User blog:Rgetar/Equal values and comparison of generalized Veblen function

I published blogs about Equal values and comparison of extended Veblen function, that is with one-dimensional array of variables (even not transfinite).

Now the same thing about generalized Veblen function.

Here is list of designations.

Comparison of arrays
Elements of an array should be arranged in descending order of coordinates. But what if coordinates are also arrays? How to compare arrays?

One-element arrays are ordinals, so, they are compared as ordinals.

Arrays may be considered as "extension" of ordinals, for example, array 1, 0 is larger than any ordinal α, array 2, 0 is larger than any array 1, α, array 1, 0, 0 is larger than any array α, 0 etc.

So, to compare two two-element arrays α1, α0 and β1, β0 we should just compare α1 and β1. And, if α1 = β1 then we should compare α0 and β0, and this is the result.

To compare two three-element arrays α2, α1, α0 and β2, β1, β0, we should compare α2 and β2, then, if they are equal, compare α1 and β1, and, if they are equal, compare α0 and β0.

Etc.

To compare two arrays of different sizes, add zeros in the beginning of array with lesser size to make them of equal sizes. (If the arrays are of different sizes, and both begin not from zero, then array with larger size is larger).

Now let's compare arrays written with coordinates, for example,

⟨γ2⟩α2, ⟨γ1⟩α1, ⟨γ0⟩α0

and

⟨δ1⟩β1, ⟨δ0⟩β0

where

α2, α1, α0, β1, β0 > 0

First, compare most left coordinates γ2 and δ1 (if γ2 > δ1, then first array is larger, γ2 < δ1, then second array is larger).

If γ2 = δ1, then compare most left elements α2 and β1 (if α2 > β1, then first array is larger, α2 < β1, then second array is larger).

If α2 = β1, then compare next coordinates γ1 and δ0, if γ1 = δ0, then compare next elements α1 and β0 etc.

If the coordinates are also arrays, we already know how to compare arrays, so, now we can compare any arrays.

Equal values
The algorithm is similar as for extended Veblen function, but now on Step 4 the variables not only should be less than the initial ordinal α, but also isobe of theirs coordinates should not contain ordinal ≥ α. And on Step 3 the variable can be set to α (but with condition isobe of its coordinates ∌ ordinal ≥ α). Or, it can be set to 1 (but with condition isobe of its coordinates ∌ ordinal ≥ α except lbeo of the coordinates should be α). Or, it can be set to 1, lbeo of the coordinates should be 1 with condition isobe of its coordinates ∌ ordinal ≥ α except lbeo of coordinates of lbeo of the coordinates should be α. Etc.

Also, there is a "standard form" of Veblen function: isobe of its array of variables ∌ ordinal ≥ the ordinal itself.

Comparison
Comparison is also similar as for extended Veblen function and reduces to test, if one of compared ordinals is an equal form of another.

So, first try to find largest coordinates with inequal elements. If all elements are equal, then the ordinals are equal. If largest coordinates with inequal elements is 0, then compare last elements, and this is the result.

Else begin to compare non-zero variables with lesser coordinates of ordinal with larger value with the with the other ordinal itself. Compare not only values of variables, but also elements of isobe of theirs coordinates. If all values and elements of isobe of theirs coordinates are less than that ordinal, then this ordinal is less than that.

(Let the other ordinal is α).

Else check, if this variable

etc.
 * = α, and isobe of its coordinates ∌ ordinal ≥ α
 * = 1, and isobe of its coordinates ∌ ordinal ≥ α except lbeo of its coordinates = α
 * = 1, lbeo of its coordinates = 1, and isobe of its coordinates ∌ ordinal ≥ α except lbeo of coordinates of lbeo of its coordinates = α
 * = 1, lbeo of its coordinates = 1, lbeo of coordinates of lbeo of its coordinates = 1, and isobe of its coordinates ∌ ordinal ≥ α except lbeo of coordinates of lbeo of coordinates of lbeo of its coordinates = α
 * = 1, lbeo of its coordinates = 1, lbeo of coordinates of lbeo of its coordinates = 1, lbeo of coordinates of lbeo of coordinates of lbeo of its coordinates = 1, and isobe of its coordinates ∌ ordinal ≥ α except lbeo of coordinates of lbeo of coordinates of lbeo of coordinates of lbeo of its coordinates = α

If no, then this ordinal is larger than that.

If yes, then check if this variable is lbeo of this ordinal. If yes, then the ordinals are equal. If no, then this ordinal is larger than that.