My system of number names (FGS)

This is system of naming of numbers, which are expressed through functions of fast-growing hierarchy, which is defined by the following rules:

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

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

Numbers are grouped in series (the fast-growing series, FGS), 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.


codes of operations (group 1)

+ add

\times ult

\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 brackets the letter is not written, if after the  code of the number  farther  is any   vowel letter

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 hekt(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)

first transfinite ordinal \omega: om

ordinal \alpha<\omega: alum

ordinal \varepsilon_0=\text{min}\{\alpha|\omega^\alpha=\alpha\}: ep

ordinal \zeta_0=\text{min}\{\alpha|\varepsilon_\alpha=\alpha\}: zet

ordinal \eta_0=\text{min}\{\alpha|\zeta_\alpha=\alpha\}: et

Veblen's function of two arguments \varphi: phi

ordinal \Gamma_0=\text{min}\{\alpha|\varphi(\alpha,0)=\alpha\}: gam

first uncountable ordinal \Omega: omm

theta-function \theta: thet

the extended Wilfried Buchholz's function \psi_0: wil

first weakly inaccessible cardinal I_1: ot

the Hypcos's function \psi_{\Omega_1}: os

the function collapsing \alpha-weakly inaccessible cardinals \psi_{I(0,0)}: ah

function of two arguments I: im


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

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.

In the complex case like, for example,

f_{\omega^3+\omega.2+3}(10) according rule 1, this is Traddbultomaddtrexom and f_{\omega^\omega+3}(10) is traddomexom. 

Argument a=10  default, in other case, write code of "a"  in the beginning of the number name, adding "argum", and separate it via "-":

for example f_{\omega+1}(10) is unaddom, but f_{\omega+1}(3) is trargum-unaddom


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 natural number b before in and a code of \alpha follows after in, then it means f_{\underbrace{\alpha(\alpha(...(\alpha(i))...))}_{b+1\quad  \alpha 's}}(10), 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

if code of natural number b before "mix" and a codes of \alpha and \beta follow one for another after mix, then it means

f_{\underbrace{\beta(\alpha(\beta(\alpha...(\beta(\alpha(1))...)))}_{b \quad\alpha'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(\varphi(\theta(\varphi(\theta(\varphi(1,0)),0)),0))}(10)

Default

If codes of \alpha and \beta follow one another without code of operation between them, then it means f_{\beta(\alpha)}(10)

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

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 - second uncountable ordinal (i.e. 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)


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.

Ra\bullet is spherical region of the universe bounded by imaginary sphere with center in the center of the Earth 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").

Example: Roctinommwil is spherical region of the universe bounded by imaginary sphere with center in the center of the Earth and with radius of octinommwil meters.

One more example. Ratrinimah is spherical region of the universe bounded by imaginary sphere with center in the center of the Earth and with radius of trinimah meters.

About trinimah and octinommwil you can read below.

See also Space scale.


Fast-growing hierarchy 

Fast-growing hierarchy is defined as follows:

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

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

Fast-growing hierarchy of functions f_\alpha is defined for \alpha<\beta, where \beta is a large countable ordinal such that a fundamental sequence is assigned to every limit ordinal less than \beta

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

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

Note: \omega^0=1 

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

See also fundamental sequences for Veblen's function.

1) alpha series

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))))))))))=

=10^{10^{10^{10^{10^{10^{10^{10^{1,0865890600\times 10^{3086}}}}}}}}} > 10\uparrow \uparrow 10 > 2\uparrow \uparrow 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} 10


Full list of the names of  the numbers in this series from 0 to 99, see here

2) Omega series 

2.1) Omega-addition series

Zeraddom f_{\omega} (10)=f_{10} (10)=f_9^{10}(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

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

(ultom=multiplication+ omega)

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

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

Quintultom f_{\omega.5 } (10)= f_{\omega.4+10} (10)>(10 \rightarrow 10 \rightarrow 10 \rightarrow 10 \rightarrow 10\rightarrow 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) >(\underbrace{10 \rightarrow 10 \rightarrow \cdots10 \rightarrow 10\rightarrow 10}_{10 \quad\rightarrow's })=(10\rightarrow_2 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)>(\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)

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

3.1.1) Epsilon(0)-addition series

Note: f_{\varepsilon_0+n}(10)=f_{\varphi(1,0)+n}(10) where n is a non-negative integer and \varphi is the Veblen's function.

See also fundamental sequences for Veblen's function.

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

Note: f_{\varepsilon_0.k}(10)=f_{\varphi(1,0)\cdot k}(10) where k is a positive integer and \varphi is the Veblen's function.

See also fundamental sequences for Veblen's function.

Bultep f_{\varepsilon(0).2 } (10)

(ultep=multiplication+ epsilon)

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)

3.1.3) Epsilon(0)-exponentiation series

Note: f_{\varepsilon_0^k}(10)=f_{\varphi(0,\varphi(1,0)\cdot k)}(10) where k is a positive integer and \varphi is the Veblen's function.

See also fundamental sequences for Veblen's function.

Bexep f_{\varepsilon(0)^2}(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

Note: for an integer k\geq 2 we can write f_{\varepsilon_0\uparrow\uparrow k}(10)=f_{\alpha[k]}(10) where \alpha[0]=\varphi(1,0)+\varphi(1,0) and \alpha[n+1]=\varphi(0,\alpha[n]) and \varphi is the Veblen's function.

See also fundamental sequences for Veblen's function.

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

Note: f_{\varepsilon_1+n}(10)=f_{\varphi(1,1)+n}(10) where n is a non-negative integer and \varphi is the Veblen's function.

See also fundamental sequences for Veblen's function.

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

Note: f_{\underbrace{\varepsilon_{\varepsilon_{\cdots_{\varepsilon_{\varepsilon_{0}}}}}}_{k\quad\varepsilon's}} (10)=f_{\varphi(2,0)[k]}(10) where k is a positive integer and \varphi is the Veblen's function.

See also fundamental sequences for Veblen's function.

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 

Note: f_{\underbrace{\zeta_{\zeta_{\cdots_{\zeta_{\zeta_{0}}}}}}_{k\quad\zeta's}} (10)=f_{\varphi(3,0)[k]}(10) where k is a positive integer and \varphi is the Veblen's function.

See also fundamental sequences for Veblen's function.

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 

Note: f_{\underbrace{\eta_{\eta_{\cdots_{\eta_{\eta_{0}}}}}}_{k\quad\eta's}} (10)=f_{\varphi(4,0)[k]}(10) where k is a positive integer and \varphi is the Veblen's function.

See also fundamental sequences for Veblen's function

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 

6.1)

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

where \varphi is the Veblen's function.

See also fundamental sequences for Veblen's function

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)

6.2) Inserted phi series

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

(inphi= insert+ phi)

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) Inserted Gamma series 

Note: f_{\underbrace{\Gamma_{\Gamma_{\cdots_{\Gamma_{\Gamma_{0}}}}}}_{k\quad\Gamma's}} (10)=f_{\varphi(1,1,0)[k]}(10) where k is a positive integer and \varphi is the Veblen's function.

See also fundamental sequences for Veblen's function.

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 

8.1) Theta- exponentiation series

For series 8.1  the Chris Bird's theta-function is used, which is defined as follows [2]:

\theta(\Omega^k \alpha_k+\cdots+\Omega^2 \alpha_2+\Omega \alpha_1+\alpha_0, \beta)=\varphi(\alpha_k, \ldots, \alpha_2, \alpha_1, \alpha_0, \beta)

where \Omega is the first uncountable ordinal and \varphi is the Veblen's function.

Note 1: in series 8.1 \theta(\alpha)=\theta(\alpha,0) where \theta is the Bird's theta-function.

Note 2: {\displaystyle f_{\theta (\Omega ^{k})}(10)=f_{\varphi (1,\underbrace {0,0,...,0} _{k+1\;\,0's})}(10)} where k is a positive integer and \varphi is the Veblen's function.

See also fundamental sequences for Veblen's function.

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)

(exommthet= exponent+ Omega+ theta)

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)

Bird's theta-function is not defined for arguments larger than \Omega^\omega and that is why for series 8.2 and 8.3  the Feferman's theta-function is used. 

In series 8.2 and 8.3: \theta(\alpha)=\theta_{\alpha}(0) where \theta_{\alpha} is the Feferman's theta-function [3] [4].

8.2) Theta- tetration series

Note: 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)

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

See also fundamental sequences for the extended Wilfried Buchholz's functions \psi_\nu

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

(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) Omega subscript series

Note: f_{\theta (\Omega _{i})}(10)=f_{\psi _{0}(\Omega _{i}^{\Omega _{i}})}(10)=f_{\psi _{0}(\psi_i^3(0))}(10) where \psi_i^3(0)=\psi_i(\psi_i(\psi_i(0))) and i is a positive integer and \psi_0\psi_i are the extended Wilfried Buchholz's functions

See also fundamental sequences for the extended Wilfried Buchholz's functions \psi_\nu

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)

9) Psi-series 

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

9.1)

Note: in series 9.1 \psi(\alpha) is a shorthand for \psi_0(\alpha) and \Omega is a shorthand for \Omega_1 

See also fundamental sequences for the extended Wilfried Buchholz's functions.

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

(mixommwil= mix+ Omega+ the extended Wilfried Buchholz's function)

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)


9.2)

Note: f_{\underbrace{\psi_0(\Omega_{\Omega_{\cdots_{\Omega }}})}_{k\quad\Omega's}}(10)=f_{\psi_0(\alpha[k])}(10) where k is a positive integer and \alpha[0]=1 and \alpha[n+1]=\Omega_{\alpha[n]}

See also fundamental sequences for the extended Wilfried Buchholz's functions.

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

(inommwil= insert+ Omega+ the extended Wilfried Buchholz's function)

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.

10.1) 

Note 1: in series 10.1 I is a shorthand for I_1 

Note 2: in series 10.1 f_{\psi(I \uparrow \uparrow k)}(10)= f_{\psi_{\Omega_1}(\sigma[k+1])}(10) where

  • \sigma[0]=0 and \sigma[n+1]=\psi_{\Omega_{I+1}}(\sigma[n])
  • integer k\geq 2
  • \psi_{\Omega_1} and \psi_{\Omega_{I+1}} are the Hypcos's functions.

See also: fundamental sequences for the Hypcos's functions.

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)

10.2) 

Note 1: in series 10.2 \psi(\lambda[n]) is a shorthand for \psi_{\Omega_1}(\lambda[n])

Note 2: in series 10.2 \lambda[0]=1 and \lambda[\eta+1]=I_{\lambda[\eta]}

See also: fundamental sequences for the Hypcos's functions.

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.

11.1)

Note: in series 11.1 \psi(I(n,0)) is a shorthand for \psi_{I(0,0)}(I(n,0))

See also: fundamental sequences for the functions collapsing \alpha-weakly inaccessible cardinals.

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)

11.2)

Note 1: in series 11.2 \psi(\tau[n]) is a shorthand for \psi_{I(0,0)}(\tau[n])

Note 2: in series 11.2 \tau[0]=I(0,0) and \tau[\eta+1]=I(\tau[\eta],0)

See also: fundamental sequences for the functions collapsing \alpha-weakly inaccessible cardinals.

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) 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(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)

The fundamental sequences of Taranovsky’s notation can be defined as follows [1]:

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\}

12.1)

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})


12.2)


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)

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