User blog:Rgetar/Simple way to create lists of ordinals

Hi!

I entered the University, so I have not much time. I was thinking about my Veblen-like functions and other notations, I changed something in the definitions, also I changed my "computer format". I tried to program something, but in the main I failed. (I'll not describe it in this blog).

Also, I simplified the way to create lists of ordinals:

Small expansion
We need a pair of ordinals a - b0 such as

a < b0

and a system of fundamental sequences. So,

b0[0], b0[1], b0[2], ... b0

Let b1 is smallest b0[n] > a. Insert b1 between a and b0:

a, b1, b0

a, b1 is small expansion of pair a - b0.

if b0 = a + 1, then small expansion of pair a - b0 is a.

Single expansion
Do the same with pair a - b1:

a, b2, b1, b0

then with pair a - b2:

a, b3, b2, b1, b0

...

Repeat until get

a, bn, bn - 1, ... , b3, b2, b1, b0

bn = a + 1

List

a, bn, bn - 1, ... , b3, b2, b1

is single expansion of pair a - b0.

Higher expansions
List of single expansion of a - bn, single expansion of bn - bn - 1, ..., single expansion of b3 - b2, single expansion of b2 - b1, single expansion of b1 - b0 is double expansion of pair a - b0.

List of double expansion of a - bn, double expansion of bn - bn - 1, ..., double expansion of b3 - b2, double expansion of b2 - b1, double expansion of b1 - b0 is triple expansion of pair a - b0.

Etc.

Examples
Single, double, triple, quadruple, pentuple expansions of 0 - ε0, Wainer hierarchy:

Single expansion of 0 - ε0
start of single expansion of 0 - ε0
 * 0
 * 1

end of single expansion of 0 - ε0

Double expansion of 0 - ε0
start of double expansion of 0 - ε0
 * start of single expansion of 0 - 1
 * 0
 * end of single expansion of 0 - 1
 * start of single expansion of 1 - ε0
 * 1
 * 2
 * ω
 * end of single expansion of 1 - ε0

end of double expansion of 0 - ε0

Triple expansion of 0 - ε0
start of triple expansion of 0 - ε0
 * start of double expansion of 0 - 1
 * start of single expansion of 0 - 1
 * 0
 * end of single expansion of 0 - 1
 * end of double expansion of 0 - 1
 * start of double expansion of 1 - ε0
 * start of single expansion of 1 - 2
 * 1
 * end of single expansion of 1 - 2
 * start of single expansion of 2 - ω
 * 2
 * 3
 * end of single expansion of 2 - ω
 * start of single expansion of ω - ε0
 * ω
 * ω + 1
 * ω2
 * ω2
 * ωω
 * end of single expansion of ω - ε0
 * end of double expansion of 1 - ε0

end of triple expansion of 0 - ε0

Quadruple expansion of 0 - ε0
start of quadruple expansion of 0 - ε0
 * start of triple expansion of 0 - 1
 * start of double expansion of 0 - 1
 * start of single expansion of 0 - 1
 * 0
 * end of single expansion of 0 - 1
 * end of double expansion of 0 - 1
 * end of triple expansion of 0 - 1
 * start of triple expansion of 1 - ε0
 * start of double expansion of 0 - ω
 * start of single expansion of 1 - 2
 * 1
 * end of single expansion of 1 - 2
 * start of single expansion of 2 - 3
 * 2
 * end of single expansion of 2 - 3
 * start of single expansion of 3 - ω
 * 3
 * 4
 * end of single expansion of 3 - ω
 * end of double expansion of 0 - ω
 * start of double expansion of ω - ε0
 * start of single expansion of ω - ω + 1
 * ω
 * end of single expansion of ω - ω + 1
 * start of single expansion of ω + 1 - ω2
 * ω + 1
 * ω + 2
 * end of single expansion of ω + 1 - ω2
 * start of single expansion of ω2 - ω2
 * ω2
 * ω2 + 1
 * ω3
 * end of single expansion of ω2 - ω2
 * start of single expansion of ω2 - ωω
 * ω2
 * ω2 + 1
 * ω2 + ω
 * ω22
 * ω3
 * end of single expansion of ω2 - ωω
 * start of single expansion of ωω - ε0
 * ωω
 * ωω + 1
 * ωω2
 * ωω + 1
 * ωω2
 * ωω 2
 * ωω ω
 * end of single expansion of ωω - ε0
 * end of double expansion of ω - ε0
 * end of triple expansion of 1 - ε0

end of quadruple expansion of 0 - ε0

Pentuple expansion of 0 - ε0
start of pentuple expansion of 0 - ε0
 * start of quadruple expansion of 0 - 1
 * start of triple expansion of 0 - 1
 * start of double expansion of 0 - 1
 * start of single expansion of 0 - 1
 * 0
 * end of single expansion of 0 - 1
 * end of double expansion of 0 - 1
 * end of triple expansion of 0 - 1
 * end of quadruple expansion of 0 - 1
 * start of quadruple expansion of 1 - ε0
 * start of triple expansion of 0 - ω
 * start of double expansion of 1 - 2
 * start of single expansion of 1 - 2
 * 1
 * end of single expansion of 1 - 2
 * end of double expansion of 1 - 2
 * start of double expansion of 2 - 3
 * start of single expansion of 2 - 3
 * 2
 * end of single expansion of 2 - 3
 * end of double expansion of 2 - 3
 * start of double expansion of 3 - ω
 * start of single expansion of 3 - 4
 * 3
 * end of single expansion of 3 - 4
 * start of single expansion of 4 - ω
 * 4
 * 5
 * end of single expansion of 4 - ω
 * end of double expansion of 3 - ω
 * end of triple expansion of 0 - ω
 * start of triple expansion of ω - ε0
 * start of double expansion of ω - ω + 1
 * start of single expansion of ω - ω + 1
 * ω
 * end of single expansion of ω - ω + 1
 * end of double expansion of ω - ω + 1
 * start of double expansion of ω + 1 - ω2
 * start of single expansion of ω + 1 - ω + 2
 * ω + 1
 * end of single expansion of ω + 1 - ω + 2
 * start of single expansion of ω + 2 - ω2
 * ω + 2
 * ω + 3
 * end of single expansion of ω + 2 - ω2
 * end of double expansion of ω + 1 - ω2
 * start of double expansion of ω2 - ω2
 * start of single expansion of ω2 - ω2 + 1
 * ω2
 * end of single expansion of ω2 - ω2 + 1
 * start of single expansion of ω2 + 1 - ω3
 * ω2 + 1
 * ω2 + 2
 * end of single expansion of ω2 + 1 - ω3
 * start of single expansion of ω3 - ω2
 * ω3
 * ω3 + 1
 * ω4
 * end of single expansion of ω3 - ω2
 * end of double expansion of ω2 - ω2
 * start of double expansion of ω2 - ωω
 * start of single expansion of ω2 - ω2 + 1
 * ω2
 * end of single expansion of ω2 - ω2 + 1
 * start of single expansion of ω2 + 1 - ω2 + ω
 * ω2 + 1
 * ω2 + 2
 * end of single expansion of ω2 + 1 - ω2 + ω
 * start of single expansion of ω2 + ω - ω22
 * ω2 + ω
 * ω2 + ω + 1
 * ω2 + ω2
 * end of single expansion of ω2 + ω - ω22
 * start of single expansion of ω22 - ω3
 * ω22
 * ω22 + 1
 * ω22 + ω
 * ω23
 * end of single expansion of ω22 - ω3
 * start of single expansion of ω3 - ωω
 * ω3
 * ω3 + 1
 * ω3 + ω
 * ω3 + ω2
 * ω32
 * ω4
 * end of single expansion of ω3 - ωω
 * end of double expansion of ω2 - ωω
 * start of double expansion of ωω - ε0
 * start of single expansion of ωω - ωω + 1
 * ωω
 * end of single expansion of ωω - ωω + 1
 * start of single expansion of ωω + 1 - ωω2
 * ωω + 1
 * ωω + 2
 * ωω + ω
 * end of single expansion of ωω + 1 - ωω2
 * start of single expansion of ωω2 - ωω + 1
 * ωω2
 * ωω2 + 1
 * ωω3
 * end of single expansion of ωω2 - ωω + 1
 * start of single expansion of ωω + 1 - ωω2
 * ωω + 1
 * ωω + 1 + 1
 * ωω + 1 + ωω
 * ωω + 12
 * ωω + 2
 * end of single expansion of ωω + 1 - ωω2
 * start of single expansion of ωω2 - ωω 2 
 * ωω2
 * ωω2 + 1
 * ωω2 + ωω
 * ωω22
 * ωω2 + 1
 * ωω3
 * end of single expansion of ωω2 - ωω 2 
 * start of single expansion of ωω 2 - ωω ω 
 * ωω 2
 * ωω 2 + 1
 * ωω 2 2
 * ωω 2 + 1
 * ωω 2 + ω
 * ωω 22
 * ωω 3
 * end of single expansion of ωω 2 - ωω ω 
 * start of single expansion of ωω ω - ε0
 * ωω ω
 * ωω ω + 1
 * ωω ω + ω
 * ωω ω 2
 * ωω ω + 1
 * ωω ω2
 * ωω ω + 1
 * ωω ω2
 * ωω ω 2
 * ωω ω ω
 * end of single expansion of ωω ω - ε0
 * end of double expansion of ωω - ε0
 * end of triple expansion of ω - ε0
 * end of quadruple expansion of 1 - ε0

end of pentuple expansion of 0 - ε0