User blog:B1mb0w/FGH of Omega

Fast Growing Hierarchy Function with \(\omega\)
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.

Summary Results (so far)
We start by recognising that:

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

Then

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

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

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

\(\omega\uparrow\uparrow\omega = \epsilon_0 = \varphi(1,0) = f_3(\omega)\)

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.

Detailed Results (so far)
We start by recognising that:

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

Then

\(\omega + 2 = f_0^2(\omega)\)

And

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

My summary results so far are:

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

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

WORK IN PROGRESS