User blog:Rgetar/Idea of program

I'm going to write a program to automatically create lists of ordinals such as this.

Few weeks ago I already tried to write such a program, but it computed fundamental sequence elements only up to ε0 (by the way, I used that program to make computations for a comment to this blog). Now I'm going to make my new program compute fundamental sequence elements of ordinals up to BHO.

I'll use such format for ordinals: Cantor normal form without coefficients, 0 for 0, (X) for generalized Veblen function φ(X), for array X pairs &lt;coordinates&gt;element, separated with ",", and if coordinates = 0, then &lt;coordinates&gt; is omitted.

So, ordinals up to BHO are represented using 8 symbols: "0", "1", "+", ",", "(", ")", "<", ">".

Examples:

0 = 0

1 = 1

2 = 1+1

5 = 1+1+1+1+1

ω = (1)

ω + 1 = (1)+1

ω2 = (1)+(1)

ω2 = (1+1)

ω6 = (1+1+1+1+1+1)

ωω = ((1))

ωω ω = (((1)))

ε0 = (<1>1)

ε1 = (<1>1,1)

εω + 1 = (<1>1,(1))

ζ0 = (<1>1+1)

φ(ω, 0) = (<1>(1))

Γ0 = (<1+1>1)

φ(1, 0, 0, 0) = (<1+1+1>1)

SVO = (<(1)>1)

LVO = (<<1>1>1)

etc.

(But after creating a list I'm going to transform ordinals to more "common" form).

For now I made transformation of an ordinal to its standard form (this was the most difficult part, but this is necessary, since for non-standard forms (in any of this and this fundamental sequence systems) fundamental sequences differ from fundamental sequence for standard form and from each other).

Example of different fundamental sequences for standard and non-standard form of ordinal:

Standard form of ε0 is ε0 = φ(1, 0), and its non-standard form is ωε0 = φ(φ(1, 0))

Fundamental sequences of standard form:

ε0[0] = 0

ε0[1] = 1

ε0[2] = ω

ε0[3] = ωω

ε0[4] = ωω ω

ε0[5] = ωω ω ω

Fundamental sequences of non-standard form:

ωε0[0] = 1

ωε0[1] = ω

ωε0[2] = ωω

ωε0[3] = ωω ω

ωε0[4] = ωω ω ω

ωε0[5] = ωω ω ω ω

I'll use my 6th fundamental sequence system.

Left to do: computation of fundamental sequence elements and generating lists. And, I hope, there will not be big problems.