War God numbers/functions are series of functions coined by FDLevi.[1][2] Although it is not clarified in the source, the functions are intended to be defined on the set of positive integers.

The system originally included three families of functions,[1] and later the creator added the fourth family.[2] As the functions are defined as the iteration of multivariable functions whose diagonalisations are bounded by constant iterations of \(f_{\omega^2}\) in Wainer hierarchy, they are bounded by \(f_{\omega^2+1}\). Since the system directly uses BEAF up to \(f_{\omega^2}\) level, the contribution of the original effort by the creator to ordinals in Wainer hierarchy is smaller than \(1\).

Classic ones

The first family is called the functions in order to distinguish it from the other families, and is based on the Arrow notation.

Eris(x)

The first one in the functions is Eris.

Eris(x) = x↑x↑xxx

Phobos(x)

The second one in the functions is Phobos. This grows faster than Eris.

Phobos(x) = Eris(x↑Eris(x)↑Eris(x)x)

Ares(x)

The third one in the functions is Ares. This grows even faster than Phobos does.

Ares(x) = Phobos(x↑Phobos(x)↑Phobos(x)↑Phobos(x)xx)

Kratos(x)

The fourth one in the functions is Kratos. This function grows faster than Ares.

Kratos(x) = Ares(x↑Ares(x)↑Ares(x)↑Ares(x)↑Ares(x)↑Ares(x)xxxx)

Hades(x)

The fifth one in the functions is Hades. This function grows faster than Kratos.

Hades(x) = Kratos(x↑Kratos(x)↑Kratos(x)↑Kratos(x)↑Kratos(x)↑Kratos(x)↑Kratos(x)↑Kratos(x)xxxxxx)

Charon(x)

After the fifth one comes Charon. This is starting to grow faster than Hades.

Charon(x) = Hades(x↑Hades(x)↑Hades(x)↑Hades(x)↑Hades(x)↑Hades(x)↑Hades(x)↑Hades(x)↑Hades(x)↑Hades(x)xxxxxxxx)

Erebus(x)

The seventh one is Erebus. It grows faster than Charon.

Erebus(x) = Charon(x↑Charon(x)↑Charon(x)↑Charon(x)↑Charon(x)↑Charon(x)↑Charon(x)↑Charon(x)↑Charon(x)↑Charon(x)↑Charon(x)↑Charon(x)xxxxxxxxxx)

Thanatos(x)

The last one yet, Thanatos. This is the fastest among all the functions in the first family.

Thanatos(x) = Erebus(x↑Erebus(x)↑Erebus(x)↑Erebus(x)↑Erebus(x)↑Erebus(x)↑Erebus(x)↑Erebus(x)↑Erebus(x)↑Erebus(x)↑Erebus(x)↑Erebus(x)↑Erebus(x)↑Erebus(x)xxxxxxxxxxxx)

Hyper ones

The second family is called the hyper-functions, and is based on the Bowers Exploding Array Function and the functions above.

ErisH(x)

The first one in the hyper-functions is ErisH.

ErisH(x) = x{Eris(x)}Eris(x)x

PhobosH(x)

The second one in the hyper-functions is PhobosH. This grows faster than ErisH.

PhobosH(x) = ErisH(x{Phobos(x)}ErisH(x){Phobos(x)}ErisH(x)x)

AresH(x)

The third one in the hyper-functions is AresH. This grows even faster than PhobosH does.

AresH(x) = PhobosH(x{Ares(x)}PhobosH(x){Ares(x)}PhobosH(x){Ares(x)}PhobosH(x)xx)

KratosH(x)

The fourth one in the hyper-functions is KratosH. This function grows faster than AresH.

KratosH(x) = AresH(x{Kratos(x)}AresH(x){Kratos(x)}AresH(x){Kratos(x)}AresH(x){Kratos(x)}AresH(x){Kratos(x)}AresH(x)xxxx)

HadesH(x)

The fifth one in the hyper-functions is HadesH. This function grows faster than KratosH.

HadesH(x) = KratosH(x{Hades(x)}KratosH(x){Hades(x)}KratosH(x){Hades(x)}KratosH(x){Hades(x)}KratosH(x){Hades(x)}KratosH(x){Hades(x)}KratosH(x){Hades(x)}KratosH(x)xxxxxx)

CharonH(x)

After the fifth one comes CharonH. This grows fatser than HadesH.

CharonH(x) = HadesH(x↑HadesH(x){Charon(x)}HadesH(x){Charon(x)}HadesH(x){Charon(x)}HadesH(x){Charon(x)}HadesH(x){Charon(x)}HadesH(x){Charon(x)}HadesH(x){Charon(x)}HadesH(x){Charon(x)}HadesH(x)xxxxxxxx)

ErebusH(x)

The seventh one is ErebusH. It grows faster than CharonH.

ErebusH(x) = CharonH(x{Erebus(x)}CharonH(x){Erebus(x)}CharonH(x){Erebus(x)}CharonH(x){Erebus(x)}CharonH(x){Erebus(x)}CharonH(x){Erebus(x)}CharonH(x){Erebus(x)}CharonH(x){Erebus(x)}CharonH(x){Erebus(x)}CharonH(x){Erebus(x)}CharonH(x){Erebus(x)}CharonH(x)xxxxxxxxxx)

ThanatosH(x)

The last one yet, ThanatosH. This is the fastest among all the hyper-functions.

ThanatosH(x) = ErebusH(x{Thanatos(x)}ErebusH(x){Thanatos(x)}ErebusH(x){Thanatos(x)}ErebusH(x){Thanatos(x)}ErebusH(x){Thanatos(x)}ErebusH(x){Thanatos(x)}ErebusH(x){Thanatos(x)}ErebusH(x){Thanatos(x)}ErebusH(x){Thanatos(x)}ErebusH(x){Thanatos(x)}ErebusH(x){Thanatos(x)}ErebusH(x){Thanatos(x)}ErebusH(x){Thanatos(x)}ErebusH(x)xxxxxxxxxxxx)

Big ones

The third family is called the big-functions, and is based on the Bowers Exploding Array Function, the functions above, and the hyper-functions above.

ERIS(x)

The first one in the big-functions is ERIS.

ERIS(x) = ErisH(Eris(x{ErisH(Eris(x))}x{ErisH(Eris(x))}ErisH(Eris(x))xx))

PHOBOS(x)

The second one in the big-functions is PHOBOS. This grows faster than ERIS.

PHOBOS(x) = ERIS(x{PhobosH(Phobos(x))}ERIS(x){PhobosH(Phobos(x))}ERIS(x){PhobosH(Phobos(x))}ERIS(x)xx)

ARES(x)

The third one in the big-functions is ARES. This grows even faster than PHOBOS does.

ARES(x) = PHOBOS(x{AresH(Ares(x))}ARES(x){AresH(Ares(x))}ARES(x){AresH(Ares(x))}ARES(x){AresH(Ares(x))}(PHOBOS(x))xxx)

KRATOS(x)

The fourth one in the big-functions is KRATOS. This function grows faster than ARES.

KRATOS(x) = ARES(x{KratosH(Kratos(x))}ARES(x){KratosH(Kratos(x))}ARES(x){KratosH(Kratos(x))}ARES(x){KratosH(Kratos(x))}ARES(x){KratosH(Kratos(x))}ARES(x){KratosH(Kratos(x))}ARES(x)xxxxx)

HADES(x)

The fifth one in the big-functions is HADES. This function grows faster than KRATOS.

HADES(x) = KRATOS(x{HadesH(Hades(x))}KRATOS(x){HadesH(Hades(x))}KRATOS(x){HadesH(Hades(x))}KRATOS(x){HadesH(Hades(x))}KRATOS(x){HadesH(Hades(x))}KRATOS(x){HadesH(Hades(x))}KRATOS(x){HadesH(Hades(x))}KRATOS(x){HadesH(Hades(x))}KRATOS(x)xxxxxxx)

CHARON(x)

After the fifth one comes CHARON. This grows faster than HADES.

CHARON(x) = HADES(x{CharonH(Charon(x))}HADES(x){CharonH(Charon(x))}HADES(x){CharonH(Charon(x))}HADES(x){CharonH(Charon(x))}HADES(x){CharonH(Charon(x))}HADES(x){CharonH(Charon(x))}HADES(x){CharonH(Charon(x))}HADES(x){CharonH(Charon(x))}HADES(x){CharonH(Charon(x))}HADES(x){CharonH(Charon(x))}(HADES(x))xxxxxxxxx)

EREBUS(x)

The seventh one is EREBUS. It grows faster than CHARON.

EREBUS(x) = CHARON(x{ErebusH(Erebus(x))}CHARON(x){ErebusH(Erebus(x))}CHARON(x){ErebusH(Erebus(x))}CHARON(x){ErebusH(Erebus(x))}CHARON(x){ErebusH(Erebus(x))}CHARON(x){ErebusH(Erebus(x))}CHARON(x){ErebusH(Erebus(x))}CHARON(x){ErebusH(Erebus(x))}CHARON(x){ErebusH(Erebus(x))}CHARON(x){ErebusH(Erebus(x))}CHARON(x){ErebusH(Erebus(x))}CHARON(x){ErebusH(Erebus(x))}CHARON(x)xxxxxxxxxxx)

THANATOS(x)

The last one yet, THANATOS. This is the fastest among all the big-functions.

THANATOS(x) = EREBUS(x{ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x){ThanatosH(Thanatos(x))}EREBUS(x)xxxxxxxxxxxxx)

Big-hyper ones.

The fourth family is called the big-hyper-functions, and is based on the Bowers Exploding Array Function, the functions above, the hyper-functions above, and the big-functions above.

ERISH(x)

The first one in the big-hyper-functions is ERISH.

ERISH(x) = ERIS(ErisH(Eris(x{ERIS(ErisH(Eris(x)))}x{ERIS(ErisH(Eris(x)))}x{ERIS(ErisH(Eris(x)))}ERIS(ErisH(Eris(x)))ERIS(x)ERIS(x)ERIS(x))))

PHOBOSH(x)

The second one in the big-hyper-functions is PHOBOSH. This grows faster than ERISH.

PHOBOSH(x) = ERISH(ERISH(x){ERISH(PhobosH(Phobos(x)))}ERISH(x){PHOBOS(PhobosH(Phobos(x)))}ERISH(x){PHOBOS(PhobosH(Phobos(x)))}ERISH(x){PHOBOS(PhobosH(Phobos(x)))}(PHOBOS(x))PHOBOS(x)PHOBOS(x)PHOBOS(x))

ARESH(x)

The third one in the big-hyper-functions is ARESH. This function grows even faster than PHOBOSH does.

ARESH(x) = PHOBOSH(x{PHOBOSH(AresH(Ares(x)))}PHOBOSH(x){ARES(AresH(Ares(x)))}PHOBOSH(x){ARES(AresH(Ares(x)))}PHOBOSH(x){ARES(AresH(Ares(x)))}PHOBOSH(x){ARES(AresH(Ares(x)))}ARES(x)ARES(x)ARES(x)ARES(x)ARES(x))

KRATOSH(x)

The fourth one in the big-hyper-functions is KRATOSH. This function grows faster than ARESH.

KRATOSH(x) = ARESH(ARESH(x){KRATOS(KratosH(Kratos(x)))}ARESH(x){KRATOS(KratosH(Kratos(x)))}ARESH(x){KRATOS(KratosH(Kratos(x)))}ARESH(x){KRATOS(KratosH(Kratos(x)))}ARESH(x){KRATOS(KratosH(Kratos(x)))}ARESH(x){KRATOS(KratosH(Kratos(x)))}ARESH(x){KRATOS(KratosH(Kratos(x)))}KRATOS(x)KRATOS(x)KRATOS(x)KRATOS(x)KRATOS(x)KRATOS(x)KRATOS(x))

HADESH(x)

The fifth one in the hyper-functions is HADESH. This function grows faster than KRATOSH.

HADESH(x) = KRATOSH(KRATOSH(x){HADES(HadesH(Hades(x)))}KRATOSH(x){HADES(HadesH(Hades(x)))}KRATOSH(x){HADES(HadesH(Hades(x)))}KRATOSH(x){HADES(HadesH(Hades(x)))}KRATOSH(x){HADES(HadesH(Hades(x)))}KRATOSH(x){HADES(HadesH(Hades(x)))}KRATOSH(x){HADES(HadesH(Hades(x)))}KRATOSH(x){HADES(HadesH(Hades(x)))}KRATOSH(x){HADES(HadesH(Hades(x)))}(HADES(x))HADES(x)HADES(x)HADES(x)HADES(x)HADES(x)HADES(x)HADES(x)HADES(x))

CHARONH(x)

After the fifth one comes CHARONH. This grows faster than HADESH.

CHARONH(x) = HADESH(HADESH(x){CHARON(CharonH(Charon(x)))}HADESH(x){CHARON(CharonH(Charon(x)))}HADESH(x){CHARON(CharonH(Charon(x)))}HADESH(x){CHARON(CharonH(Charon(x)))}HADESH(x){CHARON(CharonH(Charon(x)))}HADESH(x){CHARON(CharonH(Charon(x)))}HADESH(x){CHARON(CharonH(Charon(x)))}HADESH(x){CHARON(CharonH(Charon(x)))}HADESH(x){CHARON(CharonH(Charon(x)))}HADESH(x){CHARON(CharonH(Charon(x)))}HADESH(x){CHARON(CharonH(Charon(x)))}HADESH(x)CHARON(x)CHARON(x)CHARON(x)CHARON(x)CHARON(x)CHARON(x)CHARON(x)CHARON(x)CHARON(x)CHARON(x))

EREBUSH(x)

The seventh one is EREBUSH. It grows faster than CHARONH.

EREBUSH(x) = CHARONH(CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x){EREBUS(ErebusH(Erebus(x)))}CHARONH(x)EREBUS(x)EREBUS(x)EREBUS(x)EREBUS(x)EREBUS(x)EREBUS(x)EREBUS(x)EREBUS(x)EREBUS(x)EREBUS(x)EREBUS(x)EREBUS(x))

THANATOSH(x)

The last one yet, THANATOSH. This is the fastest among War God numbers/functions.

THANATOSH(x) = EREBUSH(EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x){THANATOS(ThanatosH(Thanatos(x)))}EREBUSH(x)THANATOS(x)THANATOS(x)THANATOS(x)THANATOS(x)THANATOS(x)THANATOS(x)THANATOS(x)THANATOS(x)THANATOS(x)THANATOS(x)THANATOS(x)THANATOS(x)THANATOS(x)THANATOS(x))

Sources

  1. 1.0 1.1 FDLevi, FDLevi's pointless numbers, Google site. (Retrieved at UTC 22:50 02/02/2021)
  2. 2.0 2.1 FDLevi, FDLevi's pointless numbers, Google site. (Retrieved at UTC 00:00 04/02/2021)
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