User blog:Rgetar/Limit ordinals in arrays in ordinal array and Veblen functions

Hi everyone!

My family of functions [X]a was not well-defined for limit ordinals.

Now I'll try to fix it.

Designations
Some designations used here such as (X; a; b), X·a, X0, X-1, X*, X' etc. are described in blog about [X]a functions.

-β+α is an ordinal such as β+(-β+α) = α (see also Ordinal subtraction and integer extension of ordinals blog).

Now new designations.

leo and lest
leo means "last element of".

leo(X) is last element of array X.

lest means "last element set to".

lest(X; α) is array X with last element set to α.

Example.

X = 1,2,5

leo(X) = 5

lest(X; 9) = 1,2,9

Particularly,

X0 = lest(X; 0)

X-1 = lest(X; leo(X)-1)

X{·}a
X{·}a is set of arrays with elements

(X·a; {βn})

where {βn} is ordered set of ordinals

β0; β1; β2; ...

such as

βn &lt; leo(Xn*)

X0 = X

Xn+1 = Xn'

Also βn may be -1 if leo(Xn*) = 0

{βn+1} is

β1; β2; β3; ...

Definition of (X·a; {βn}):

(X·a; {βn}) = lest(X*; β0)&lt;-β1+leo(X'*)&lt;X"&gt;&gt;(X"; 1; a)&lt;(X'·a; {βn+1})&gt;

Particularly,

X·a = (X·a; {leo(Xn*)-1})

However, it works only if all leo(Xn*) are successors ordinals or zeros.

[X]a rules
So, rules.

1. [0]a = a+1

2. [X]a = sup([(X; -1; X0)][Y]a), Y ∈ X{·}a

i. e. [X]a is least ordinal such as there is no ordinal among [(X; -1; X0)][Y]a larger than [X]a

Now, I think, these rules work for both limit and successor ordinals in X.

Veblen function
Extended Veblen function has only one row of variables. It may be generalized to have multi-dimensional, multi-trimensional etc. arrays of variables using arrays of ordinals with &lt;&gt; separators.

Designations
X - array of ordinals

Y(α) - function of ordinal α to array of ordinals

ato means "arrays to ordinal"

ato(X; Y(α)) is least ordinal β such as β = φ(Y(β)) and β > (X; -1; φ(lest(X; γ))) for all γ < leo(X)

Note: ato(X; Y(α)) does not depend on α. It depends only on array X and function Y(α) itself.

Veblen function definition
1. φ(α) = ωα

2. If X ≠ α then φ(X) = sup(ato(X; Y(α))), Y(α) ∈ X0{·}α

i. e. φ(X) is least ordinal such as there is no ordinal among ato(X; Y(α)) larger than φ(X).