FANDOM

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Fast growing series are sets of numbers names, which were proposed for marking members of some sequences of fast growing hierarchy.

Aims of this system of names generation:

1. name of the number must be connected with well-known FGH,

2. name of the number must allow easily restore mathematical expression, which defines this number.

Note: theta-function used below was described by Chris Bird (defenition of fast growing hierarchy and theta-function here)

codes of operations (group 1)

$+$ add

$\times$ mult

$\uparrow$ ex

$\uparrow^2$ tetr

$\uparrow^3$ pent

$\uparrow^4$ hex

$\uparrow^5$ hept

and so on

Codes of some natural numbers (group 2)

$0$ zer(o) - in bracket the letter is not written if code of operation begin from Vowel

$1$ un(i)

$2$ b(i)

$3$ tr(i)

$4$ quadr(i)

$5$ quint(i)

$6$ sext(i)

$7$ sept(i)

$8$ oct(i)

$9$ non(i)

$10$ dek(o)

$10^2$ hect(o)

$10^3$ kil(o)

$10^6$ meg(o)

$10^9$ gig(o)

$10^{12}$ ter(o)

$10^{15}$ pet(o)

$10^{18}$ ex(o)

$10^{21}$ zett(o)

$10^{24}$ yott(o)

codes of ordinals and functions (group 3)

$\alpha$ (if $\alpha<\omega$) alum

first transfinite ordinal $\omega$: om

ordinal $\varepsilon$: ep

ordinal $\zeta$: zet

ordinal $\eta$: et

ordinal $\Gamma$: gam

and so on

smallest uncountable ordinal $\Omega$: omm

Veblen's function of two-argument $\phi$: phi

Chris Bird's theta-function $\theta$: thet

To generation names of numbers in general case use next rule:

rule 1)

if $f_\lambda(a)=f_{\alpha_1^{b_1}+\alpha_2^{b_2}+...+\alpha_n^{b_n}}(a)$ read the subscript from right to left and then use codes of numbers, codes of operations and codes of ordinals,

$a=10$ default, in other case, write code of a in the beginning of number name and separate via "-"

$f_{\omega+1}(10)$ is unaddom

but

$f_{\omega+1}(3)$ is tri-unaddom.

in the complex case like, for example,

$f_{\omega^3+\omega.2+3}(10)$ according rule 1, this is Traddbimultomaddtriexom.

$f_{\omega^\omega+3}(10)$ is traddomexom

rule 2)

if no codes of group 3 inside name of number, then number should write in terms of up-arrow-notation:

quadritetr is $10\uparrow^2 4$, quintipent is $10\uparrow^3 5$, nonihex is $10\uparrow^4 9$

Special operations

1)in- operation

if code of number b before in and a code of $\bullet$ follows after in, then it means $f_{\bullet\underbrace{(\bullet(\bullet(...(\bullet}_{b \bullet 's}(0))...))}(10)$

example 1. "Trinep" means $f_{\varepsilon(\varepsilon(\varepsilon(\varepsilon(0))))}(10)=f_{\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}}(10)$

example 2. "Trinphi" means $f_{\phi(\phi(\phi(\phi(0,0),0),0),0)}(10)$, where $\phi(0,0)=\omega^0=1$

2)mix- operation

if code of number b before "mix" and a codes of $\bullet_1$ and $\bullet_2$ follow one for another after mix, then it means

$f_{\underbrace{\bullet_2(\bullet_1(\bullet_2(\bullet_1...(\bullet_2(\bullet_1(1))...)))}_{2\times b \bullet's}}(10)$

example 1. Trimixommthet

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

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

example 2. trimixphithet

$f_{\theta(\phi(\theta(\phi(\theta(\phi(1,0)),0)),0))}(10)$

Default

If codes of $\bullet_1$ and $\bullet_2$ follow one another without code of operation between them

it means $f_{\bullet_2(\bullet_1)}(10)$,

example 1. "triphi" means $f_{\phi(3)}(10)=f_{\phi(3,0)}(10)$,

where $\phi(\alpha,\beta)$ is Veblen function

example 2. "ommthet" means $f_{\theta(\Omega)}(10)$

example 3. "trommthet" means $f_{\theta(\Omega(3))}(10)=f_{\theta(\Omega_3)}(10)$

example 4. "Trinommthet" means $f_{\theta(\Omega(\Omega(\Omega(\Omega(0)))))}(10)=$

$=f_{\theta(\Omega_{\Omega_{\Omega_{\omega}}})}(10)$

Note:om is $\omega=\Omega(0)$, omm is $\Omega=\Omega(1)$, bomm is $\Omega_2=\Omega(2)$, 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

multom is $f_{\omega.10}(10)$=dekomultom

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

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

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

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

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

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

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= $f_{\theta(\underbrace{\Omega_{\Omega_{..._{\Omega_\omega}}}}_{10 \Omega 's})}(10)$ meters

1)alpha series Edit

FGH is defined by next rules

$f_0(a)=a+1$

$f_{\alpha+1}(a)= f_\alpha^a(a)$

$f_\alpha(a)= f_{\alpha[a]}(a)$ iff $\alpha$ is a limit ordinal

Zeralum (Zero+alpha) $f_0(10)=10+1=11$

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 $f_3(10)=f_2^{10} (10)= f_2(f_2(f_2(f_2(f_2(f_2(f_2(f_2(f_2(f_2(10))))))))))>2\uparrow\uparrow 10=\underbrace{2^{2^{2^{\cdots^{2^{2}}}}}}_{10}$

Quadralum $f_4(10)=f_3^{10} >2\uparrow\uparrow\uparrow 10=2\uparrow^3 10$

Quintalum $f_5(10)=f_4^{10} >2\uparrow\uparrow\uparrow\uparrow 10=2\uparrow^4 10$

Sextalum $f_6(10)=f_5^{10} >2\uparrow^5 10$

Septalum $f_7(10)=f_6^{10} >2\uparrow^6 10$

Octalum $f_8(10)=f_7^{10} >2\uparrow^7 10$

Nonalum $f_9(10)=f_8^{10} >2\uparrow^8 10$

Dekalum $f_{10} (10)=f_9^{10} > 2\uparrow^9 10=2\underbrace{\uparrow\uparrow...\uparrow\uparrow}_{9 \quad \uparrow 's} 10=(2 \rightarrow 10 \rightarrow 9)$

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 $f_{10^{24}} (10)> 2\uparrow^{10^{24}-1}$

2) Omega series Edit

Zeraddom $f_{\omega} (10)=f_{10} (10)=f_9^{10} >2\uparrow^9 10=(2 \rightarrow 10 \rightarrow 9)$

Unaddom $f_{\omega+1} (10)=f_{\omega } ^{10} (10) >(2 \rightarrow 10 \rightarrow 9\rightarrow 2)$,

already $f_{\omega } ^{2} (10) >\underbrace{2 \uparrow \uparrow \cdots\uparrow\uparrow}_{f_{\omega}(10)-1} f_{\omega}(10)$,

Baddom $f_{\omega+2 }(10) = f_{\omega+1}^{10} (10)>(2 \rightarrow 10 \rightarrow 9\rightarrow 3)$,

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 $f_{\omega+10^{24} } (10)= f_{\omega+(10^{24}-1) } ^{10} (10)>(2 \rightarrow 10 \rightarrow 9\rightarrow 10^{24}+1)$

2.2) Omega- multiplication series

Bimultom $f_{\omega.2 } (10)= f_{\omega+10} (10)>(2 \rightarrow 10 \rightarrow 9\rightarrow 11)$

(multom=multiplication+ omega)

Trimultom $f_{\omega.3 } (10)= f_{\omega.2+10} (10)>(10 \rightarrow 10 \rightarrow 10\rightarrow 10)$

Quadrimultom $f_{\omega.4 } (10)= f_{\omega.3+10} (10)>(10 \rightarrow 10 \rightarrow 10 \rightarrow 10\rightarrow 10)$

Quintimultom $f_{\omega.5 } (10)= f_{\omega.4+10} (10)>(10 \rightarrow 10 \rightarrow 10 \rightarrow 10 \rightarrow 10\rightarrow 10)$

Sextimultom $f_{\omega.6 } (10)$

Septimultom $f_{\omega.7 } (10)$

Octimultom $f_{\omega.8 } (10)$

Nonimultom $f_{\omega.9 } (10)$

Dekomultom $f_{\omega.10 } (10) )>(\underbrace{10 \rightarrow 10 \rightarrow \cdots10 \rightarrow 10\rightarrow 10}_{10 \quad\rightarrow's })=(10\rightarrow_2 10)$

Hektomultom $f_{\omega.100 } (10)$

Kilomultom $f_{\omega.1000 } (10)$

Megomultom $f_{\omega.10^{6} } (10)$

Gigomultom $f_{\omega.10^{9} } (10)$

Teromultom $f_{\omega.10^{12} } (10)$

Petomultom $f_{\omega.10^{15} } (10)$

Exomultom $f_{\omega.10^{18} } (10)$

Zettomultom $f_{\omega.10^{21} } (10)$

Yottomultom $f_{\omega.10^{24} } (10)>(\underbrace{10 \rightarrow 10 \rightarrow \cdots10 \rightarrow 10\rightarrow 10}_{10^{24} \quad\rightarrow's })$

2.3) Omega- exponentiation series

Bexom $f_{\omega^2} (10)=f_{\omega.10 } (10)>(\underbrace{10 \rightarrow 10 \rightarrow \cdots10 \rightarrow 10\rightarrow 10}_{10 \quad\rightarrow 's })=(10\rightarrow_2 10)$

(exom=exponent+ omega)

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

in comparing with BEAF : $f_{\omega^{3}} (10)>\{10,10,10,10,10\}\approx(10\rightarrow_{10} 10)$

Quadrexom $f_{\omega^{4}} (10)>\{10,10,10,10,10,10\}$

Quintexom $f_{\omega^{5}} (10)>\{10,10,10,10,10,10,10\}$

Sextexom $f_{\omega^{6}} (10)>\{10,10,10,10,10,10,10,10\}$

Septexom $f_{\omega^{7}} (10)>\{10,9(1)2 \}$

Octexom $f_{\omega^{8}} (10)>\{10,10(1)2 \}$

Nonexom $f_{\omega^{9}} (10)>\{10,11(1)2 \}$

Dekexom $f_{\omega^{10}} (10)>\{10,12(1)2 \}$

Hektexom $f_{\omega^{100}} (10)>\{10,102(1)2 \}$

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

2.4) Omega- tetration series

Bitetrom $f_{\omega\uparrow\uparrow 2 } (10)= f_{\omega^{\omega }} (10)=f_{\omega^{10}} (10)>\{10,12(1)2 \}$

(tetrom= tetration+ omega)

Tritetrom $f_{\omega\uparrow\uparrow 3 } (10)= f_{\omega^{\omega^{\omega }}} (10) >\{10,10(10)2 \}=>\{10,10(0,1)2 \}$

Quadritetrom $f_{\omega\uparrow\uparrow 4 } (10) >\{10,10((1)1)2 \}=10\uparrow\uparrow 3 \& 10$

Quintitetrom $f_{\omega\uparrow\uparrow 5 } (10) >\{10,10((0,1)1)2 \}=10\uparrow\uparrow 4 \& 10$

Sextitetrom $f_{\omega\uparrow\uparrow 6 } (10) >\{10,10(((1)1)1)2 \}=10\uparrow\uparrow 5 \& 10$

Septitetrom $f_{\omega\uparrow\uparrow 7 } (10) >10\uparrow\uparrow 6 \& 10$

Octitetrom $f_{\omega\uparrow\uparrow 8 } (10) >10\uparrow\uparrow 7 \&1 0$

Nonitetrom $f_{\omega\uparrow\uparrow 9 } (10) >10\uparrow\uparrow 8 \& 10$

Dekotetrom $f_{\omega\uparrow\uparrow 10 } (10) >10\uparrow\uparrow 9 \& 10$

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 Edit

Epsilon(0) series

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

3.1.2) Epsilon(0)-multiplication series

Bimultep $f_{\varepsilon(0).2 } (10)$

(multep=multiplication+ epsilon)

Trimultep $f_{\varepsilon(0).3 } (10)$

Quadrimultep $f_{\varepsilon(0).4 } (10)$

Quintimultep $f_{\varepsilon(0).5 } (10)$

Sextimultep $f_{\varepsilon(0).6 } (10)$

Septimultep $f_{\varepsilon(0).7 } (10)$

Octimultep $f_{\varepsilon(0).8 } (10)$

Nonimultep $f_{\varepsilon(0).9 } (10)$

Dekomultep $f_{\varepsilon(0).10 } (10)$

Hektomultep $f_{\varepsilon(0).100 } (10)$

Kilomultep $f_{\varepsilon(0).1000 } (10)$

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

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

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

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

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

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

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

3.1.3) Epsilon(0)-exponentiation series

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

(exep=exponent+ epsilon)

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

3.1.4) Epsilon(0)-tetration series

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

(tetrep= tetration+ epsilon)

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

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

3.3) Inserted epsilon series

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

(inep=insert+ epsilon)

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) Inserted zeta series Edit

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

(inzet=insert+ zeta)

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) Inserted eta series Edit

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

(inet=insert+ eta)

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 Edit

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

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

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

Quadriphi $f_{\phi(4,0)}(10)$

Quintiphi $f_{\phi(5,0)}(10)$

Sextiphi $f_{\phi(6,0)}(10)$

Septiphi $f_{\phi(7,0)}(10)$

Octiphi $f_{\phi(8,0)}(10)$

Noniphi $f_{\phi(9,0)}(10)$

Dekophi $f_{\phi(10,0)}(10)$

Hektophi $f_{\phi(100,0)}(10)$

Kilophi $f_{\phi(1000,0)}(10)$

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

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

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

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

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

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

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

Inserted phi series

Uninphi $f_{\phi(\phi(0,0),0)} (10)$

(inphi= insert+ phi) Binphi $f_{\phi(\phi(\phi(0,0),0),0)} (10)$

Trinphi $f_{\phi(\phi(\phi(\phi(0,0),0),0),0)} (10)$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

7) Inserted Gamma series Edit

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

(ingam=insert+ gamma)

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 Edit

Below used single-argument theta-function $\theta(\alpha)=\theta(\alpha,0)$.

8.1) Theta- exponentiation series

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

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

(exommthet= exponent+ Omega+ theta) Bexomthet $f_{\theta(\Omega^2)}(10)=f_{\phi(1,0,0,0)}(10)=$

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

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

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

Decexommthet $f_{\theta(\Omega^{10})}(10)$

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

Bird's theta-function is not defined for arguments larger than $\Omega^\omega$. To go further it is possible to use next definition of theta-function.

8.2) Theta- tetration series

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

$=f_{\underbrace{\theta(\Omega^{\theta(\Omega^{\cdots^{ \theta(\Omega)}}}}_{\omega\quad\theta's})}(10)$

where $\theta(\Omega)=\underbrace{(\theta(\theta(\cdots(\theta}_{\omega\quad\theta's}(0))\cdots)))$

(tetrommthet = tetration + Omega+ theta)

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

8.3)

Bommthet $f_{\theta(\Omega_2)}(10)$ (ommthet= Omega+ theta)

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

8.4)

Unimixommthet $f_{\theta(\Omega)}(10)$

Bimixommthet $f_{\theta(\Omega_{\theta(\Omega) })}(10)$

(mixommthet= mix+ Omega+ theta)

Trimixommthet $f_{\theta(\Omega_{\theta(\Omega_{\theta(\Omega) }) })}(10)$

Quadrimixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{4\quad\Omega's}}(10)$

Quintimixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{5\quad\Omega's}}(10)$

Sextimixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{6\quad\Omega's}}(10)$

Septimixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{7\quad\Omega's}}(10)$

Octimixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{8\quad\Omega's}}(10)$

Nonimixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{9\quad\Omega's}}(10)$

Dekomixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{10\quad\Omega's}}(10)$

Hektomixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{100\quad\Omega's}}(10)$

Kilomixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{10^{3}\quad\Omega's}}(10)$

Megomixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{10^{6}\quad\Omega's}}(10)$

Gigomixomthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{10^{9}\quad\Omega's}}(10)$

Teromixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{10^{12}\quad\Omega's}}(10)$

Petomixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{10^{15}\quad\Omega's}}(10)$

Exomixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{10^{18}\quad\Omega's}}(10)$

Zettomixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{10^{21}\quad\Omega's}}(10)$

Yottomixommthet $f_{\underbrace{\theta(\Omega_{\theta(\Omega_{\cdots_{\theta(\Omega) }\cdots})})}_{10^{24}\quad\Omega's}}(10)$

8.5)

Uninommthet $f_{\theta(\Omega_{\omega}) }(10)$

(inommthet= insert+ Omega+ theta)

Binomthet $f_{\theta(\Omega_{\Omega_{\omega}}) }(10)$

Trinommthet $f_{\theta(\Omega_{\Omega_{\Omega_{\omega}}}) }(10)$

Quadrinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{4\quad\Omega's}}(10)$

Quintinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{5\quad\Omega's}}(10)$

Sextinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{6\quad\Omega's}}(10)$

Septinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{7\quad\Omega's}}(10)$

Octinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{8\quad\Omega's}}(10)$

Noninommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{9\quad\Omega's}}(10)$

Dekinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{10\quad\Omega's}}(10)$

Hektinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{100\quad\Omega's}}(10)$

Kilinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{10^{3}\quad\Omega's}}(10)$

Meginommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{10^{6}\quad\Omega's}}(10)$

Giginommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{10^{9}\quad\Omega's}}(10)$

Terinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{10^{12}\quad\Omega's}}(10)$

Petinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{10^{15}\quad\Omega's}}(10)$

Exinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{10^{18}\quad\Omega's}}(10)$

Zettinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{10^{21}\quad\Omega's}}(10)$

Yottinommthet $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{10^{24}\quad\Omega's}}(10)$

Yottinomthetplex $f_{\underbrace{\theta(\Omega_{\Omega_{\cdots_{\Omega_{\omega}) }}}}_{a\quad\Omega's}}(a)$,

where a= Yottinommthet

9) Tar seriesEdit

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

Taranovsky's notation is very powerfull:

if $C(0,0)=1$

already $C(C(\Omega_2 2,0),0)$ is the limit of theta function $\theta(\Omega_{\Omega_{\Omega_{\cdots}}})$

according to this nomenclature for such expressions the names will sound too long

and by this reason let's define the auxiliary function (Tar) to simplify the generation of numbers of names

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

9.1)

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

9.2)

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 $\underbrace{Tar(Tar(\cdots(Tar)\cdots))}_{Tar\quad pairs \quad of \quad brackets}=\underbrace{Tar(Tar(\cdots(Tar(Dekotar))\cdots))}_{Dekotar\quad Tar's}$