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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)

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

### Charon(x)

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

### 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)

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

### CharonH(x)

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

### 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)

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

### CHARON(x)

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

### 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))

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

### CHARONH(x)

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

### 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. FDLevi, FDLevi's pointless numbers, Google site. (Retrieved at UTC 22:50 02/02/2021)
2. FDLevi, FDLevi's pointless numbers, Google site. (Retrieved at UTC 00:00 04/02/2021)
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