User blog:Rgetar/Comparison of ordinals expressed through extended Veblen function

Let we have two ordinals expressed through extended Veblen function with equal number of variables. For example,

α1 = φ(β10, β9, β8, β7, β6, β5, β4, β3, β2, β1, β0)

α2 = φ(γ10, γ9, γ8, γ7, γ6, γ5, γ4, γ3, γ2, γ1, γ0)

If number of variables is not equal, we can make them equal by adding zeros. (For example, φ(β1, β0) → φ(0, 0, 0, β1, β0)).

So, the alrogithm.

Step 1. Compare values of variables, starting from the left, until we find not equal values. If all values are equal, then α1 = α2. If we find not equal values, and if this is the last variable, then if β0 > γ0, then α1 > α2, and if β0 < γ0, then α1 < α2. If we find not equal values, and if this is not the last variable, then continue with Step 2.

Step 2. Rest of variables of ordinal with larger value compare with the other ordinal itself, also starting from the left, until we find value larger or equal to that ordinal. If all values are less than that ordinal, then this ordinal is less than that. If we find value larger then that ordinal, then this ordinal is larger than that. If we find value equal to that ordinal, then continue with Step 3.

Step 3. Rest of variables of this ordinal compare with 0, starting from the left. If we find value larger than 0, then this ordinal is larger then that. If all values are equal to 0, then α1 = α2.

Exmaples
Exmaples with three steps

Exmaple #1

Step 1

β10 = γ10

β9 = γ9

β8 = γ8

β7 > γ7

Step 2

β6 < α2

β5 < α2

β4 < α2

β3 = α2

Step 3

β2 = 0

β1 = 0

β0 = 0

Result: α1 = α2

Exmaple #2

Step 1

β10 < γ10

Step 2

γ9 < α1

γ8 < α1

γ7 < α1

γ6 < α1

γ5 = α1

Step 3

γ4 = 0

γ3 = 0

γ2 > 0

Result: α1 < α2

Exmaple #3

Step 1

β10 = γ10

β9 = γ9

β8 = γ8

β7 = γ7

β6 = γ6

β5 = γ5

β4 = γ4

β3 = γ3

β2 = γ2

β1 > γ1

Step 2

β0 = α2

Step 3

Result: α1 = α2

Exmaples with two steps

Exmaple #4

Step 1

β10 = γ10

β9 = γ9

β8 = γ8

β7 = γ7

β6 = γ6

β5 < γ5

Step 2

γ4 < α1

γ3 < α1

γ2 > α1

Result: α1 < α2

Exmaple #5

Step 1

β10 = γ10

β9 = γ9

β8 > γ8

Step 2

β7 < α2

β6 < α2

β5 < α2

β4 < α2

β3 < α2

β2 < α2

β1 < α2

β0 < α2

Result: α1 < α2

Exmaple #6

Step 1

β10 = γ10

β9 < γ9

Step 2

γ8 > α1

Result: α1 < α2

Exmaples with one step

Exmaple #7

Step 1

β10 = γ10

β9 = γ9

β8 = γ8

β7 = γ7

β6 = γ6

β5 = γ5

β4 = γ4

β3 = γ3

β2 = γ2

β1 = γ1

β0 > γ0

Result: α1 > α2

Exmaple #8

Step 1

β10 = γ10

β9 = γ9

β8 = γ8

β7 = γ7

β6 = γ6

β5 = γ5

β4 = γ4

β3 = γ3

β2 = γ2

β1 = γ1

β0 = γ0

Result: α1 = α2