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## Compact Veblen Notation

A new notation is proposed to simplify the presentation of complex Fast-growing hierarchy (FGH) functions using the full power of the Veblen Hierarchy. This blog introduces a new notation (similar to the Compact FGH Notation) that will simplify the work required to evaluate and investigate the patterns and properties of FGH functions using complex Veblen Hierarchy ordinals. This notation is used in my other blogs on a fast growing J Function that I am designing.

The notation is meant to be easy to read, but also, easy to translate back into LaTeX math notation so that the relevant FGH function can be presented visually in its usual form.

## Notation Example

Here is the first example, which shows the key notational change. The use of [ square ] brackets implicitly notates an instance of $$\phi$$ function ordinal and makes it possible to compactly notate various permutations of these ordinals.

f[2~2.1]^2(4) $$= f_{\phi(2)^{2}.1}^2(4) = f_{\phi(2)^{2}}^2(4) = f_{\omega^{\omega^2}}^2(4)$$

Every instance of [ square ] brackets requires the delimiters "~" representing exponentiation and "." representing multiplication. Here is another more complex example using nested ordinals:

f[1,0~[1~1.1].[1~1.1]]^2(4) $$= f_{\phi(1,0)^{\phi(1)}.\phi(1)}^{2}(4) = f_{\epsilon_0^{\omega}.\omega}^{2}(4)$$

## Notation Definition

The notation uses "[" square "]" brackets to implicitly notate an instance of a $$\phi$$ function ordinal. The general notation definition is as follows:

$$v_i = [\alpha$$~$$\beta.\gamma]$$ where

$$v = v_0 + v_1 + v_2 + ... + v_i$$ and $$v_n > v_{n+1}$$ for all $$n$$

Next the definition of $$\alpha$$ is a series of column delimited parameters using this general definition:

$$\alpha = \alpha_0, \alpha_1, \alpha_2, ..., \alpha_j$$ and $$\alpha_n > \alpha_{n+1}$$ for all $$n$$

and each instance of $$\alpha_n$$ is recursively defined using the same general notation $$v$$.

Then $$\beta$$ and $$\gamma$$ are also recursively defined using the same general notation $$v$$ where

$$\alpha > \beta$$ and $$\alpha > \gamma$$ at all times

Finally an FGH function is constructed with the following notation:

$$fv$$^$$p(n)$$ where p and n are finite integers, and n >= p and n > "any finite integer used in the construction of v"

Here is a complex example of this notation:

f[1,[1~1.1],3~[1,4~[1~1.1].[1~1.1]].[1,0~[1~1.1].[1~1.1]]]^3(5) $$= f_{\phi(1,\phi(1),3)^{\phi(1,4)^{\phi(1)}.\phi(1)}.\phi(1,0)^{\phi(1)}.\phi(1)}^{3}(5)$$

$$= f_{\phi(1,\omega,3)^{\epsilon_4^{\omega}.\omega}.\epsilon_0^{\omega}.\omega}^{3}(5)$$

Note that $$\phi(1,\omega,3)$$ is well beyond the Feferman–Schütte ordinal $$\Gamma_0$$ which is only $$\phi(1,0,0)$$, In fact this notation can easily reach the Small Veblen Ordinal (SVO) for any finite value of n as in these examples:

f[1,0,0~1.1]^1(2) $$= f_{\phi(1,0,0)}(2) = f_{\Gamma_0}(2) = f_{SVO}(2)$$

f[1,0,0,0~1.1]^1(3) $$= f_{\phi(1,0,0,0)}(3) = f_{SVO}(3)$$

and

f[1,0,0,0,0~1.1]^1(4) $$= f_{\phi(1,0,0,0,0)}(4) = f_{SVO}(4)$$

Here is another complex example of this notation:

f[3,[1~1.1],1~[3,0~[3~1.1].[3~1.3]].[1,3~[1~1.1].[1~1.1]]]^5(5) $$= f_{\phi(3,\phi(1),1)^{\phi(3,0)^{\phi(3)}.\phi(3).3}.\phi(1,3)^{\phi(1)}.\phi(1)}^{5}(5)$$

$$= f_{\phi(3,\omega,1)^{\phi(3,0)^{\omega^{\omega^{\omega}}}.\omega^{\omega^{\omega}}.3}.\epsilon_3^{\omega}.\omega}^{5}(5)$$

Work In Progress

## Ruleset

The following rules apply to general notation example of the form:

[a,b,c,d,e~f.g]^h(n) $$= f_{\phi(a,b,c,d,e)^f.g}^h(n)$$

Rule n: n is any natural integer

Rule h: h can be either n, or, any number in the range 1 to n. This rule simply means

h = n = 0

or

when n > 0 then h = 1 to n

Rule g: g can be either 0 to n when n < 2, or any value from 1 to n-1 when n > 1, and then continuing from [1~1.1] to [a,b,c,d,e-1~g'.h'], including all addition sequences allowed in the Notation Definition. This rule means:

when n = 0 then g = 0

when n = 1 then g = 0 to 1

when n > 1 then g = 1 to n-1

and g can also equal

[1~1.1]

[1~1.1]+[1~0.1]

[1~1.1]+[1~0.2]

and up to

[a,b,c,d,e-1~g'.h'] where g' and h' are recursively defined by the value of g (i.e. square brackets).

Rule f one: when f is zero, then the square brackets collapses to the value of g. For example:

[a,b,c,d,e~0.g] = g for any values of a,b,c,d and e

Rule f two: f can be any value from 0 to n-1, and continuing from [1~1.1] to [a,b,c,d,e-1], including all addition sequences allowed in the Notation Definition, similar to Rule g.

Rule e one: e can be any value from 0 to n-1, and if any parameter a,b,c or d is non-zero it can continue from [1~1.1] to [a,b,c,d,0], including all addition sequences allowed in the Notation Definition, similar to Rule g.

Rule e two: All other leading parameters can also have any value from 0 to n-1, and (in the case of d for instance) if any parameter a,b or c is non-zero d can continue from [1~1.1] to [a,b,c,0,0], including all addition sequences allowed in the Notation Definition, similar to Rule g.

Rule a one: At least one of the parameters a,b,c,d or e must be non-zero.

Rule a two: All leading zeros amongst the parameters a,b,c,d and e can be dropped. For example:

[0,0,c,d,e~f.g]^h(n) = [c,d,e~f.g]^h(n)

Work In Progress

## Notation Translation

Compact Veblen Notation can be translated back into LaTeX math notation using the following string replacements:

Replace "^" with "}^"

Work In progress