User blog:Rgetar/Square brackets OCF

This is modified ordinal collapsing function from my blog OCF based on ordinal array function.

The notation is

[X1][X2][X3]...[Xn - 1][Xn]α

where Xi, α - ordinals, n - natural number.

[Y][X]α = [Y]([X]α)

"Initial" ordinals: 0, I.

An ordinal should be represented in form

[X1][X2][X3]...[Xn - 1][Xn]0

or

[X1][X2][X3]...[Xn - 1][Xn]I

Rules
Rule 1. [I]α is least uncountable cardinal larger than α.

Rule 2. [X]α should not contain ordinal β such as β < [I]α, β > α.

Rule 3. [X]α is least ordinal larger than all [X1][X2][X3]...[Xn - 1][Xn]α such as Xi < X.

Examples
[0]α = α + 1, since only [X1][X2][X3]...[Xn - 1][Xn]0 with Xi < 0 is α, and least ordinal larger than α is α + 1 (Rule 3).

0

[0]0 = 1

[0][0]0 = 2

[0][0][0]0 = 3

[0][0][0][0]0 = 4

[0][0][0][0][0]0 = 5

[0][0][0][0][0][0]0 = 6

"[1]0" is incorrect, since 1 < [I]α = Ω, 1 > 0 (Rule 2).

[1][0]0 = [1]1 = ω, since [X1][X2][X3]...[Xn - 1][Xn]0 with Xi < 1 are [0][0][0]...[0][0]1, and least ordinal larger than all [0][0][0]...[0][0]1 is ω (Rule 3).

[0][1]1 = ω + 1

[0][0][1]1 = ω + 2

[0][0][0][1]1 = ω + 3

[1][1]1 = ω2

[0][1][1]1 = ω2 + 1

[1][1][1]1 = ω3

[1][1][1][1]1 = ω4

[1][1][1][1][1]1 = ω5

Generally, [X + 1]α is limit of α, [X]α, [X][X]α, [X][X][X]α, [X][X][X][X]α, [X][X][X][X][X]α, ...

"[2]0", "[2]1" are incorrect (Rule 2).

[2][1][0]0 = [2][1]1 = [2]ω = ω2

[0][2]ω = ω2 + 1

[1][2]ω = ω2 + ω

[0][1][2]ω = ω2 + ω + 1

[1][1][2]ω = ω2 + ω2

[2][2]ω = ω22

[2][2][2]ω = ω23

[3]ω = ω3

[4]ω = ω4

[5]ω = ω5

[ω]ω = ωω

[ωω][ω]ω = [ωω]ωω = ωω ω

[[I]0]0 = [Ω]0 = ε0, since Ω is larger then any countable ordinal, but according Rule 2 we can use in [X]α for countable α only countalbe or finite X not larger than α. So, [Ω]0 should be limit of

[0]0 = 1

[1]1 = ω

[ω]ω = ωω

[ωω]ωω = ωω ω

[ωω ω ]ωω ω = ωω ω ω

[ωω ω ω ]ωω ω ω  = ωω ω ω ω

[ωω ω ω ω  ]ωω ω ω ω   = ωω ω ω ω ω

...

that is ε0.

[Ω][Ω]0 = ε1

[Ω][Ω][Ω]0 = ε2

[[0][I]0]0 = [[0]Ω]0 = [Ω + 1]0 = εω

[[[I]0][I]0]0 = [[Ω]Ω]0 = [Ω2]0 is limit of

[Ω]0 = ε0

[[ε0]Ω]0 = [Ω + ε0]0 = εε 0

[[εε 0 ]Ω]0 = [Ω + εε 0 ]0 = εε ε 0

...

that is ζ0.

[[I][I]0]0 = [Ω2]0 is limit of

[Ω]0

[[Ω]Ω]0

[[[Ω]Ω][Ω]Ω]0

[[[[Ω]Ω][Ω]Ω][[Ω]Ω][Ω]Ω]0

[[[[[Ω]Ω][Ω]Ω][[Ω]Ω][Ω]Ω][[[Ω]Ω][Ω]Ω][[Ω]Ω][Ω]Ω]0

...

BHO