\)
• \(\psi_0\) is the extended Wilfried Buchholz's function.

Names of all numbers of series 9.2 contain following codes: in, omm, wil.

Binommwil \(f_{\psi_0(\Omega_{\Omega}) }(10)\)

Trinommwil \(f_{\psi_0(\Omega_{\Omega_{\Omega}}) }(10)\)

Quadrinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{4\quad\Omega's}}(10)\)

Quintinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{5\quad\Omega's}}(10)\)

Sextinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{6\quad\Omega's}}(10)\)

Septinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{7\quad\Omega's}}(10)\)

Octinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{8\quad\Omega's}}(10)\)

Noninommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{9\quad\Omega's}}(10)\)

Dekinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{10\quad\Omega's}}(10)\)

Hektinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{100\quad\Omega's}}(10)\)

Kilinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{10^{3}\quad\Omega's}}(10)\)

Meginommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{10^{6}\quad\Omega's}}(10)\)

Giginommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{10^{9}\quad\Omega's}}(10)\)

Terinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{10^{12}\quad\Omega's}}(10)\)

Petinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{10^{15}\quad\Omega's}}(10)\)

Exinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{10^{18}\quad\Omega's}}(10)\)

Zettinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{10^{21}\quad\Omega's}}(10)\)

Yottinommwil \(f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{10^{24}\quad\Omega's}}(10)\)

10) \(I_\alpha\)-series

The extended Wilfried Buchholz's functions are not defined for arguments larger than first omega fixed point. That is why for series 10.1 and 10.2 the Hypcos's function \(\psi_{\Omega_1}\) is used.

Series 10.1

In series 10.1 \(f_{\psi(I \uparrow \uparrow k)}(10)= f_{\psi_{\Omega_1}(\sigma[k+1])}(10)\) using the fast-growing hierarchy with fundamental sequences for the Hypcos's functions, where:

• \(\psi_{\Omega_1}\) is the Hypcos's function
• \(k>1\) is an integer
• \(\sigma[0]=0\)
• \(\sigma[n+1]=\psi_{\Omega_{I+1}}(\sigma[n])\) where \(n\geq0\) is an integer and \(I\) is a shorthand for \(I_1\)

Names of all numbers of series 10.1 contain following codes: tetr, ot, os.

Bitetrotos \(f_{\psi(I\uparrow\uparrow 2)}(10)\)

Tritetrotos \(f_{\psi(I\uparrow\uparrow 3)}(10)\)

Quadritetrotos \(f_{\psi(I\uparrow\uparrow 4)}(10)\)

Quintitetrotos \(f_{\psi(I\uparrow\uparrow 5)}(10)\)

Sextitetrotos \(f_{\psi(I\uparrow\uparrow 6)}(10)\)

Septitetrotos \(f_{\psi(I\uparrow\uparrow 7)}(10)\)

Octitetrotos \(f_{\psi(I\uparrow\uparrow 8)}(10)\)

Nonitetrotos \(f_{\psi(I\uparrow\uparrow 9)}(10)\)

Dekotetrotos \(f_{\psi(I\uparrow\uparrow 10)}(10)\)

Hektotetrotos \(f_{\psi(I\uparrow\uparrow 100)}(10)\)

Kilotetrotos \(f_{\psi(I\uparrow\uparrow 10^{3})}(10)\)

Megotetrotos \(f_{\psi(I\uparrow\uparrow 10^{6})}(10)\)

Gigotetrotos \(f_{\psi(I\uparrow\uparrow 10^{9})}(10)\)

Terotetrotos \(f_{\psi(I\uparrow\uparrow 10^{12})}(10)\)

Petotetrotos \(f_{\psi(I\uparrow\uparrow 10^{15})}(10)\)

Exotetrotos \(f_{\psi(I\uparrow\uparrow 10^{18})}(10)\)

Zettotetrotos \(f_{\psi(I\uparrow\uparrow 10^{21})}(10)\)

Yottotetrotos \(f_{\psi(I\uparrow\uparrow 10^{24})}(10)\)

Series 10.2

In series 10.2 \(f_{\psi(\lambda[k])}(10)=f_{\psi_{\Omega_1}(\lambda[k])}(10)\) using the fast-growing hierarchy with fundamental sequences for the Hypcos's functions, where:

• \(k\) is a natural number
• \(\lambda[0]=1\)
• \(\lambda[n+1]=I_{\lambda[n]}\) for all integers \(n\geq0\)
• \(\psi_{\Omega_1}\) is the Hypcos's function.

Names of all numbers of series 10.2 contain following codes: in, ot, os.

Uninotos \(f_{\psi(\lambda[2])}(10)\)

Binotos \(f_{\psi(\lambda[3])}(10)\)

Trinotos \(f_{\psi(\lambda[4])}(10)\)

Quadrinotos \(f_{\psi(\lambda[5])}(10)\)

Quintinotos \(f_{\psi(\lambda[6])}(10)\)

Sextinotos \(f_{\psi(\lambda[7])}(10)\)

Septinotos \(f_{\psi(\lambda[8])}(10)\)

Octinotos \(f_{\psi(\lambda[9])}(10)\)

Noninotos \(f_{\psi(\lambda[10])}(10)\)

Dekinotos \(f_{\psi(\lambda[11])}(10)\)

Hektinotos \(f_{\psi(\lambda[101])}(10)\)

Kilinotos \(f_{\psi(\lambda[1001])}(10)\)

Meginotos \(f_{\psi(\lambda[10^{6}+1])}(10)\)

Giginotos \(f_{\psi(\lambda[10^{9}+1])}(10)\)

Terinotos \(f_{\psi(\lambda[10^{12}+1])}(10)\)

Petinotos \(f_{\psi(\lambda[10^{15}+1])}(10)\)

Exinotos \(f_{\psi(\lambda[10^{18}+1])}(10)\)

Zettinotos \(f_{\psi(\lambda[10^{21}+1])}(10)\)

Yottinotos \(f_{\psi(\lambda[10^{24}+1])}(10)\)

11) \(I(\alpha,\beta)\)-series

For series 11.1 and 11.2 the function collapsing \(\alpha\)-weakly inaccessible cardinals \(\psi_{I(0,0)}\) is used.

Series 11.1

In series 11.1 \(f_{\psi(I(n,0))}(10)=f_{\psi_{I(0,0)}(I(n,0))}(10)\) using the fast-growing hierarchy with fundamental sequences for the functions collapsing \(\alpha\)-weakly inaccessible cardinals, where:

• \(n\) is a natural number
• \(\psi_{I(0,0)}\) is the function collapsing \(\alpha\)-weakly inaccessible cardinals.

Names of all numbers of series 11.1 contain following codes: im, ah.

Unimah \(f_{\psi(I(1,0))}(10)\)

Bimah \(f_{\psi(I(2,0))}(10)\)

Trimah \(f_{\psi(I(3,0))}(10)\)

Quadrimah \(f_{\psi(I(4,0))}(10)\)

Quintimah \(f_{\psi(I(5,0))}(10)\)

Sextimah \(f_{\psi(I(6,0))}(10)\)

Septimah \(f_{\psi(I(7,0))}(10)\)

Octimah \(f_{\psi(I(8,0))}(10)\)

Nonimah \(f_{\psi(I(9,0))}(10)\)

Dekimah \(f_{\psi(I(10,0))}(10)\)

Hektimah \(f_{\psi(I(100,0))}(10)\)

Kilimah \(f_{\psi(I(10^{3},0))}(10)\)

Megimah \(f_{\psi(I(10^{6},0))}(10)\)

Gigimah \(f_{\psi(I(10^{9},0))}(10)\)

Terimah \(f_{\psi(I(10^{12},0))}(10)\)

Petimah \(f_{\psi(I(10^{15},0))}(10)\)

Eximah \(f_{\psi(I(10^{18},0))}(10)\)

Zettimah \(f_{\psi(I(10^{21},0))}(10)\)

Yottimah \(f_{\psi(I(10^{24},0))}(10)\)

Series 11.2

In series 11.2 \(f_{\psi(\tau[k])}(10)=f_{\psi_{I(0,0)}(\tau[k])}(10)\) using the fast-growing hierarchy with fundamental sequences for the functions collapsing \(\alpha\)-weakly inaccessible cardinals, where:

• \(k\) is a natural number
• \(\tau[0]=I(0,0)\)
• \(\tau[n+1]=I(\tau[n],0)\) for all integers \(n\geq0\)
• \(\psi_{I(0,0)}\) is the function collapsing \(\alpha\)-weakly inaccessible cardinals.

Names of all numbers of series 11.2 contain following codes: in, im, ah.

Uninimah \(f_{\psi(\tau[1])}(10)\)

Binimah \(f_{\psi(\tau[2])}(10)\)

Trinimah \(f_{\psi(\tau[3])}(10)\)

Quadrinimah \(f_{\psi(\tau[4])}(10)\)

Quintinimah \(f_{\psi(\tau[5])}(10)\)

Sextinimah \(f_{\psi(\tau[6])}(10)\)

Septinimah \(f_{\psi(\tau[7])}(10)\)

Octinimah \(f_{\psi(\tau[8])}(10)\)

Noninimah \(f_{\psi(\tau[9])}(10)\)

Dekinimah \(f_{\psi(\tau[10])}(10)\)

Hektinimah \(f_{\psi(\tau[100])}(10)\)

Kilinimah \(f_{\psi(\tau[10^{3}])}(10)\)

Meginimah \(f_{\psi(\tau[10^{6}])}(10)\)

Giginimah \(f_{\psi(\tau[10^{9}])}(10)\)

Terinimah \(f_{\psi(\tau[10^{12}])}(10)\)

Petinimah \(f_{\psi(\tau[10^{15}])}(10)\)

Exinimah \(f_{\psi(\tau[10^{18}])}(10)\)

Zettinimah \(f_{\psi(\tau[10^{21}])}(10)\)

Yottinimah \(f_{\psi(\tau[10^{24}])}(10)\)

12) M-series

For series 12.1 and 12.2 the function collapsing weakly Mahlo cardinals \(\psi_{\alpha}\) where \(\alpha=\chi_{M_1}(0)\) is used.

Series 12.1

In series 12.1 \(f_{\psi (M\uparrow \uparrow k)}(10)=f_{\psi _{\alpha }(\beta [k])}(10)\) using the fast-growing hierarchy with fundamental sequences for the function collapsing weakly Mahlo cardinals, where:

• \(k>1\) is an integer
• \(\alpha=\chi_{M_1}(0)\) and \(\psi_\alpha\) is the function collapsing weakly Mahlo cardinals
• \(\beta[0]=M_1\)
• \(\beta[n+1]=\psi_{\gamma}(\beta[n])\) where \(n\geq0\) is an integer and \(\gamma=\chi_{M_2}(1)=\Omega_{M+1}\)

Names of all numbers of series 12.1 contain following codes: tetr, em, ar.

Bitetremar \(f_{\psi(M\uparrow\uparrow 2)}(10)\)

Tritetremar \(f_{\psi(M\uparrow\uparrow 3)}(10)\)

Quadritetremar \(f_{\psi(M\uparrow\uparrow 4)}(10)\)

Quintitetremar \(f_{\psi(M\uparrow\uparrow 5)}(10)\)

Sextitetremar \(f_{\psi(M\uparrow\uparrow 6)}(10)\)

Septitetremar \(f_{\psi(M\uparrow\uparrow 7)}(10)\)

Octitetremar \(f_{\psi(M\uparrow\uparrow 8)}(10)\)

Nonitetremar \(f_{\psi(M\uparrow\uparrow 9)}(10)\)

Dekotetremar \(f_{\psi(M\uparrow\uparrow 10)}(10)\)

Hektotetremar \(f_{\psi(M\uparrow\uparrow 100)}(10)\)

Kilotetremar \(f_{\psi(M\uparrow\uparrow 10^{3})}(10)\)

Megotetremar \(f_{\psi(M\uparrow\uparrow 10^{6})}(10)\)

Gigotetremar \(f_{\psi(M\uparrow\uparrow 10^{9})}(10)\)

Terotetremar \(f_{\psi(M\uparrow\uparrow 10^{12})}(10)\)

Petotetremar \(f_{\psi(M\uparrow\uparrow 10^{15})}(10)\)

Exotetremar \(f_{\psi(M\uparrow\uparrow 10^{18})}(10)\)

Zettotetremar \(f_{\psi(M\uparrow\uparrow 10^{21})}(10)\)

Yottotetremar \(f_{\psi(M\uparrow\uparrow 10^{24})}(10)\)

Series 12.2

In series 12.2 \(f_{\underbrace {\psi (M_{M_{..._{M}}})} _{k\quad M's}}(10)=f_{\psi _{\alpha }(\beta [k])}(10)\) using the fast-growing hierarchy with fundamental sequences for the function collapsing weakly Mahlo cardinals, where:

• \(k\) is a positive integer
• \(\alpha=\chi_{M_1}(0)\) and \(\psi_\alpha\) is the function collapsing weakly Mahlo cardinals
• \(\beta[0]=1\)
• \(\beta[n+1]=M_{\beta[n]}\) for all integers \(n\geq0\)

Names of all numbers of series 12.2 contain following codes: in, em, ar.

Uninemar \(f_{\psi(M_{M})}(10)\)

Binemar \(f_{\psi(M_{M_{M}})}(10)\)

Trinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{4 \quad M's}}(10)\)

Quadrinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{5 \quad M's}}(10)\)

Quintinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{6 \quad M's}}(10)\)

Sextinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{7 \quad M's}}(10)\)

Septinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{8 \quad M's}}(10)\)

Octinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{9 \quad M's}}(10)\)

Noninemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{10 \quad M's}}(10)\)

Dekinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{11 \quad M's}}(10)\)

Hektinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{101 \quad M's}}(10)\)

Kilinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{10^{3}+1 \quad M's}}(10)\)

Meginemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{10^{6}+1 \quad M's}}(10)\)

Giginemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{10^{9}+1 \quad M's}}(10)\)

Terinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{10^{12}+1 \quad M's}}(10)\)

Petinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{10^{15}+1 \quad M's}}(10)\)

Exinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{10^{18}+1 \quad M's}}(10)\)

Zettinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{10^{21}+1 \quad M's}}(10)\)

Yottinemar \(f_{\underbrace{\psi(M_{M_{..._{M}}})}_{10^{24}+1 \quad M's}}(10)\)

Series 12.3

In series 12.3 \(f_{\psi (\upsilon [k])}(10)=f_{\psi _{M(0;0)}(\upsilon [k])}(10)\) using the fast-growing hierarchy with fundamental sequences for the function collapsing \(M(\alpha;\beta)\), where:

• \(k\) is a natural number
• \(M(0;0)\) is the first 0-weakly Mahlo cardinal
• \(\upsilon[0]=M(0;0)\)
• \(\upsilon[n+1]=M(\upsilon[n];0)\) for all integers \(n\geq0\)
• \(\psi_{M(0;0)}\) is the function collapsing \(M(\alpha;\beta)\)

Names of all numbers of series 12.3 contain following codes: in, am, us.

Uninamus \(f_{\psi(\upsilon[1])}(10)\)

Binamus \(f_{\psi(\upsilon[2])}(10)\)

Trinamus \(f_{\psi(\upsilon[3])}(10)\)

Quadrinamus \(f_{\psi(\upsilon[4])}(10)\)

Quintinamus \(f_{\psi(\upsilon[5])}(10)\)

Sextinamus \(f_{\psi(\upsilon[6])}(10)\)

Septinamus \(f_{\psi(\upsilon[7])}(10)\)

Octinamus \(f_{\psi(\upsilon[8])}(10)\)

Noninamus \(f_{\psi(\upsilon[9])}(10)\)

Dekinamus \(f_{\psi(\upsilon[10])}(10)\)

Hektinamus \(f_{\psi(\upsilon[100])}(10)\)

Kilinamus \(f_{\psi(\upsilon[10^{3}])}(10)\)

Meginamus \(f_{\psi(\upsilon[10^{6}])}(10)\)

Giginamus \(f_{\psi(\upsilon[10^{9}])}(10)\)

Terinamus \(f_{\psi(\upsilon[10^{12}])}(10)\)

Petinamus \(f_{\psi(\upsilon[10^{15}])}(10)\)

Exinamus \(f_{\psi(\upsilon[10^{18}])}(10)\)

Zettinamus \(f_{\psi(\upsilon[10^{21}])}(10)\)

Yottinamus \(f_{\psi(\upsilon[10^{24}])}(10)\)

13) Tar series

To go even further let's use Taranovsky's notation. Definition of the notation was published here and here

My system of number names generates too complicated names for case of using of Taranovsky's notation and that is why all names of numbers from series 13.1 and 13.2 although were created according this system, but, unlike previous series, without strict compliance with its rules.

Also let's define the auxiliary function \(Tar(a)\) to simplify the generation of names of numbers.

Let \(Tar(a)=f_{\underbrace{C(C(\cdots(C(C(\Omega_{a}2,0),0),\cdots ),0)}_{a\quad C's}}(a)\) using the fast-growing hierarchy and the folowing fundamental sequences for Taranovsky’s notation:

Let \(L(\alpha)\) be the amount of C's in standard representation of \(\alpha\), then \(\alpha[n]=\max\{\beta|\beta<\alpha\land L(\beta)\le L(\alpha)+n\}\)

Series 13.1

Tritar \(Tar(3)=f_{C(C(C(\Omega_{3} 2,0),0),0)}(3)\)

Quadritar \(Tar(4)=f_{C(C(C(C(\Omega_{4} 2,0),0),0),0)}(4)\)

Quintitar \(Tar(5)=f_{\underbrace{C(C(\cdots C(\Omega_{5} 2,0)\cdots,0),0)}_{5 \quad C's}}(5)\)

Sextitar \(f_{\underbrace{C(C(\cdots C(\Omega_{6} 2,0)\cdots,0),0)}_{6 \quad C's}}(6)\)

Septitar \(f_{\underbrace{C(C(\cdots C(\Omega_{7} 2,0)\cdots,0),0)}_{7 \quad C's}}(7)\)

Octitar \(f_{\underbrace{C(C(\cdots C(\Omega_{8} 2,0)\cdots,0),0)}_{8 \quad C's}}(8)\)

Nonitar \(f_{\underbrace{C(C(\cdots C(\Omega_{9} 2,0)\cdots,0),0)}_{9 \quad C's}}(9)\)

Dekotar \(f_{\underbrace{C(C(\cdots C(\Omega_{10} 2,0)\cdots,0),0)}_{10 \quad C's}}(10)\)

Hektotar \(f_{\underbrace{C(C(\cdots C(\Omega_{100} 2,0)\cdots,0),0)}_{100 \quad C's}}(100)\)

Kilotar \(f_{\underbrace{C(C(\cdots C(\Omega_{10^{3}} 2,0)\cdots,0),0)}_{10^{3} \quad C's}}(10^{3})\)

Megotar \(f_{\underbrace{C(C(\cdots C(\Omega_{10^{6}} 2,0)\cdots,0),0)}_{10^{6} \quad C's}}(10^{6})\)

Gigotar \(f_{\underbrace{C(C(\cdots C(\Omega_{10^{9}} 2,0)\cdots,0),0)}_{10^{9} \quad C's}}(10^{9})\)

Terotar \(f_{\underbrace{C(C(\cdots C(\Omega_{10^{12}} 2,0)\cdots,0),0)}_{10^{12} \quad C's}}(10^{12})\)

Petotar \(f_{\underbrace{C(C(\cdots C(\Omega_{10^{15}} 2,0)\cdots,0),0)}_{10^{15} \quad C's}}(10^{15})\)

Exotar \(f_{\underbrace{C(C(\cdots C(\Omega_{10^{18}} 2,0)\cdots,0),0)}_{10^{18} \quad C's}}(10^{18})\)

Zettotar \(f_{\underbrace{C(C(\cdots C(\Omega_{10^{21}} 2,0)\cdots,0),0)}_{10^{21} \quad C's}}(10^{21})\)

Yottotar \(f_{\underbrace{C(C(\cdots C(\Omega_{10^{24}} 2,0)\cdots,0),0)}_{10^{24} \quad C's}}(10^{24})\)

Series 13.2

Let \(Tar=Tar(10)=f_{\underbrace{C(C(\cdots(C(C(\Omega_{10}2,0),0),\cdots ),0)}_{10\quad C's}}(10)= Dekotar\)

Unintar \(Tar(Tar)=f_{\underbrace{C(C(\cdots(C(C(\Omega_{Dekotar}2,0),0),\cdots ),0)}_{Dekotar\quad C's}}(Dekotar)\)

Bintar \(Tar(Tar(Tar))=f_{\underbrace{C(C(\cdots(C(C(\Omega_{Unintar}2,0),0),\cdots ),0)}_{Unintar\quad C's}}(Unintar)\)

Trintar \(\underbrace{Tar(\cdots (Tar)\cdots)}_{3\quad pairs \quad of \quad brackets}=f_{\underbrace{C(C(\cdots(C(C(\Omega_{Bintar}2,0),0),\cdots ),0)}_{Bintar\quad C's}}(Bintar)\)

Quadrintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{4\quad pairs \quad of \quad brackets}=\)

\(=f_{\underbrace{C(C(\cdots(C(C(\Omega_{Trintar}2,0),0),\cdots ),0)}_{Trintar\quad C's}}(Trintar)\)

Quintintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{5\quad pairs \quad of \quad brackets}\)

Sextintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{6\quad pairs \quad of \quad brackets}\)

Septintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{7\quad pairs \quad of \quad brackets}\)

Octintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{8\quad pairs \quad of \quad brackets}\)

Nonintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{9\quad pairs \quad of \quad brackets}\)

Dekintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{10\quad pairs \quad of \quad brackets}\)

Hektintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{100\quad pairs \quad of \quad brackets}\)

Kilintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{10^{3}\quad pairs \quad of \quad brackets}\)

Megintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{10^{6}\quad pairs \quad of \quad brackets}\)

Gigintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{10^{9}\quad pairs \quad of \quad brackets}\)

Terintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{10^{12}\quad pairs \quad of \quad brackets}\)

Petintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{10^{15}\quad pairs \quad of \quad brackets}\)

Exintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{10^{18}\quad pairs \quad of \quad brackets}\)

Zettintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{10^{21}\quad pairs \quad of \quad brackets}\)

Yottintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{10^{24}\quad pairs \quad of \quad brackets}\)

Tarintar \(\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{Tar\quad pairs \quad of \quad brackets}=\underbrace{Tar(Tar(\cdots(Tar}_{Dekotar\quad Tar's}(Dekotar))\cdots))\)