User blog comment:Scorcher007/Biggest ordinal's table/@comment-32213734-20180529083234/@comment-32213734-20180530231508

It would be great! I like how you explained meaning of Graham's number.

I too noticed some incorrect data in your blogs.

Do you need some corrections? If yes, then: it is said that set of functions has cardinality ℶ2, and sets of cardinality ℶ3 and higher are unknown and not used in practice. But a function may have anything as its input and output, including other functions (see higher-order function). I think that set of functions has cardinality card(set of outputs)card(set of inputs). Set of first-order functions, say, real number to real number, has cardinality ℶ2. Set of functionals (function to number) has cardinality ℶ3. Set of functional operators (function to function) also has cardinality ℶ3. Set of "third-order functions" (functional operator to functional operator) has cardinality ℶ4 etc.

I'll tell how few years ago I used second-, third- and even fourth-order functions (set of which has cardinality ℶ5) in order to define imaginary order derivatives.

Let Fn is functional operator (second-order function) such as Fn(f) = fn (iterated function). So, its input is first-order function f, and output is first-order function fn.

n may be negative:

f-1(f(x)) = x (inverse function)

and even fractional:

f1/2(f1/2(x)) = f(x) (functional root)

Let Gn is third-order function such as Gn(g) = gn, where g is second-order function, that is functional operator.

Let Hn is fourth-order function such as Hn(h) = hn, where h is third-order function.

n in Gn, Hn also may be negative and fractional.

What if we apply Fn twice? We get

Fn(Fn(f)) = Fn(fn) = fn 2

So,

G2(Fn) = Fn2

Genarally,

Gm(Fn) = Fnm

Similarly,

Hm(Gn) = Gnm

Derivative d/dx is functional operator (second-order function). Indeed, its input is function f, and its output is function f'.

What is higher-order derivative? It is d/dx applied many times. So,

Gn(d/dx) is n-th derivative.

To get i-th derivative, use G with negative subscript and H with fractional subscript:

H1/2(G-1(d/dx)) = G(-1)1/2(d/dx) = Gi(d/dx) - it is i-th derivative.

Also, BHO should be θ(εΩ + 1), not θ(εΩ) and the like.