### My system of number names (FGS)

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

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

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

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

Codes of operations (group 1)

 codes operations add Addition ult Multiplication ex Exponentiation tetr Tetration pent Pentation hex Hexation hept Heptation

and so on

Codes of some natural numbers (group 2)

 codes natural numbers zer(o) un(i) b(i) tr(i) quadr(i) quint(i) sext(i) sept(i) oct(i) non(i) dek(o) hekt(o) kil(o) meg(o) gig(o) ter(o) pet(o) ex(o) zett(o) yott(o)

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

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

Codes of ordinals and functions (group 3)

 codes ordinals and functions alum a finite ordinal om the first transfinite ordinal ep ordinal zet ordinal et ordinal phi the Veblen function  of two variables gam ordinal omm the first uncountable ordinal thet Feferman's theta-function wil the extended Wilfried Buchholz's function ot the first weakly inaccessible cardinal os the Hypcos's function ah the function collapsing -weakly inaccessible cardinals im the first 0-weakly inaccessible cardinal em the first weakly Mahlo cardinal er the function collapsing weakly Mahlo cardinals  where am the first 0-weakly Mahlo cardinal us the function collapsing M(α;β) into countable ordinals

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

rule 1.

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

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

rule 2.

If name of a number  does not contain codes from group 3, then  where  at the reading from right to left corresponds to sequence of codes from groups 1-2 inside name of the number . Default

For example: quadritetr is , quintipent is , nonihex is

Special operations

1) in- operation (code: in)

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

where

example 1. "Trinep" is equal to

example 2. "Trinphi" is equal to , where

2) mix- operation (code: mix)

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

example 1. Trimixommthet

example 2. trimixphithet

Default

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

example 1. "triphi" is equal to  where  is the Veblen function of two variables

example 2. "ommthet" is equal to

example 3. "trommthet" is equal to

example 4. "Trinommwil" is equal to  since for the extended Wilfried Buchholz's functions  but "Trinommos" is equal to  since for the Hypcos's functions

Note: zeromm is , omm=unomm is  - first uncountable ordinal (i.e. the smallest ordinal that has cardinality ), bomm is  -  the smallest ordinal that has cardinality , tromm is  and so on; ep is  and gamm is  but unep is  and unigam is  and so on.

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

ultom is =dekultom

exom is =dekexom

tetrom is =dekotetrom

inphi is =dekinphi

mixommthet is = dekomixommthet

inommthet is =dekinommthet

Appendix 1: huge units of measurement

l- is the distance  meters,

c- is the -dimensional hypercube with side length  meters,

t- is the time interval  seconds,

m- is the mass  kg.

Example: l-inommthet = inommthet meters.

Appendix 2: googological regions

Let's define: Ra is spherical region of the universe bounded by imaginary sphere with center at the Earth's center and with radius of  meters, where  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 at the Earth's center and with radius of octinommwil meters. Octinommwil is equal to  using the fast-growing hierarchy with fundamental sequences for the extended Wilfried Buchholz's functions, where  and  for all integers

One more example. Ratrinimah is spherical region of the universe bounded by imaginary sphere with center at the Earth's center and with radius of trinimah meters. Trinimah is equal to  using the fast-growing hierarchy with fundamental sequences for the functions collapsing -weakly inaccessible cardinals.

List of my nymbers

## 1) Series with finite ordinals

Series 1.1  (Alpha series)

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

Zeralum

Unalum

Balum

Tralum

Quintalum

Sextalum

Septalum

Octalum

Nonalum

Dekalum

Hektalum

Kilalum

Megalum

Gigalum

Teralum

Petalum

Exalum

Zettalum

Yottalum

Full list of the names of  the numbers in this series, see here

## 2) Omega series

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

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

Series 2.2 (Omega- multiplication series)

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

Bultom

Trultom

Quintultom

Sextultom

Septultom

Octultom

Nonultom

Dekultom

Hektultom

Kilultom

Megultom

Gigultom

Terultom

Petultom

Exultom

Zettultom

Yottultom

Series 2.3 (Omega- exponentiation series)

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

Bexom

Trexom

Quintexom

Sextexom

Septexom

Octexom

Nonexom

Dekexom

Hektexom

Kilexom

Megexom

Gigexom

Terexom

Petexom

Exexom

Zettexom

Yottexom

Series 2.4 (Omega- tetration series)

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

Bitetrom

Tritetrom

Quintitetrom

Sextitetrom

Septitetrom

Octitetrom

Nonitetrom

Dekotetrom

Hektotetrom

Kilotetrom

Megotetrom

Gigotetrom

Terotetrom

Petotetrom

Exotetrom

Zettotetrom

Yottotetrom

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

In series 3.1.1   using the fast-growing hierarchy with fundamental sequences for the Veblen functionwhere:

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

Series 3.1.2 (Epsilon(0)-multiplication series)

In series 3.1.2    using the fast-growing hierarchy with fundamental sequences for the Veblen functionwhere:

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

Bultep

Trultep

Quintultep

Sextultep

Septultep

Octultep

Nonultep

Dekultep

Hektultep

Kilultep

Megultep

Gigultep

Terultep

Petultep

Exultep

Zettultep

Yottultep

Series 3.1.3 (Epsilon(0)-exponentiation series)

In series 3.1.3   using the fast-growing hierarchy with fundamental sequences for the Veblen functionwhere:

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

Bexep

Trexep

Quintexep

Sextexep

Septexep

Octexep

Nonexep

Dekexep

Hektexep

Kilexep

Megexep

Gigexep

Terexep

Petexep

Exexep

Zettexep

Yottexep

Series 3.1.4 (Epsilon(0)-tetration series)

In series 3.1.4   using the fast-growing hierarchy with fundamental sequences for the Veblen functionwhere:

•  is an integer
•   and  where  is an integer and  is the Veblen function of two variables.

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

Bitetrep

Tritetrep

Quintitetrep

Sextitetrep

Septitetrep

Octitetrep

Nonitetrep

Dekotetrep

Hektotetrep

Kilotetrep

Megotetrep

Gigotetrep

Terotetrep

Petotetrep

Exotetrep

Zettotetrep

Yottotetrep

3.2) Epsilon(1) series

In series 3.2.1   using the fast-growing hierarchy with fundamental sequences for the Veblen functionwhere:

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

Series 3.3 (Inserted epsilon series)

In series 3.3    using the fast-growing hierarchy with fundamental sequences for the Veblen functionwhere:

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

Uninep

Binep

Trinep

Quintinep

Sextinep

Septinep

Octinep

Noninep

Dekinep

Hektinep

Kilinep

Meginep

Giginep

Terinep

Petinep

Exinep

Zettinep

Yottinep

## 4) Zeta series

Series 4.1 (Inserted zeta series)

In series 4.1    using the fast-growing hierarchy with fundamental sequences for the Veblen functionwhere:

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

Uninzet

Binzet

Trinzet

Quintinzet

Sextinzet

Septinzet

Octinzet

Noninzet

Dekinzet

Hektinzet

Kilinzet

Meginzet

Giginzet

Terinzet

Petinzet

Exinzet

Zettinzet

Yottinzet

## 5) Eta series

Series 5.1 (Inserted eta series)

In series 5.1    using the fast-growing hierarchy with fundamental sequences for the Veblen functionwhere:

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

Uninet

Binet

Trinet

Quintinet

Sextinet

Septinet

Octinet

Noninet

Dekinet

Hektinet

Kilinet

Meginet

Giginet

Terinet

Petinet

Exinet

Zettinet

Yottinet

## 6) Phi-series

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

Series 6.1

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

Uniphi

Biphi

Triphi

Quintiphi

Sextiphi

Septiphi

Octiphi

Noniphi

Dekophi

Hektophi

Kilophi

Megophi

Gigophi

Terophip

Petophi

Exophi

Zettophi

Yottophi

Series 6.2 (Inserted phi series)

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

Uninphi

Binphi

Trinphi

Quintinphi

Sextinphi

Septinphi

Octinphi

Noninphi

Dekinphi

Hektinphi

Kilinphi

Meginphi

Giginphi

Terinphi

Petinphi

Exinphi

Zettinphi

Yottinphi

## 7) Gamma series

Series 7.1 (Inserted gamma series)

In series 7.1     using the fast-growing hierarchy with fundamental sequences for the Veblen functionwhere:

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

Uningam

Bingam

Tringam

Quintingam

Sextingam

Septingam

Octingam

Noningam

Dekingam

Hektingam

Kilingam

Megingam

Gigingam

Teringam

Petingam

Exingam

Zettingam

Yottingam

## 8) Theta-series

In these series 8.1-8.3 where  is a Feferman's theta-function.

Series 8.1 (Theta- exponentiation series)

In series 8.1   using the fast-growing hierarchy with fundamental sequences for the Veblen functionwhere:

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

Unexommthet

Bexommthet

Trexommthet

Quintexommthet

Sextexommthet

Septexommthet

Octexommthet

Nonexommthet

Dekexommthet

Small Veblen ordinal level

Hektexommthet

Kilexommthet

Megexommthet

Gigexommthet

Terexommthet

Petexommthet

Exexommthet

Zettexommthet

Yottexommthet

Series 8.2 (Theta- tetration series)

In series 8.2   using the fast-growing hierarchy with fundamental sequences for the extended Wilfried Buchholz's functions, where:

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

Bitetrommthet

Large Veblen ordinal level

Tritetrommthet

Quintitetrommthet

Sextitetrommthet

Septitetrommthet

Octitetrommthet

Nonitetrommthet

Dekotetrommthet

Hektotetrommthet

Kilotetrommthet

Megotetrommthet

Gigotetrommthet

Terotetrommthet

Petotetrommthet

Exotetrommthet

Zettotetrommthet

Yottotetrommthet

Series 8.3 (Omega subscript series)

In series 8.3    using the fast-growing hierarchy with fundamental sequences for the extended Wilfried Buchholz's functions, where:

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

Bommthet

Trommthet

Quintommthet

Sextommthet

Septommthet

Octommthet

Nonommthet

Dekommthet

Hektommthet

Kilommthet

Megommthet

Gigommthet

Terommthet

Petommthet

Exommthet

Zettommthet

Yottommthet

## 9) Psi-series

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

Series 9.1

In series 9.1    using the fast-growing hierarchy with fundamental sequences for the extended Wilfried Buchholz's functions, where:

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

Bimixommwil

Trimixommwil

Quintimixommwil

Sextimixommwil

Septimixommwil

Octimixommwil

Nonimixommwil

Dekomixommwil

Hektomixommwil

Kilomixommwil

Megomixommwil

Gigomixommwil

Teromixommwil

Petomixommwil

Exomixommwil

Zettomixommwil

Yottomixommwil

Series 9.2

In series 9.2    using the fast-growing hierarchy with fundamental sequences for the extended Wilfried Buchholz's functions, where:

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

Binommwil

Trinommwil

Quintinommwil

Sextinommwil

Septinommwil

Octinommwil

Noninommwil

Dekinommwil

Hektinommwil

Kilinommwil

Meginommwil

Giginommwil

Terinommwil

Petinommwil

Exinommwil

Zettinommwil

Yottinommwil

## 10) -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  is used.

Series 10.1

In series 10.1   using the fast-growing hierarchy with fundamental sequences for the Hypcos's functionswhere:

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

Bitetrotos

Tritetrotos

Quintitetrotos

Sextitetrotos

Septitetrotos

Octitetrotos

Nonitetrotos

Dekotetrotos

Hektotetrotos

Kilotetrotos

Megotetrotos

Gigotetrotos

Terotetrotos

Petotetrotos

Exotetrotos

Zettotetrotos

Yottotetrotos

Series 10.2

In series 10.2   using the fast-growing hierarchy with fundamental sequences for the Hypcos's functionswhere:

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

Uninotos

Binotos

Trinotos

Quintinotos

Sextinotos

Septinotos

Octinotos

Noninotos

Dekinotos

Hektinotos

Kilinotos

Meginotos

Giginotos

Terinotos

Petinotos

Exinotos

Zettinotos

Yottinotos

## For series 12.1 and 12.2 the function collapsing weakly Mahlo cardinals  where  is used.Series 12.1In series 12.1   using the fast-growing hierarchy with fundamental sequences for the function collapsing weakly Mahlo cardinals, where: is an integer and  is the function collapsing weakly Mahlo cardinals  where  is an integer and Names of all numbers of series 12.1 contain following codes: tetr, em, ar.Bitetremar Tritetremar Quadritetremar Quintitetremar Sextitetremar Septitetremar Octitetremar Nonitetremar Dekotetremar Hektotetremar Kilotetremar Megotetremar Gigotetremar Terotetremar Petotetremar Exotetremar Zettotetremar Yottotetremar Series 12.2In series 12.2   using the fast-growing hierarchy with fundamental sequences for the function collapsing weakly Mahlo cardinals, where: is a positive integer and  is the function collapsing weakly Mahlo cardinals  for all integers Names of all numbers of series 12.2 contain following codes: in, em, ar.Uninemar Binemar Trinemar Quadrinemar Quintinemar Sextinemar Septinemar Octinemar Noninemar Dekinemar Hektinemar Kilinemar Meginemar Giginemar Terinemar Petinemar Exinemar Zettinemar Yottinemar Series 12.3In series 12.3  using the fast-growing hierarchy with fundamental sequences for the function collapsing M(α;β), where: is a natural number is the first 0-weakly Mahlo cardinal for all integers  is the function collapsing M(α;β) Names of all numbers of series 12.3 contain following codes: in, am, us.Uninamus Binamus Trinamus Quadrinamus Quintinamus Sextinamus Septinamus Octinamus Noninamus Dekinamus Hektinamus Kilinamus Meginamus Giginamus Terinamus Petinamus Exinamus Zettinamus Yottinamus  13) Tar series

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

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

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

Let  using the fast-growing hierarchy and the folowing fundamental sequences for Taranovsky’s notation:

Let  be the amount of C’s in standard representation of , then

Series 13.1

Tritar

Quintitar

Sextitar

Septitar

Octitar

Nonitar

Dekotar

Hektotar

Kilotar

Megotar

Gigotar

Terotar

Petotar

Exotar

Zettotar

Yottotar

Series 13.2

Let

Unintar

Bintar

Trintar

Quintintar

Sextintar

Septintar

Octintar

Nonintar

Dekintar

Hektintar

Kilintar

Megintar

Gigintar

Terintar

Petintar

Exintar

Zettintar

Yottintar

Tarintar