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Fundamental Sequences (used by The Alpha Function)

This blog is out-dated and no longer relevant to my Alpha Function blogs. Please keep this in mind if you refer to that blog.

This blog provides a precise rule-set for the Fundamental Sequences used by in my Alpha Function blogs. It relies upon the use of the Fast-growing hierarchy functions applied to transfinite ordinals (e.g. \(\omega\)). Refer to my blog on FGH Function with Omega for more information.

This blog is a significant change to previous blogs on Fundamental Sequences. Please keep this in mind if you refer to that blog.


Rule-set (used by The Alpha Function)

I am exploring how my Alpha Function could be changed to use a new Fundamental Sequence. The Fundamental Sequence is intended to be clearly distinguishable from other rule-set definitions. It also has some added benefits in simplicity and faster growth rates.

It relies upon the use of the Fast-growing hierarchy functions applied to transfinite ordinals (e.g. \(\omega\)). This offers a simpler ordinal notation which may avoid the need for Veblen Function (and possibly other Ordinal Collapsing Functions)

The following results are presented and explained on my FGH of Omega blog:

\(\omega + 1 = f_0(\omega)\)

\(\omega.2 = f_0^{\omega}(\omega) = f_1(\omega)\)

\(\omega^2 = f_1^{\omega}(\omega) = f_2(\omega)\)

\(\omega\uparrow\uparrow 2 << f_2^2(\omega)\)

\(\epsilon_0 = \varphi(1,0) << f_3^2(\omega)\)

\(\zeta_0 = \varphi(2,0) << f_5(\omega)\)

\(\varphi(m,0) << f_{m.2+1}(\omega)\)

The Rule-set for this Fundamental Sequence can be applied to an arbitrary FGH function:

\(h = f_{\lambda}^u(v)\)

\(\lambda = f_{\gamma}^{\delta}(\beta)\)

The Rule-set is very easy to describe and apply:

\(\delta\) \(\gamma\) \(\beta\) \(\lambda[n] = f_{\gamma}^{\delta}(\beta)[n]\)
limit any any \(= f_{\gamma}^{\delta[n]}(\beta)\)
\(= 1\) limit any \(= f_{\gamma}^1(\beta)[n] = f_{\gamma[n]}(\beta)\)
\(= 1\) any \(= f_1(\beta)[n] = f_0^{\beta[n]}(\beta) = \beta + \beta[n]\)
successor any \(= f_{\gamma-1}^{\beta[n]}(\beta)\)
successor any any \(= f_{\gamma[n]}(f_{\gamma}^{\delta-1}(\beta))\)


Cantor's Normal Form

Cantor's Normal Form applies as usual to the FGH functions with \(\omega\), as seen in these examples for an arbitrary ordinal \(\lambda\) made up of two other arbitrary transfinite \(\gamma\) and \(\delta\) defined with FGH functions:

\(\gamma + \delta = f_{\gamma_0}^{\gamma_1}(\omega) + f_{\delta_0}^{\delta_1}(\omega)\)

\(\gamma.\delta = f_{\gamma_0}^{\gamma_1}(\omega).f_{\delta_0}^{\delta_1}(\omega)\)

\(\gamma^{\delta} = f_{\gamma_0}^{\gamma_1}(\omega)^{f_{\delta_0}^{\delta_1}(\omega)} < f_{\gamma_0}^{\gamma_1}(\omega)\uparrow\uparrow 2\)

Only If

\(\gamma_0 > \delta_0\)

Or

\(\gamma_0 = \delta_0\) and \(\gamma_1 >= \delta_1\)


Veblen Hierarchy

This Fundamental Sequence has the benefits of being simpler to explain and apply, and, avoids the need for more Theoretic notation to access larger transfinite ordinals. Specifically, this Fundamental Sequence avoids the use of some of the Veblen Hierarchy to access very large transfinite ordinals.

Let's refer back to these speculative results (to be confirmed):

\(\epsilon_0 = \varphi(1,0) = f_3^2(\omega)\)

\(\zeta_0 = \varphi(2,0) << f_5(\omega)\)

\(\varphi(m,0) << f_{m.2+1}(\omega)\)


Growth Rate of this Fundamental Sequence

When this Fundamental Sequence is applied to The Alpha Function, it results in a very fast growing function. This is because of two effects:

FGH of Omega functions eventually dominate Veblen Functions

See above example in the Veblen Hierarchy section. A given FGH of Omega function like \(f_5(\omega)\) is nominally equivalent to \(\zeta_0\) but is actually a much larger ordinal.

These Fundamental Sequences diagonalise at a faster rate

Even when an FGH of Omega function and a given Veblen Hierarchy function are equivalent, e.g.:

WORK IN PROGRESS

When these are diagonalised for a given n (i.e. \([n]\)), this Fundamental Sequence will result in a much larger ordinal and subsequently larger finite number:

WORK IN PROGRESS


Calculated Examples

WORK IN PROGRESS

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