User blog:Rgetar/Classes of ordinals ｛-｝a, ｛*｝a, ｛+｝a

Recently I was trying to extend set X{·}a (see Definitions update blog) beyond ΩΩ Ω Ω ...  . (Now I like to represent arrays of ordinals as "larger" ordinals).

I did not want to give up X{·}a, since it was used in short and independent of fundamental sequence systems definitions of ordinal array functions [X]a and generalized Veblen function φ(X). Also, fundamental sequences was not used in these definitions.

When I failed to extend X{·}a, I started to formulate equal values and comparison of generalized Veblen function in terms of "larger" ordinals instead of arrays of ordinals. I noticed that apparently it does work beyond ΩΩ Ω Ω ... .

Then I suddenly realized, that sets used there can be used instead of X{·}a. (Later I found out that I already used something like this in one of my earliest blogs, but I almost forgot it).

{-}βa
{-}βa, where a < Ωβ, is class of ordinals. It depends on ordinal a and ordinal β. And it is intersection of classes {-n}βa for all n.

{-n}βa are also classes of ordinals such as {-1}βa is subclass of {-0}βa, {-2}βa is subclass of {-1}βa, {-3}βa is subclass of {-2}βa etc. But currently I do not define {-n}βa for all n. Some {-n}βa for large n may be defined later. But for not very lagre ordinals we can use {-n}βa only for small n.

Any ordinal can be represented as sum of Ωβici terms, ci < Ωβ. (Special case for β = 0 is Cantor normal form).

{-0}βa is class of ordinals such as any ci < a

{-1}βa is class of ordinals such as any ci < a, and i ∈ {-1}βa

For an ordinal less than Ωβ + 1 term Ωβici can be represented as

φβγ(X)c

where φβγ(X) is Veblen-like function.

For an ordinal of cardinality Ωδ > Ωβ term Ωδici can be represented as

φδγ(X)c

then we can represent coefficient c the same way and so on until we get all coefficients less than Ωβ.

{-2}βa is class of ordinals such as any c < a, and any X ∈ {-2}βa

{-3}βa is class of ordinals such as any c < a, and any X and γ ∈ {-3}βa

Currently I did not define {-n}βa for n > 3.

{*}βa
{*}βa is intersection of {*n}βa for all n.

{*0}βa is class of ordinals such as it is sum of an ordinal of class {-0}βa and Ωβba for any b

{*1}βa is class of ordinals such as it is sum of an ordinal of class {-1}βa and ((Ωβba for b of class {-1}βa) or (Ωβb for b of class {*1}βa))

{*2}βa is class of ordinals such as it is sum of an ordinal of class {-2}βa and ((φδγ(X)c for c of class {*2}βa and X of class {-2}βa) or (φδγ(X) for X of class {*2}βa))

{*3}βa is class of ordinals such as it is sum of an ordinal of class {-3}βa and ((φδγ(X)c for c of class {*2}βa and X and γ of class {-2}βa) or (φδγ(X) for X of class {*2}βa and γ of class {-2}βa) or (φδγ(0) for γ of class {*2}βa))

Currently I did not define {*n}βa for n > 3.

{+}βa
{+}βa is intersection of {+n}βa for all n.

{+n}βa is union of {-n}βa and {*n}βa.

X{-}βa
X{-}βa is set of ordinals of class {-}βa less than X.

X{*}βa
X{*}βa is set of ordinals of class {*}βa less than X.

X{+}βa
X{+}βa is set of ordinals of class {+}βa less than X.

And this set can replace X{·}a.

Ordinal array function
[0]βa = a + 1

[X]βa = sup([(X; -1; X0)]β[Y]βa), Y ∈ X{+}βa

Veblen-like function
φαβ(X) = φαγ(φγβ(X))

φαα + 1(X) = ΩαX, if card(X) < Ωα + 1

φαα + 1(X) = α is (1 + leo(X))-th common fixed point of all functions α = φαα + 1(Y), Y ∈ X{+}αa, if card(X) = Ωα + 1

Note: I am not sure, if this enough for limit β. But I guess yes.

Equal values
φαβ(X) = φαβ(Y)

where

Y ∈ X{*}αφαβ(X)

Comparison
If

Y ∈ X{-}αφαβ(X)

then

φαβ(X) > φαβ(Y)

Standard form
φαβ(X) is in standard form, if X is of class {-}αφαβ(X).

Otherwise X is of class {*}αφαβ(X), and φαβ(X) is in non-standard form.

Fundamental sequences
In previous blog I defined fundamental sequence system (fss)-dependent ordinal array function

[0]βα = α + 1

[X + 1]βα = [X0][X]α

[X]βα = sup([X[n]]βα), 1 < cof(X) < Ωβ

[X]βα = [X[α]]α, cof(X) = Ωβ

Here I defined fss-independent ordinal array function.

Maybe, this can help to define certain fss (or class of fss's) such as both ordinal array function coincide.