User blog:Rgetar/Way of making lists of ordinals

I thought of how to make lists of ordinals.

Making a list
We need:


 * Initial ordinal
 * Fundamental sequence system
 * natural number n - maximal level
 * natural number m - fundamental sequence excess

Also we have a list of ordinals (initially empty), and each ordinal in this list has two properties:


 * level (a natural number)
 * expandable (boolean, i. e. true or false)

So, initially our list is empty.

First, we add to the list the initial ordinal. Its level we set to 1, and its expandable we set to true, if the initial ordinal is limit, else to false.

Then, while the list contain an ordinal with expandable = true and level <= n we do the following:

(start of the cycle)

We take least ordinal in the list α with expandable(α) = true and level(α) = k, where k is least level such as there is an ordinal with expandable = true and level = k in the list.

We set expandable(α) to false.

We get set of elements of fundamental sequence of α:

...
 * α[0]
 * α[1]
 * α[2]

We exclude from this set all ordinals, which exist in our main list, then we make a list of first n+m elements of this set. For example, let α[0] and α[1] already exist in the main list, so, we exclude them from the set, then write first n+m its elements:

...
 * α[2]
 * α[3]
 * α[4]
 * α[n+m+1]

We set level of the first of these ordinals to k, of the second - to k+1, and so on:

...
 * level(α[2]) = k
 * level(α[3]) = k+1
 * level(α[4]) = k+2
 * level(α[n+m+1]) = k+n+m-1

For each ordinal of this list we set its expandable to true, if the it is limit, else (if it is successor or zero) to false.

We add all these ordinals to the main list.

(end of the cycle)

Then we sort the list in the ascending order of ordinals.

Examples
Designation of an ordinal in the list and its properties:

k?; α

where α is the ordinal, k is level(α), ? is + if expandable(α) = true, and ? is - if expandable(α) = false

One level
Let initial ordinal is ε0, the fundamental sequence system is Wainer hierarchy, n = 1, m = 9.

FS #1

1-; 0

2-; 1

3+; ω

4+; ωω

5+; ωω ω

6+; ωω ω ω

7+; ωω ω ω ω

8+; ωω ω ω ω ω

9+; ωω ω ω ω ω ω

10+; ωω ω ω ω ω ω ω

Initial ordinal

1-; ε0

(Total 11 ordinals, 1 fundamental sequence).

Two levels
The same input data, but now n = 2, m = 8.

Note: we may use list for maximal level = n-1 and fundamental sequence excess m+1 as initial list instead of empty list since we start making list for n as for n-1, and then, when we reach level n, we cannot get new ordinals with level < n. New elements of old fundamental sequences does not appear since number of elements (n+m) remains the same.

However, now we cannot add to the previous list any ordinals since the list does not contain ordinals with 2+ properties.

So, the list remains the same - 10 ordinals, 1 fundamental sequence.

Three levels
Set n = 3, m = 7.

Now we can expand 3+; ω.

FS #1

1-; 0

2-; 1

3-; 2

4-; 3

5-; 4

6-; 5

7-; 6

8-; 7

9-; 8

10-; 9

11-; 10

12-; 11

FS #2

3-; ω

4+; ωω

5+; ωω ω

6+; ωω ω ω

7+; ωω ω ω ω

8+; ωω ω ω ω ω

9+; ωω ω ω ω ω ω

10+; ωω ω ω ω ω ω ω

Initial ordinal

1-; ε0

(Total 21 ordinals, 2 fundamental sequences).

Four levels
Set n = 4, m = 6.

FS #1

1-; 0

2-; 1

3-; 2

4-; 3

5-; 4

6-; 5

7-; 6

8-; 7

9-; 8

10-; 9

11-; 10

12-; 11

FS #2

3-; ω

4-; ω + 1

5-; ω + 2

6-; ω + 3

7-; ω + 4

8-; ω + 5

9-; ω + 6

10-; ω + 7

11-; ω + 8

12-; ω + 9

13-; ω + 10

FS #3

4-; ω2

5+; ω3

6+; ω4

7+; ω5

8+; ω6

9+; ω7

10+; ω8

11+; ω9

12+; ω10

13+; ω11

FS #4

4-; ω2

5+; ω3

6+; ω4

7+; ω5

8+; ω6

9+; ω7

10+; ω8

11+; ω9

12+; ω10

13+; ω11

FS #5

4-; ωω

5+; ωω ω

6+; ωω ω ω

7+; ωω ω ω ω

8+; ωω ω ω ω ω

9+; ωω ω ω ω ω ω

10+; ωω ω ω ω ω ω ω

Initial ordinal

1-; ε0

(Total 51 ordinals, 5 fundamental sequences).

Five levels
Set n = 5, m = 5.

FS #1

1-; 0

2-; 1

3-; 2

4-; 3

5-; 4

6-; 5

7-; 6

8-; 7

9-; 8

10-; 9

11-; 10

12-; 11

FS #2

3-; ω

4-; ω + 1

5-; ω + 2

6-; ω + 3

7-; ω + 4

8-; ω + 5

9-; ω + 6

10-; ω + 7

11-; ω + 8

12-; ω + 9

13-; ω + 10

FS #3

4-; ω2

5-; ω2 + 1

6-; ω2 + 2

7-; ω2 + 3

8-; ω2 + 4

9-; ω2 + 5

10-; ω2 + 6

11-; ω2 + 7

12-; ω2 + 8

13-; ω2 + 9

14-; ω2 + 10

FS #4

5-; ω3

6+; ω4

7+; ω5

8+; ω6

9+; ω7

10+; ω8

11+; ω9

12+; ω10

13+; ω11

FS #5

4-; ω2

5-; ω2 + 1

6-; ω2 + 2

7-; ω2 + 3

8-; ω2 + 4

9-; ω2 + 5

10-; ω2 + 6

11-; ω2 + 7

12-; ω2 + 8

13-; ω2 + 9

14-; ω2 + 10

FS #6

5-; ω2 + ω

6+; ω2 + ω2

7+; ω2 + ω3

8+; ω2 + ω4

9+; ω2 + ω5

10+; ω2 + ω6

11+; ω2 + ω7

12+; ω2 + ω8

13+; ω2 + ω9

14+; ω2 + ω10

FS #7

5-; ω22

6+; ω23

7+; ω24

8+; ω25

9+; ω26

10+; ω27

11+; ω28

12+; ω29

13+; ω210

14+; ω211

FS #8

5-; ω3

6+; ω4

7+; ω5

8+; ω6

9+; ω7

10+; ω8

11+; ω9

12+; ω10

13+; ω11

FS #9

4-; ωω

5-; ωω + 1

6+; ωω + ω

7+; ωω + ω2

8+; ωω + ω3

9+; ωω + ω4

10+; ωω + ω5

11+; ωω + ω6

12+; ωω + ω7

13+; ωω + ω8

14+; ωω + ω9

FS #10

5-; ωω2

6+; ωω3

7+; ωω4

8+; ωω5

9+; ωω6

10+; ωω7

11+; ωω8

12+; ωω9

13+; ωω10

14+; ωω11

FS #11

5-; ωω + 1

6+; ωω + 2

7+; ωω + 3

8+; ωω + 4

9+; ωω + 5

10+; ωω + 6

11+; ωω + 7

12+; ωω + 8

13+; ωω + 9

14+; ωω + 10

FS #12

5-; ωω2

6+; ωω3

7+; ωω4

8+; ωω5

9+; ωω6

10+; ωω7

11+; ωω8

12+; ωω9

13+; ωω10

14+; ωω11

FS #13

5-; ωω 2

6+; ωω 3

7+; ωω 4

8+; ωω 5

9+; ωω 6

10+; ωω 7

11+; ωω 8

12+; ωω 9

13+; ωω 10

14+; ωω 11

FS #14

5-; ωω ω

6+; ωω ω ω

7+; ωω ω ω ω

8+; ωω ω ω ω ω

9+; ωω ω ω ω ω ω

10+; ωω ω ω ω ω ω ω

Initial ordinal

1-; ε0

(Total 141 ordinals, 14 fundamental sequences).

Six levels
Set n = 6, m = 4.

FS #1

1-; 0

2-; 1

3-; 2

4-; 3

5-; 4

6-; 5

7-; 6

8-; 7

9-; 8

10-; 9

11-; 10

12-; 11

FS #2

3-; ω

4-; ω + 1

5-; ω + 2

6-; ω + 3

7-; ω + 4

8-; ω + 5

9-; ω + 6

10-; ω + 7

11-; ω + 8

12-; ω + 9

13-; ω + 10

FS #3

4-; ω2

5-; ω2 + 1

6-; ω2 + 2

7-; ω2 + 3

8-; ω2 + 4

9-; ω2 + 5

10-; ω2 + 6

11-; ω2 + 7

12-; ω2 + 8

13-; ω2 + 9

14-; ω2 + 10

FS #4

5-; ω3

6-; ω3 + 1

7-; ω3 + 2

8-; ω3 + 3

9-; ω3 + 4

10-; ω3 + 5

11-; ω3 + 6

12-; ω3 + 7

13-; ω3 + 8

14-; ω3 + 9

15-; ω3 + 10

FS #5

6-; ω4

7+; ω5

8+; ω6

9+; ω7

10+; ω8

11+; ω9

12+; ω10

13+; ω11

FS #6

4-; ω2

5-; ω2 + 1

6-; ω2 + 2

7-; ω2 + 3

8-; ω2 + 4

9-; ω2 + 5

10-; ω2 + 6

11-; ω2 + 7

12-; ω2 + 8

13-; ω2 + 9

14-; ω2 + 10

FS #7

5-; ω2 + ω

6-; ω2 + ω + 1

7-; ω2 + ω + 2

8-; ω2 + ω + 3

9-; ω2 + ω + 4

10-; ω2 + ω + 5

11-; ω2 + ω + 6

12-; ω2 + ω + 7

13-; ω2 + ω + 8

14-; ω2 + ω + 9

15-; ω2 + ω + 10

FS #8

6-; ω2 + ω2

7+; ω2 + ω3

8+; ω2 + ω4

9+; ω2 + ω5

10+; ω2 + ω6

11+; ω2 + ω7

12+; ω2 + ω8

13+; ω2 + ω9

14+; ω2 + ω10

FS #9

5-; ω22

6-; ω22 + 1

7-; ω22 + 2

8-; ω22 + 3

9-; ω22 + 4

10-; ω22 + 5

11-; ω22 + 6

12-; ω22 + 7

13-; ω22 + 8

14-; ω22 + 9

15-; ω22 + 10

FS #10

6-; ω22 + ω

7+; ω22 + ω2

8+; ω22 + ω3

9+; ω22 + ω4

10+; ω22 + ω5

11+; ω22 + ω6

12+; ω22 + ω7

13+; ω22 + ω8

14+; ω22 + ω9

15+; ω22 + ω10

FS #11

6-; ω23

7+; ω24

8+; ω25

9+; ω26

10+; ω27

11+; ω28

12+; ω29

13+; ω210

14+; ω211

FS #12

5-; ω3

6-; ω3 + 1

7-; ω3 + 2

8-; ω3 + 3

9-; ω3 + 4

10-; ω3 + 5

11-; ω3 + 6

12-; ω3 + 7

13-; ω3 + 8

14-; ω3 + 9

15-; ω3 + 10

FS #13

6-; ω3 + ω

7+; ω3 + ω2

8+; ω3 + ω3

9+; ω3 + ω4

10+; ω3 + ω5

11+; ω3 + ω6

12+; ω3 + ω7

13+; ω3 + ω8

14+; ω3 + ω9

15+; ω3 + ω10

FS #14

6-; ω3 + ω2

7+; ω3 + ω22

8+; ω3 + ω23

9+; ω3 + ω24

10+; ω3 + ω25

11+; ω3 + ω26

12+; ω3 + ω27

13+; ω3 + ω28

14+; ω3 + ω29

15+; ω3 + ω210

FS #15

6-; ω32

7+; ω33

8+; ω34

9+; ω35

10+; ω36

11+; ω37

12+; ω38

13+; ω39

14+; ω310

15+; ω311

FS #16

6-; ω4

7+; ω5

8+; ω6

9+; ω7

10+; ω8

11+; ω9

12+; ω10

13+; ω11

FS #17

4-; ωω

5-; ωω + 1

6-; ωω + 2

7-; ωω + 3

8-; ωω + 4

9-; ωω + 5

10-; ωω + 6

11-; ωω + 7

12-; ωω + 8

13-; ωω + 9

14-; ωω + 10

15-; ωω + 11

FS #18

6-; ωω + ω

7+; ωω + ω2

8+; ωω + ω3

9+; ωω + ω4

10+; ωω + ω5

11+; ωω + ω6

12+; ωω + ω7

13+; ωω + ω8

14+; ωω + ω9

FS #19

5-; ωω2

6-; ωω2 + 1

7+; ωω2 + ω

8+; ωω2 + ω2

9+; ωω2 + ω3

10+; ωω2 + ω4

11+; ωω2 + ω5

12+; ωω2 + ω6

13+; ωω2 + ω7

14+; ωω2 + ω8

15+; ωω2 + ω9

FS #20

6-; ωω3

7+; ωω4

8+; ωω5

9+; ωω6

10+; ωω7

11+; ωω8

12+; ωω9

13+; ωω10

14+; ωω11

FS #21

5-; ωω + 1

6-; ωω + 1 + 1

7+; ωω + 1 + ω

8+; ωω + 1 + ω2

9+; ωω + 1 + ω3

10+; ωω + 1 + ω4

11+; ωω + 1 + ω5

12+; ωω + 1 + ω6

13+; ωω + 1 + ω7

14+; ωω + 1 + ω8

15+; ωω + 1 + ω9

FS #22

6-; ωω + 1 + ωω

7+; ωω + 1 + ωω2

8+; ωω + 1 + ωω3

9+; ωω + 1 + ωω4

10+; ωω + 1 + ωω5

11+; ωω + 1 + ωω6

12+; ωω + 1 + ωω7

13+; ωω + 1 + ωω8

14+; ωω + 1 + ωω9

15+; ωω + 1 + ωω10

FS #23

6-; ωω + 12

7+; ωω + 13

8+; ωω + 14

9+; ωω + 15

10+; ωω + 16

11+; ωω + 17

12+; ωω + 18

13+; ωω + 19

14+; ωω + 110

15+; ωω + 111

FS #24

6-; ωω + 2

7+; ωω + 3

8+; ωω + 4

9+; ωω + 5

10+; ωω + 6

11+; ωω + 7

12+; ωω + 8

13+; ωω + 9

14+; ωω + 10

FS #25

5-; ωω2

6-; ωω2 + 1

7+; ωω2 + ω

8+; ωω2 + ω2

9+; ωω2 + ω3

10+; ωω2 + ω4

11+; ωω2 + ω5

12+; ωω2 + ω6

13+; ωω2 + ω7

14+; ωω2 + ω8

15+; ωω2 + ω9

FS #26

6-; ωω2 + ωω

7+; ωω2 + ωω + 1

8+; ωω2 + ωω + 2

9+; ωω2 + ωω + 3

10+; ωω2 + ωω + 4

11+; ωω2 + ωω + 5

12+; ωω2 + ωω + 6

13+; ωω2 + ωω + 7

14+; ωω2 + ωω + 8

15+; ωω2 + ωω + 9

FS #27

6-; ωω22

7+; ωω23

8+; ωω24

9+; ωω25

10+; ωω26

11+; ωω27

12+; ωω28

13+; ωω29

14+; ωω210

15+; ωω211

FS #28

6-; ωω2+1

7+; ωω2+2

8+; ωω2+3

9+; ωω2+4

10+; ωω2+5

11+; ωω2+6

12+; ωω2+7

13+; ωω2+8

14+; ωω2+9

15+; ωω2+10

FS #29

6-; ωω3

7+; ωω4

8+; ωω5

9+; ωω6

10+; ωω7

11+; ωω8

12+; ωω9

13+; ωω10

14+; ωω11

FS #30

5-; ωω 2

6-; ωω 2 + 1

7+; ωω 2 + ωω

8+; ωω 2 + ωω2

9+; ωω 2 + ωω3

10+; ωω 2 + ωω4

11+; ωω 2 + ωω5

12+; ωω 2 + ωω6

13+; ωω 2 + ωω7

14+; ωω 2 + ωω8

15+; ωω 2 + ωω9

FS #31

6-; ωω 2 2

7+; ωω 2 3

8+; ωω 2 4

9+; ωω 2 5

10+; ωω 2 6

11+; ωω 2 7

12+; ωω 2 8

13+; ωω 2 9

14+; ωω 2 10

15+; ωω 2 11

FS #32

6-; ωω 2 + 1

7+; ωω 2 + 2

8+; ωω 2 + 3

9+; ωω 2 + 4

10+; ωω 2 + 5

11+; ωω 2 + 6

12+; ωω 2 + 7

13+; ωω 2 + 8

14+; ωω 2 + 9

15+; ωω 2 + 10

FS #33

6-; ωω 2 + ω

7+; ωω 2 + ω2

8+; ωω 2 + ω3

9+; ωω 2 + ω4

10+; ωω 2 + ω5

11+; ωω 2 + ω6

12+; ωω 2 + ω7

13+; ωω 2 + ω8

14+; ωω 2 + ω9

15+; ωω 2 + ω10

FS #34

6-; ωω 22

7+; ωω 23

8+; ωω 24

9+; ωω 25

10+; ωω 26

11+; ωω 27

12+; ωω 28

13+; ωω 29

14+; ωω 210

15+; ωω 211

FS #35

6-; ωω 3

7+; ωω 4

8+; ωω 5

9+; ωω 6

10+; ωω 7

11+; ωω 8

12+; ωω 9

13+; ωω 10

14+; ωω 11

FS #36

5-; ωω ω

6-; ωω ω + 1

7-; ωω ω + 2

8-; ωω ω + 3

9-; ωω ω + 4

10-; ωω ω + 5

11-; ωω ω + 6

12-; ωω ω + 7

13-; ωω ω + 8

14-; ωω ω + 9

15-; ωω ω + 10

FS #37

6-; ωω ω + ω

7+; ωω ω + ωω

8+; ωω ω + ωω 2

9+; ωω ω + ωω 3

10+; ωω ω + ωω 4

11+; ωω ω + ωω 5

12+; ωω ω + ωω 6

13+; ωω ω + ωω 7

14+; ωω ω + ωω 8

15+; ωω ω + ωω 9

FS #38

6-; ωω ω 2

7+; ωω ω 3

8+; ωω ω 4

9+; ωω ω 5

10+; ωω ω 6

11+; ωω ω 7

12+; ωω ω 8

13+; ωω ω 9

14+; ωω ω 10

15+; ωω ω 11

FS #39

6-; ωω ω + 1

7+; ωω ω + ω

8+; ωω ω + ω2

9+; ωω ω + ω3

10+; ωω ω + ω4

11+; ωω ω + ω5

12+; ωω ω + ω6

13+; ωω ω + ω7

14+; ωω ω + ω8

15+; ωω ω + ω9

FS #40

6-; ωω ω2

7+; ωω ω3

8+; ωω ω4

9+; ωω ω5

10+; ωω ω6

11+; ωω ω7

12+; ωω ω8

13+; ωω ω9

14+; ωω ω10

15+; ωω ω11

FS #41

6-; ωω ω + 1

7+; ωω ω + 2

8+; ωω ω + 3

9+; ωω ω + 4

10+; ωω ω + 5

11+; ωω ω + 6

12+; ωω ω + 7

13+; ωω ω + 8

14+; ωω ω + 9

15+; ωω ω + 10

FS #42

6-; ωω ω2

7+; ωω ω3

8+; ωω ω4

9+; ωω ω5

10+; ωω ω6

11+; ωω ω7

12+; ωω ω8

13+; ωω ω9

14+; ωω ω10

15+; ωω ω11

FS #43

6-; ωω ω 2

7+; ωω ω 3

8+; ωω ω 4

9+; ωω ω 5

10+; ωω ω 6

11+; ωω ω 7

12+; ωω ω 8

13+; ωω ω 9

14+; ωω ω 10

15+; ωω ω 11

FS #44

6-; ωω ω ω

7+; ωω ω ω ω

8+; ωω ω ω ω ω

9+; ωω ω ω ω ω ω

10+; ωω ω ω ω ω ω ω

Initial ordinal

1-; ε0

(Total 441 ordinals, 44 fundamental sequences).

Analysis
Number of ordinals is one more than number of fundamental sequences multiplied by n+m.

Number of fundamental sequences does not depend on m.

So, number of fundamental sequences is determined by initial ordinal, fundamental sequence system and n.

There is a thought that for given initial ordinal and n should be a class of fundamental sequence systems producing lists with minimal number of fundamental sequences.

There should be only one fundamental sequence, if fundamental sequence of initial ordinal does not contain limit ordinals. For example, for ε0

ω + 1

ωω + 1

ωω ω + 1

ωω ω ω + 1

ωω ω ω ω  + 1

...

Such fundamental sequence system does not show all diversity of ordinals less-than ε0 in corresponding lists (it generates only one fundamental sequence for any n).

We may define a subclass of fundamental sequence systems: if there is fundamental sequence for a ordinal containing only limit ordinals, then fundamental sequence for this ordinal should contain only limit ordinals.

Note: Wainer hierarchy does not possess this property. For example, in Wainer hierarchy ωω has fundamental sequence 1; ω; ω2; ω3; ω4; ω5... (0-th element is not limit ordinal, the rest are limit ordinals), and ε0 has fundamental sequence 0; 1; ω; ωω; ωω ω ; ωω ω ω ... (0-th and 1-st elements are not limit ordinals, the rest are  limit ordinals).

So, Wainer hierarchy does not show in our lists all diversity of ordinals less-than ε0. For example, in five levels list:

FS #9

4-; ωω

5-; ωω + 1

6+; ωω + ω

7+; ωω + ω2

8+; ωω + ω3

...

Actually, here could be two fundamental sequences instead of one:

ωω

ωω + 1

ωω + 2

...

and

6+; ωω + ω

7+; ωω + ω2

8+; ωω + ω3

...

We may define modified Wainer hierarchy: it is Wainer hierarchy, but if a fundamental sequence contain infinite number of limit ordinals, all non-limit ordinals are excluded from this fundamental sequence.

For example, in modified Wainer hierarchy ωω has fundamental sequence ω; ω2; ω3; ω4; ω5..., and ε0 has fundamental sequence ω; ωω; ωω ω ; ωω ω ω ...

(Maybe, I'll publish other lists for modified Wainer hierarchy).

So, for given initial ordinal and n should be a class of fundamental sequence systems with this property producing lists with minimal number of fundamental sequences.

Also, I suppose, there should be another class of fundamental sequence systems such as iterating fundamental sequence for any limit ordinal this system can deal with covers all ordinals less-than this ordinal.

For such fundamental sequence systems my algorithm of making lists assign a level to all ordinals less-than initial ordinal. This number is less for "simple" ordinals. Ordinals with lesser level are more rare than ordinals with greater level.

Actually, these lists not so about ordinals as about fundamental sequences.

We may even set m to infinity, so, in these lists will be full fundamental sequences, not cut versions.

I'd name these lists "iterated fundamental sequences". For example, list in "Six levels" section is "iterated fundamental sequence of ε0 in Wainer hierarchy for 6 levels and excess 4".

Conclusion
I think it would be interesting to make such lists for large ordinals.