FANDOM


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

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

2.1) Omega-addition series

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

(addom=addition+ omega)

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

Epsilon(0) series

3.1.1) Epsilon(0)-addition series

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

(addep=addition+ epsilon)

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

3.2.1) Epsilon(1)-addition 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

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

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

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

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

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 series

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}

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