User blog:Rgetar/How to compare ordinals expressed in the terms of the Veblen function

Hi!

This blog is about how to compare two ordinals written using (generalized) Veblen function φ.

That is there are two ordinals

α1 = φ(X1)

α2 = φ(X2)

X1, X2 - arrays of ordinals.

"Generalized" means that X1, X2 may be multi-dimensional, multi-trimensional etc. arrays (whereas for extended Veblen function φ(X) X is one-dimensional array).

The task is to find out, whether α1 > α2 or α1 < α2 or α1 = α2.

For some designations used here see two older blogs: 1, 2.

Coordinates of array elements
An element of an array X has "coordinates" of this element in this array.

For example, in array

X = β9,β8,β7,β6,β5,β4,β3,β2,β1,β0

an element βn has coordinate n (last element always has zero coordinates). Here X is one-dimensional array, so its elements have only one coordinate.

In multi-dimensional arrays its elements have multiple coordinates.

For example,

X = β2,1,1,β2,1,0<1,0>β2,0,2,β2,0,1,β2,0,0<1,0,0>β1,2,2,β1,2,1,β1,2,0<1,0>β1,1,5,β1,1,4,β1,1,3,β1,1,2,β1,1,1,β1,1,0<1,0>β1,0,5,β1,0,4,β1,0,3,β1,0,2,β1,0,1,β1,0,0<1,0,0>β2,5,β2,4,β2,3,β2,2,β2,1,β2,0<1,0>β1,5,β1,4,β1,3,β1,2,β1,1,β1,0<1,0>β5,β4,β3,β2,β1,β0

, = <1> - separator between variables

<1,0> - separator between rows

<1,0,0> - separator between planes

An element βY has coordinates Y. Y is array of ordinals itself.

In multi-trimensional array X its element βY has coordinates Y, and Y is multi-dimensional array; in multi-quadramensional array X its element βY has coordinates Y, and Y is multi-trimensional array etc.

Five parts of two arrays of ordinals
X1, X2 are arrays of ordinals

There are five parts of X1 and X2: part 1 (P1), part 2 (P2), part 3 (P3), part 4 (P4), part 5 (P5).

part 2 (P2(X1; X2) and P2(X2; X1)) are first left elements of X1 and X2 with equal coordinates but different values

part 1 (P1(X1; X2) and P1(X2; X1)) are part left to part 2

P1(X1; X2) = P1(X2; X1)

part 4 (P4(X1) and P4(X2)) are leo(X1*) and leo(X2*)

P4(X1) = leo(X1*)

P4(X2) = leo(X2*)

part 3 (P3(X1; X2) and P3(X2; X1)) are part between part 2 and part 4 (not including them)

part 5 (P5(X1) and P5(X2)) are part right to part 4

P5(X1) = 

P5(X2) = 

Example:

X1 = β9,β8,β7,β6,β5,β4,β3,0,0,0

X2 = β9,β8,β7,γ6,γ5,γ4,γ3,γ2,0,0

β6 ≠ γ6

part 1:

P1(X1; X2) = P1(X2; X1) = β9,β8,β7,

part 2:

P2(X1; X2) = β6

P2(X2; X1) = γ6

part 3:

P3(X1; X2) = ,β5,β4,

P3(X2; X1) = ,γ5,γ4,γ3,

part 4:

P4(X1) = β3

P4(X2) = γ2

part 5:

P5(X1) = ,0,0,0

P5(X2) = ,0,0

Some parts may be missing.

If X1 = X2 then parts 2—5 are missing.

If part 2 is last element of an array then parts 3—5 are missing.

If part 4 is last element of an array then part 5 is missing.

If parts 2 and 4 exist, part 3 may contain no one element, only separators.

Part 1 always exists, but it may contain no one non-zero element.

sonze(X; Y)
sonze means "set of non-zero elements"

X is array of ordinals

Y is part of X

sonze(X; Y) is set of values of non-zero-values elements of part Y of array X.

For example,

X = 1,2,3,0,1,3,1,0,0,0,ω+1,5,0,1,0,0

sonze(X; X) is set of ordinals {1, 2, 3, 5, ω+1}

If Y is part of X from 6th to 10th elements of X (that is "3,1,0,0,0"), then

sonze(X; Y) is set of ordinals {1, 3}

soconze(X; Y)
soconze means "set of coordinates of non-zero elements"

X is array of ordinals

Y is part of X

soconze(X; Y) is set of coordinates (which are arrays of ordinals) of elements with non-zero values of part Y of array X.

For example,

X = 1,0,1<1,0>1,2,3,0,1,3,1,0,0,0,ω+1,5,0,1,0,0

soconze(X; X) is set of arrays of ordinals {2; 4; 5; 9; 10; 11; 13; 14; 15; 1,0; 1,2}

If Y is part of X from 6th to 10th elements of X (that is "3,1,0,0,0"), then

soconze(X; Y) is set of arrays of ordinals {9; 10}

isonze(X; Y)
isonze means "iterated set of non-zero elements"

X is array of ordinals

Y is part of X

Let

C1(X; Y) = soconze(X; Y)

Cn+1(X; Y) is union of soconze(Z; Z) for all Z ∈ Cn(X; Y)

isonze(X; Y) = sonze(X; Y) ∪ sonze(Z; Z) for all Z such as Z ∈ union of Cn(X; Y), n = 1, 2, 3, ...

For example,

X = 1,0,1<1,0>1,2,3,0,1,3,1,0,0,0,ω+1,5,0,1,0,0

C1(X; X) = soconze(X; X) is set of arrays of ordinals {2; 4; 5; 9; 10; 11; 13; 14; 15; 1,0; 1,2}

C2(X; X) is set of arrays of ordinals {0; 1}

C3(X; X) is set of arrays of ordinals {0}

For n > 3 Cn(X; X) is empty set

Union of Cn(X; X), n = 1, 2, 3, ... is set of arrays of ordinals {0; 1; 2; 4; 5; 9; 10; 11; 13; 14; 15; 1,0; 1,2}

sonze(X; X) is set of ordinals {1, 2, 3, 5, ω+1}

isonze(X; X) is set of ordinals {1, 2, 3, 4, 5, 9, 10, 11, 13, 14, 15, ω+1}

If Y is part of X from 6th to 10th elements of X (that is "3,1,0,0,0"), then

C1(X; Y) = soconze(X; Y) is set of arrays of ordinals {9; 10}

C2(X; Y) is set of arrays of ordinals {0}

For n > 2 Cn(X; Y) is empty set

Union of Cn(X; Y), n = 1, 2, 3, ... is set of arrays of ordinals {0; 9; 10}

sonze(X; Y) is set of ordinals {1, 3}

isonze(X; Y) is set of ordinals {1, 3, 9, 10}

cofrewnzeloi(X)
cofrewnzeloi means "coordinates of first right element without non-zero elements left of it"

X is array of ordinals

cofrewnzeloi(X) is coordinates of first right element of X without non-zero elements left of it

(cofrewnzeloi(X) is array of ordinals itself)

Another definition of cofrewnzeloi(X):

if X ≠ 0 then cofrewnzeloi(X) is coordinates of first left non-zero element of X else cofrewnzeloi(X) = 0

For example,

X = 9,1<1,0>ω<1,0>1<1,0,0>1,0,1<1,0>1,2,3,0,1,3,1,0,0,0,ω+1,5,0,1,0,0

X ≠ 0 and first non-zero element of X is "9" with coordinates "1,2,1"

So,

cofrewnzeloi(X) = 1,2,1

cofrewnzeloi2(X) = 2

cofrewnzeloi3(X) = 0

it is 0, so its cofrewnzeloi is also 0:

cofrewnzeloi4(X) = 0

So,

For n > 2 cofrewnzeloin(X) = 0

X{··}α
X is array of ordinals

α is ordinal

X{··}α is set of arrays of ordinals depending on X and α

Let

C1(X; α) = X{·}α

Cn+1(X; α) is union of Y{·}α for all Y ∈ Cn(X; α)

X{··}α is union of Cn(X; α), n = 1, 2, 3, ...

X{{{sup|0}}··}α
X is array of ordinals

α is ordinal

X0{··}α is set of arrays of ordinals depending on X and α

Let

C1(X; α) = X0{·}α

Cn+1(X; α) is union of Y0{·}α for all Y ∈ Cn(X; α)

X{0··}α is union of Cn(X; α), n = 1, 2, 3, ...

X{0··}α is subset of X{··}α

Equal ordinals expressed in the terms of the Veblen function
Some ordinals may be expressed as φ(X) in different ways.

If an ordinal may be expressed as α = φ(X) then isonze(X; X) contains only ordinals less-than or equal to α, and there is only X such as isonze(X; X) contains only ordinals less-than α.

So, α can be uniquely expressed as φ(X) so as isonze(X; X) ∌ λ ≥ α.

Also, some ordinals may be expressed (sometimes even in different ways) as φ(X) so as isonze(X; X) ∋ α. If so then

leo(X*) = α or leo(X'*) = α or leo(X"*) or ...

and there are no more element in X, X', X", ... equal to α.

An ordinal cannot be expressed as φ(X) so as isonze(X; X) ∋ λ > α.

Comparing ordinals expressed in the terms of the Veblen function
By the definition of the Veblen function (see the second of two aforementioned blogs)

φ(lest(X; λ)) > φ(X) for all λ > leo(x)

and

φ(X) = φ(Y) for all Y ∈ X{{{sup|0}}··}φ(X)

If P2(X; Y) > P2(Y; X) then φ(X) > φ(Y)

φ(X) > φ(Y) for all Y ∈ X{··}φ(X) \ X{{{sup|0}}··}φ(X)

Rules of comparing:

if Y ∉ X ∪ X{··}φ(X) then φ(X) < φ(Y)

if Y ∈ X ∪ X{{{sup|0}}··}φ(X) then φ(X) = φ(Y)

if Y ∈ X{··}φ(X) \ X{{{sup|0}}··}φ(X) then φ(X) > φ(Y)

So, we need find out, is Y = X and is Y element of X{··}φ(X) and X{{{sup|0}}··}φ(X)

(will be continued tomorrow)