User blog:Rgetar/Generalizations of Ordinal array functions and FGH

For Ordinal aray functions see User blog:Rgetar/Definitions update.

Here are my Ordinal aray functions: [X]a, where a is ordinal, X is array of ordinals, and [X]a is ordinal.

Then I started to designate arrays of ordinals as "larger" ordinals, for example, array of finite and countable ordinals

1, ω + 1, 10, ε0, 5

as uncountable ordinal

Ω4 + Ω3(ω + 1) + Ω210 + Ωε0 + 5

etc.

cp(X), leo(X), lest(X; α), X0
Here are new function cp(X) and new definitions of functions leo(X), lest(X; α), X0

cp means "cardinality part"

leo means "last element of"

lest means "last element set to"

("Element", since I named leo and lest when I considered X as array, now I consider X as "larger" ordinal, but I didn't change names of these functions).

cp(X), leo(X)
Ordinal X of cardinality card(X) can be represented as sum

X = cp(X) + leo(X)

such as cp(X) ≥ card(X), leo(X) < card(X), and X cannot be represented as X = α + β such as 0 < β < card(X).

Examples.

For finite X cp(X) = X, leo(X) = 0.

For countable X in my blog Ordinal arithmetic I designated cp(X) as lp(X), that is "limit part", and leo(X) as fp(X), that is "finite part".

For example,

X = ω4 + ω32 + ω23 + ω5 + 15

cp(X) = ω4 + ω32 + ω23 + ω5

leo(X) = 15

Example for card(X) = Ω:

X = Ω2 + Ω + ω + 1

cp(X) = Ω2 + Ω

leo(X) = ω + 1

Generally, ordinal X of cardinality Ωβ can be represented as sum of terms Ωβiαi, where card(i) ≤ Ωβ, card(αi) < Ωβ, i are decreasing.

leo(X) = Ωβ0α0 = α0

cp(X) is rest of this sum without α0.

lest(X; α)
lest(X; α), where card(α) < card(X), is X with leo(X) replaced with α.

That is

cp(lest(X; α)) = cp(X)

leo(lest(X; α)) = α

X0
X0 = lest(X; 0)

That is

cp(X0) = cp(X)

leo(X0) = 0

Generalization of Ordinal aray function
Now Ordinal aray function has subscript β:

[X]βα

where α < Ωβ, X < Ωβ + 1, [X]βα < Ωβ.

Definition:

[0]βα = α + 1

[X + 1]βα = [X0][X]α

[X]βα = sup([X[n]]βα), 1 < cof(X) < Ωβ

[X]βα = [X[α]]α, cof(X) = Ωβ

where X[n] is n-th element of fundamental sequence of X. So, Ordinal aray function becomes dependent on fundamental sequence system (fss).

β = 0
Special case of [X]βα for β = 0:

[X]0n

where X is countable or finite ordinal, n is natural number, [X]0n is also natural number. Examples for Wainer hierarchy:

[0]0n = n + 1

[1]0n = n + 2

[2]0n = n + 3

[3]0n = n + 4

[4]0n = n + 5

Generally, for natural m

[m]0n = n + m + 1

[ω]0n = [n]n0 = 2n + 1

[ω + 1]0n = [ω]0[ω]0n = [ω]0(2n + 1) = 4n + 3

[ω + 2]0n = [ω]0[ω + 1]0n = [ω]0(4n + 3) = 8n + 7

[ω + 3]0n = [ω]0[ω + 2]0n = [ω]0(8n + 7) = 16n + 15

Generally, for natural m

[ω + m]0n = 2m + 1(n + 1) - 1

[ω2]0n = [ω + n]0n = 2n + 1(n + 1) - 1

[ω2 + 1]0n = [ω2]0[ω2]0n

[ω2 + 2]0n = [ω2]0[ω2 + 1]0n

[ω2 + 3]0n = [ω2]0[ω2 + 2]0n

[ω3]0n = [ω2 + n]0n

[ω3 + 1]0n = [ω3]0[ω3]0n

[ω3 + 2]0n = [ω3]0[ω3 + 1]0n

[ω3 + 3]0n = [ω3]0[ω3 + 2]0n

[ω4]0n = [ω3 + n]0n

[ω5]0n = [ω4 + n]0n

[ω2]0n = [ωn]0n

[ω2 + 1]0n = [ω2]0[ω2]0n

[ω2 + 2]0n = [ω2]0[ω2 + 1]0n

[ω2 + ω]0n = [ω2 + n]0n

[ω2 + ω + 1]0n = [ω2 + ω]0[ω2 + ω]0n

[ω2 + ω2]0n = [ω2 + ω + n]0n

[ω22]0n = [ω2 + ωn]0n

[ω23]0n = [ω22 + ωn]0n

[ω3]0n = [ω2n]0n

[ω4]0n = [ω3n]0n

[ω5]0n = [ω4n]0n

[ωω]0n = [ωn]0n

[ωω + 1]0n = [ωω]0[ωω]0n

[ωω + 2]0n = [ωω]0[ωω + 1]0n

[ωω + ω]0n = [ωω + n]0n

[ωω + ω2]0n = [ωω + ω + n]0n

[ωω + ω2]0n = [ωω + ωn]0n

[ωω2]0n = [ωω + ωn]0n

[ωω + 1]0n = [ωωn]0n

[ωω2]0n = [ωω + n]0n

[ωω 2 ]0n = [ωωn]0n

[ωω ω ]0n = [ωω n ]0n

Generalization of FGH
[X]0n is similar to FGH (theirs inputs and outputs are natural numbers, their parameter is finite or countable ordinal, and they are both fss-dependent).

And, as we have Ordinal array functions for β > 0, maybe, we can generalize FGH the same way? Yes, we can.

Here is FGH:

f0(n) = n + 1

fα + 1(n) = fαn(n)

fα(n) = fα[n](n), iff α is limit ordinal

where n is natural number, α is countable ordinal or natural number, α[n] is n-th element of fundamental sequence of α, fαm + 1(n) = fα(fαm(n)), where m is natural number. fα(n) is also natural number.

Generalization of FGH has two subscripts:

fβ, X(α)

where α < Ωβ, X < Ωβ + 1, fβ, X(α) < Ωβ.

Definition:

fβ, 0(α) = α + 1

fβ, X + 1(α) = fβ, Xα(α)

fβ, X(α) = sup(fβ, X[n](α)), 1 < cof(X) < Ωβ

fβ, X(α) = fβ, X[α](α), cof(X) = Ωβ

where X[n] is n-th element of fundamental sequence of X. And now may be infinite iteration:

fβ, Xγ + 1(α) = fβ, X(fβ, Xγ(α))

fβ, Xγ(α) = sup(fβ, Xγ[n](α)), iff γ is limit ordinal.

Special case of fβ, X(α) for β = 0:

f0, X(α) = fX(α) — it is FGH.