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This latest version of The Alpha Function has been aligned to my latest blog on The S Function (Version 2). The growth rate has been significantly increased to well beyond \(f_{LVO}(n)\).
Version 7 has been completely modified from Version 6 to use my latest work on The S Function.
The key change has been to generalise the string substitution procedures introduced by the S Function. The S Function is recursively defined with a sibling T Function. The T Function has been generalised to support significantly more recursive behaviour. This means the S Function can generate very long strings of S and T function combinations. Each string can be translated to a finite integer and the S Function has a growth rate well above \(f_{LVO}(n)\).
The Alphaâ€¦
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This blog is outdated and no longer relevant to my Alpha Function blogs.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 ruleset for the Fundamental Sequences used by in my Alpha Function blogs. It relies upon the use of the Fastgrowing 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.
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 ruleset definitions. It also has some added benefiâ€¦
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I am exploring how the FGH functions will behave with an transfinite ordinal e.g. \(\omega\) as the input parameter.
The results seem to be very interesting. The notation grows at the same rate as some Veblen Functions and it may be possible to extend it to grow at a comparable rate to Ordinal Collapsing Functions.
Also refer to my other Fundamental Sequences blogs for more information on my work.
There are many blogs out there that cover this some material. Here are the ones I am aware of:
 FGH_with_ordinals by SuperJedi224
 Ordinal_FGH,_with_an_actual_definition! by Googology_Noob
 FGH_Things by King2218
 FGH_with_transfinite_ordinals by Wythagoras
 BHOlevel_Hardy_Hierarchy_with_ordinals by Emlightened
We start by recognising that:
\(f_0(\omega) = \omâ€¦
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This blog will cover the standard definitions on Fundamental Sequences for Ordinals. It will also provide a precise ruleset for the Fundamental Sequences used by in my Beta Function blogs.
This blog is a complete update of my previous blog on Fundamental Sequences. Please keep this in mind if you refer to that blog.
Let \(\gamma\) and \(\delta\) be two arbitrary transfinite ordinals, \(\lambda\) is an arbitrary limit ordinal, and \(n\) is a finite integer. Then:
\((\gamma + 1)[n] = \gamma\)
\((\gamma + \lambda)[n] = \gamma + \lambda[n]\) when \(\gamma >= \lambda\)
\(\lambda.(\delta + 1)[n] = \lambda.\delta + \lambda[n]\)
\(\gamma.\lambda[n] = \gamma.(\lambda[n])\) when \(\gamma >= \lambda\)
\(\lambda^{\delta + 1}[n] = \lambda^{\delta}.(\lambda[n])â€¦
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The Alpha Function has been defined using Sequence Generator code shown below. A separate blog has been written to explain how to read Sequence Generator Code and how it works.
Version 6 has been completely modified from Version 5 to align it completely to my my other work on Beta Function blogs.
The key change has been to use the power of the The Beta Function to access every Veblen ordinal and every FGH function and therefore every finite integer (up to the size of \(f_{SVO}(v)\) for a given base \(v\)).
The Alpha Function is a one input parameter version of the Beta Function that can access every finite integer up to \(f_{SVO}(v)\) for any n.
The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. The real number is manipulaâ€¦
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