User blog:B1mb0w/The S Function (substitution function)

The S Function (substitution function)
The S function offers a way to generate very large string sequences (representing large numbers) which grows at a faster rate than \(f_{SVO}(n)\), and may be possible to extend its range to grow at a comparable rate to Ordinal Collapsing Functions.

Refer to my The Alpha Function blogs for more information on my work.

Summary Results
We start by recognising that:

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

Then

\(f_1(\omega) = \omega.2\)

\(f_2(\omega) >> \omega^2\) because \(f_1^m(\omega) = \omega.2^m\)

\(f_2^2(\omega) >> \omega\uparrow\uparrow 2\) because \(f_1^{m-2}(f_2(\omega)) >> \omega^m\)

\(f_3^2(\omega) >> \omega\uparrow\uparrow\omega = \varphi(1,0) = \epsilon_0\) because \(f_2^{m.2 + 2}(\omega) >> \omega\uparrow\uparrow m\)

\(f_3^4(\omega) >> \varphi(1,1)\)

\(f_3^{f_0^2(f_1(m))}(\omega) >> \varphi(1,m)\)

\(f_4^2(\omega) >> f_3^{f_0^2(f_1(f_3^2(\omega))) - \omega}f_4(\omega) = f_3^{f_0^2(f_1(f_3^2(\omega)))}(\omega) >> \varphi(1,\varphi(1,0)) = \varphi^2(1,0_*)\)

\(f_4^3(\omega) >> f_3^{f_0^2(f_1(f_3^{f_1(f_3^2(\omega)))}(\omega))}(\omega) >> \varphi^3(1,0_*)\)

\(f_5(\omega) = f_4^{\omega}(\omega) >> \varphi^{\omega}(1,0_*) = \varphi(2,0) = \zeta_0\)

Without any proof, this notation seems to reach:

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

Limitations of this Notation
There seems to be no limitation to using \(\omega\) into the FGH functions and many other transfinite input ordinals can be used as seen above when the functions are nested.

The notation has a growth rate comparable to \(\varphi(\omega,0)\)

However it does not seem possible to extend these FGH functions to \(f_{\omega}(\gamma)\) or beyond for any transfinite ordinal \(\gamma\). This is the limit at which inconsistent results can not be avoided. Here is one example:

Let

\(\gamma = \omega + 1\)

Then

\(f_{\omega}(\gamma) = f_{\omega}(\omega + 1) = f_{\omega + 1}(\omega + 1) = f_{\omega}^{\omega + 1}(\omega + 1)\)

The problem here seems to be a lack of definition on how to diagonalise this step:

\(f_{\omega}(\omega + 1) = f_{\omega + 1}(\omega + 1)\)

It may be necessary to force the subscript of the FGH function to be finite if the input is transfinite. Of course, if the input if finite then the subscript can be transfinite as is the normal case for FGH functions.

Questions and Comments
Really appreciate some feedback on this notation. I see significant benefits from this approach. It removes the need to introduce more complex notation (e.g. the Veblen Functions) and I believe it can grow at a comparable rate to Collapsing Ordinal Functions.

Cheers B1mB0w.