User blog:ArtismScrub/Ordinal factorial notation?

This notation involves iteration by means of calling infinite ordinals similar to how the hierarchies do.

The definition might not be that coherent or consistent, but it's just some ideas.

'In case anyone already designed something like this, let me point it out that I came up with this idea independently. If I'm not the first to do so, I'm sorry.'

Finite numbers
n!0 = n*(n-1)*(n-2)*...*3*2*1 (normal factorial )

n!1 = n^(n-1)^(n-2)^...^3^2^1 (exponential factorial )

n!2 = n↑↑(n-1)↑↑(n-2)↑↑...↑↑3↑↑2↑↑1 (tetrational factorial )

n!3 = n↑↑↑(n-1)↑↑↑(n-2)↑↑↑...↑↑↑3↑↑↑2↑↑↑1 (pentational factorial )

...

n!x = n{x}(n-1){x}(n-2){x}...{x}3{x}2{x}1 (where {x} represents the number of up-arrows)

Omega and addition
n!ω = n!n

n!(ω+1) = n!(n!)

n!(ω+2) = n!(n!1)

n!(ω+3) = n!(n!2)

...

n!(ω+x) = n!(n!(x-1))

for future reference:

n!(Y)+1 = n!(n!(Y)) where Y is any ordinal

n!(Y)+x = n!(n!(Y+(x-1)))

Omega and multiplication
n!ω*2 =  n!( ω+n)

n!ω*3 = n!( ω*2+n)

n!ω*4 =  n!( ω*3+n)

...

n!ω*x =  n!( ω*(x-1)+n)

Omega and exponents
n!(ω^2) = n!(ω*n)

n!(ω^2*2) = n!(ω*n!)

n!(ω^2*3) = n!(ω*n!1)

n!(ω^2*4) = n!(ω*n!2)

...

n!(ω^2*x) = n!(ω*n!(x-2))

n!(ω^3) = n!(ω^2*n!ω)

n!(ω^4) = n!(ω^3*n!ω)

n!(ω^x) = n!(ω^(x-1)*n!ω) multiplication will continue to work similarly

Omega and tetration
n!(ω ↑↑2)  = n!(ω^n) (also can be denoted n!ω^ ω )

n!(ω ^ ω^ 2) = n!(ω^n!ω)

n!(ω^ω^3) = n!(ω^n!(ω^2))

...

n!(ω^ω^x) = n!(ω^n!(ω^(x-1))) power towers of  ω with a finite number at the end will continue to work similarly

n!(ω↑↑3) = n!(ω^n!(ω ↑↑2 ))

n!(ω↑↑4) =  n!(ω^n!(ω ↑↑3 ))

...

n!(ω↑↑x) =  n!(ω^n!(ω ↑↑(x-1) ))

Epsilon-zero
n!(ε0) = n!(ω↑↑n)

n!(ε0^2) = n!(ω↑↑n!( ε0 ))

n!(ε0^3) = n!(ω↑↑n!( ε0^2 ))

...

n!(ε0^x) =  n!(ω↑↑n!( ε0^(x-1) ))

n!(ε0^ω) = n!(ε0^n)

n!(ε0^ω^2) = n!(ε0^n!(ε0^ ω ))

n!(ε0^ω^3) = n!(ε0^n!(ε0^ ω^2 ))

...

n!(ε0^ω^x) =  n!(ε0^n!(ε0^ ω^(x-1) ))

n!(ε0^ω↑↑2) = n!(ε0^ω^n)

n!(ε0^ω↑↑3) = n!(ε0^ω^ω^n)

...

n!(ε0^ω↑↑x) = n!(ε0^ω^ω^...^ω^ω^n) (power tower of  ωs  x terms high)

n!(ε0↑↑2) = n!(ε0^ω↑↑n)

n!(ε0↑↑2^2) = n!(ε0^ω↑↑n!(ε0))

n!(ε0↑↑2^3) = n!(ε0^ω↑↑n!(ε0^2))

...

n!(ε0↑↑2^x) = n!(ε0^ω↑↑n!(ε0^(x-1))) exponentiation will continue to work similarly

n!(ε0↑↑3) = n!(ε0^ω↑↑n!(ε0↑↑2))

n!(ε0↑↑4) = n!(ε0^ω↑↑n!(ε0↑↑3))

...

n!(ε0↑↑x) = n!(ε0^ω↑↑n!(ε0↑↑(x-1)))

Further epsilons
n!(ε1) = n!(ε0↑↑n)

n!(ε1↑↑2) = n!(ε0↑↑n!(ε0↑↑n))

n!(ε1↑↑3) = n!(ε0↑↑n!(ε0↑↑n!(ε0↑↑n))

n!(ε1↑↑x) = n!(ε0↑↑n!(ε0↑↑n!(ε0↑↑...↑↑n!(ε0↑↑n!(ε0↑↑n))...))) nested x times

n!(ε2) = n!(ε1↑↑n)

continue in similar fashion for ε2

n!(ε3) = n!(ε2↑↑n)

n!(ε4) = n!(ε3↑↑n)

...

n!(εx) = n!(ε(x-1)↑↑n)

n!(εω) = n!(εn)

n!(ε(ω^2)) = n!(εn!(εω) (not to be confused with ( εω)^2)

n!(ε(ω^3)) = n!(εn!(ε(ω^2))

n!(ε(ω^4)) = n!(εn!(ε(ω^3))

...

n!(ε(ω^x)) = n!(εn!(ε(ω^(x-1)))

n!(ε(ω↑↑2)) = n!(ε(ω^n))

n!(ε(ω↑↑3)) = n!(ε(ω^n!(ε(ω↑↑2))))

...

n!(ε(ω↑↑x)) = n!(ε(ω^n!(ε(ω↑↑(x-1)))))

Recursive epsilons
n!(εε0) = n!(ε(ω↑↑n))

n!(εε1) = n!(ε(ω↑↑n!(εε0)))

n!(εε2) = n!(ε(ω↑↑n!(εε1)))

n!(εε3) = n!ε((ω↑↑n!(εε2)))

...

n!(εεx) = n!(ε(ω↑↑n!(εε(x-1))))

n!(εεω) = n!(εεn)

n!(εε(ω^2)) = n!(εεn!(εεω)

n!(εε(ω^3)) = n!(εεn!(εε(ω^2)))

n!(εε(ω^4)) = n!(εεn!(εε(ω^3)))

...

n!(εε(ω^x)) = n!(εε(n!εε(ω^(x-1))))

n!(εε(ω↑↑2)) = n!(εε(ω^n))

n!(εε(ω↑↑3) = n!(εε(ω^n !(εε(ω↑↑2))))

...

n!(εε(ω↑↑x)) = n!(εε(ω^ n!(εε(ω↑↑(x-1)))))

n!(εεε0) = n!(εε(ω↑↑n))

continue in a similar fashion for further nested epsilons...

n!(εεεε0) = n!(εεε(ω↑↑n))

n!(ε[5]0) = n!(ε[4](ω↑↑n)) where [x] means x nested epsilons

n!(ε[x]0) = n!(ε[x-1](ω↑↑n))

n!(ζ0) = n!(ε[n]0)

and so on.

I really, really hope I defined this well, so that it can safely continue into further ordinals. I don't want this to be strictly limited at n!(ζ0). If I failed to define this well, I don't know what to say. The idea seems fairly straightforward anyway, so it may be useful through interpretation. If not, I guess I failed this one.