My system of number names (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.


Codes of operations (group 1)

codesoperations
addAddition +
ultMultiplication \times
exExponentiation \uparrow
tetrTetration \uparrow^2
pentPentation \uparrow^3
hexHexation \uparrow^4
heptHeptation \uparrow^5


and so on


Codes of some natural numbers (group 2)

codesnatural 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)

codesordinals and functions
aluma finite ordinal
omthe first transfinite ordinal \omega
epordinal \varepsilon_0=\text{min}\{\alpha|\omega^\alpha=\alpha\}
zetordinal \zeta_0=\text{min}\{\alpha|\varepsilon_\alpha=\alpha\}
etordinal \eta_0=\text{min}\{\alpha|\zeta_\alpha=\alpha\}
phithe Veblen function \varphi of two variables
gamordinal \Gamma_0=\text{min}\{\alpha|\varphi(\alpha,0)=\alpha\}
ommthe first uncountable ordinal \Omega
thetFeferman's theta-function \theta
wilthe extended Wilfried Buchholz's function \psi_0
otthe first weakly inaccessible cardinal I_1
osthe Hypcos's function \psi_{\Omega_1}
ahthe function collapsing \alpha-weakly inaccessible cardinals \psi_{I(0,0)}
imthe first 0-weakly inaccessible cardinal I(0,0)
emthe first weakly Mahlo cardinal M_1
erthe function collapsing weakly Mahlo cardinals \psi_{\alpha} where \alpha=\chi_{M_1}(0)
amthe first 0-weakly Mahlo cardinal M(0;0) 
usthe function collapsing M(α;β) 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").

See more about googological regions.

Example: Roctinommwil is spherical region of the universe bounded by imaginary sphere with center at the Earth's center and with radius of octinommwil meters. Octinommwil is equal to f_{\psi_{0}(\alpha[8])}(10) using the fast-growing hierarchy with fundamental sequences for the extended Wilfried Buchholz's functions, where \alpha[0]=1 and \alpha[n+1]=\Omega_{\alpha[n]} for all integers n\geq0

One more example. Ratrinimah is spherical region of the universe bounded by imaginary sphere with center at the Earth's center and with radius of trinimah meters. Trinimah is equal to f_{\psi_{I(0,0)}(I(I(I(I(0,0),0),0),0))}(10) using the fast-growing hierarchy with fundamental sequences for the functions collapsing \alpha-weakly inaccessible cardinals.


List of my nymbers

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

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)

Below you can read links to the lists containing   8000 names of numbers (googologisms) from "omega-series"

GOOGOLOGISMS w^1-w^3

GOOGOLOGISMS w^4-w^6

GOOGOLOGISMS w^7-w^9

GOOGOLOGISMS w^10-w^12

GOOGOLOGISMS w^13-w^15

GOOGOLOGISMS w^16-w^18

GOOGOLOGISMS w^19-w^20


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 functionwhere:

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 functionwhere:

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 functionwhere:

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 functionwhere:

  • 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 functionwhere:

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 functionwhere:

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 functionwhere:

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 functionwhere:

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 functionwhere:

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  {\displaystyle 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 functionwhere:

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:

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:

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:

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:

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 functionswhere:

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 functionswhere:

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 cardinalswhere:

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 cardinalswhere:

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  {\displaystyle 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 cardinalswhere:

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  {\displaystyle 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:


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 {\displaystyle 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(α;β), where:


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))


original of publication (31 may 2016, my blog in googology.wikia)