User blog:Rgetar/Definition of Veblen-like functions for natural superscripts and subscripts

For designations used here see User_blog:Rgetar/Definitions update.

Arrays
I designate arrays as ordinals, for example:

1, 2, 3 = Ω2 + Ω2 + 3

Ordinal X can be represented as sum of terms

Ωαγβγ

in order of decreasing γ, where

Ωα ≤ X < Ωα + 1,

βγ < Ωα,

γ < Ωα + 1.

(Note: for ω ≤ X < Ω it is Cantor normal form).

leo(X), lest(X; α)
New designation of leo(X): leo(X) is last term of this sum.

New designation of lest(X; α): lest(X; α) is this sum with leo(X) replaced with α.

(Note: for Ω ≤ X < Ω2 these new designations coincide with old designations).

Cofinality
cof(X) is cofinality of X, that is minimal length of increasing sequence X[n] such as sup(X[n]) = X.

n < cof(X)

Veblen-like functions
Here is Veblen-like function:

φαβ(X),

where

α < β,

X < Ωβ + 1,

φαβ(X) < Ωα + 1.

Omitting superscript and subscript

φα(X) = φαα + 1(X)

φβ(X) = φ0β(X)

Definition of Veblen-like functions for natural α and β

φαβ(X) = φαγ(φγβ(X)),

where α < γ < β

φα(0) = 1

φα(X) = sup(φα(X)[n])

X[n] = lest(X; leo(X)[n])

\((φ^α(X)β)[n] = \left\{\begin{array}{lcr} (φ^α(X)β[n] \quad \text{if} \; cof(β) > 1\\ (φ^α(X)(β-1) + φ^α(X)[n] \quad \text{if} \; cof(β) = 1\\ \end{array}\right. \)

\(φ^α(X)[n] = \left\{\begin{array}{lcr} φ^α(X-1)·n \quad \text{if} \; X \; \text{- s.,} \; X<Ω_{α+1}\\ φ^α(X[n]) \quad \text{if} \; X \; \text{- l.}\\ \left.\begin{array}{lcr} φ^α(lest(X^0[n]; δ)) \quad \text{if} \; cof(X^0)<Ω_{α+1}\\ \left.\begin{array}{lcr} φ^α(X^0[φ^α(X)[n-1]]) \quad \text{if} \; n>0\\ φ^α(X^0[0]) \quad \text{if} \; n=0\\ \end{array}\right\} \; \text{if} \; cof(X^0)=Ω_{α+1}\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.} \; X≥Ω_{α+1}\\ \end{array}\right. \)