User blog:B1mb0w/Unique Ordinal Representation

Unique Ordinal Representation
This blog will demonstrate how to create a unique representation of ordinals with arbitrary complexity using the definitions from my blog on Extended Normal Form. In turn this blog will be referred to in my other blogs for the J Function.

Extended Normal Form
An arbitrary ordinal can be defined as follows using the definitions of Extended Normal Form.

\(\gamma_m = \lambda^{\gamma_{m-1}}.\gamma_c + \gamma_a = \varphi(\delta_{[q]})^{\gamma_{m-1}}.\gamma_c + \gamma_a\)

Where \(\gamma\) and \(\delta\) are arbitrary transfinite ordinals, \(\lambda\) is an arbitrary limit ordinal, and \(m, q\) are finite integers.

Unique Representation
A unique representation for any such ordinal can be demonstrated as a sequence of finite integers defined as \(\). The sequence is constructed in the following order:

Let \(q = s_0\) any finite integer

Recursively define \(\lambda = <,,,...,> = \)

Let \(m = s_1\) any finite integer

Recursively define \(\gamma_{m-1} = \)

Assign \( = ,s_1,>\)

Recursively define \(\gamma_c = \)

Assign \( = <,>\)

Recursively define \(\gamma_a = \)

Assign \( = <<g_b>,<g_a>>\)

Unique Representation and Monotonically Ascending
The resulting sequence of finite integers \(<g>\) will be unique for any ordinal. The sequence will also impose monotonically ascending representations, such that if \(<x> > <y>\) then \(\gamma_x > \gamma_y\).

For one sequence \(<x>\) to be greater than another sequence \(<y>\), then the first non-identical integer (from left to right) between the sequences must be greater in sequence \(<x>\), as shown here:

\(<x> > <y>\) if:

\(<x> = <x_{[i-1]},x_i,x_{[j]}>\)

\(<y> = <y_{[i-1]},y_i,y_{[k]}>\)

and \(x_{[i-1]} = y_{[i-1]}\) and \(x_i > y_i\)

Unique Big Number Representation
A further extension to the above can be made to create a sequence of finite integers \(<f>\) as follows:

Assign \(<f> = <<g>,n,p>\) where \(n, p\) are finite integers and \(p > max(max(<g>)+1,n)\)

This sequence will represent any Big Number (Googolism) of the form:

\(f_g^n(p)\) where \(g < SVO\) the Small Veblen Ordinal

Unique Representation for \(\omega^{\omega} + \omega.3 + 2\)
Here is a unique representation for this arbitrary ordinal:

\(\gamma_m = \omega^{\omega} + \omega.3 + 2 = \varphi(\delta_{[q]})^{\gamma_{m-1}}.\gamma_c + \gamma_a\)

Let \(q = 1\)

This will collapse \(\lambda = \varphi(\delta_{[1]} = \varphi(\delta_1) = \varphi(1) = \omega\)

Let \(m = 2\)

Recursively define \(\gamma_1 = <g_1>\)
 * \(\gamma_1 = \varphi(\delta_{[q]})^{\gamma_0}.\gamma_c + \gamma_a\)
 * Let \(q = 1\)
 * This will collapse \(\lambda = \varphi(\delta_{[1]} = \varphi(\delta_1) = \varphi(1) = \omega\)
 * Recursively define \(\gamma_0 = 1\)
 * Recursively define \(\gamma_c = <g_c>\)
 * \(\gamma_c = \varphi(\delta_{[q]})^{\gamma_{c-1}}.\gamma_cc + \gamma_a\)
 * Let \(q = 0\)
 * This will collapse \(c = 0\) and \(\gamma_c = p\)
 * Let \(p = 1\)
 * Recursively define \(\gamma_a = <g_a>\)
 * \(\gamma_a = \varphi(\delta_{[q]})^{\gamma_{a-1}}.\gamma_c + \gamma_aa\)
 * Let \(q = 0\)
 * This will collapse \(a = 0\) and \(\gamma_a = p\)
 * Let \(p = 0\)

WORK IN PROGRESS

Assign \(<g_d> = <1,,2,<g_1>> = <1,2,<1,1,0,1,0,0>> = <1,2,1,1,0,1,0,0>\)

Recursively define \(\gamma_c = <g_c>\)
 * \(\gamma_c = \varphi(\delta_{[q]})^{\gamma_{c-1}}.\gamma_cc + \gamma_a\)
 * Let \(q = 1\)
 * This will collapse \(\lambda = \varphi(\delta_{[1]} = \varphi(\delta_1) = \varphi(1) = \omega\)
 * Let \(c = 1\)
 * Recursively define \(\gamma_0 = 1\)
 * Recursively define \(\gamma_c = <g_c>\)
 * \(\gamma_c = \varphi(\delta_{[q]})^{\gamma_{c-1}}.\gamma_cc + \gamma_a\)
 * Let \(q = 0\)
 * This will collapse \(c = 0\) and \(\gamma_c = p\)
 * Let \(p = 1\)
 * Recursively define \(\gamma_a = <g_a>\)
 * \(\gamma_a = \varphi(\delta_{[q]})^{\gamma_{a-1}}.\gamma_c + \gamma_aa\)
 * Let \(q = 0\)
 * This will collapse \(a = 0\) and \(\gamma_a = p\)
 * Let \(p = 0\)

Assign \(<g_d> = <1,,2,<g_1>> = <1,2,<1,1,0,1,0,0>> = <1,2,1,1,0,1,0,0>\)

Assign \(<g_b> = <<g_d>,<g_c>>\)

Recursively define \(\gamma_a = <g_a>\)

Assign \(<g> = <<g_b>,<g_a>>\)

WORK IN PROGRESS