User:Username5243/Pi notation

This is a page where I make a new, probably strong ordinal notation, pi notation. It relies on extending fundamental sequences to ordinal lengths.

How it works
The pi notation looks like \(\pi(\alpha,\beta)\). If \(\beta = 0\), then we could rewrite \(\pi(\alpha,\beta)\) as just \(\pi(\alpha)\).

When \(\alpha = 0\), \(\pi(0,\beta) = \omega^\beta\).

for limit \(\beta\), \(\pi(\alpha,\beta)\) is always the supremum of \(\pi(\alpha,\gamma)\) for \(\gamma < \alpha\).

For limit \(\alpha\), \(\pi(\alpha,0)\) is the supremum of all the \(\pi(\gamma,0)\) for \(\gamma < \alpha\). and \(\pi(\alpha,\beta+1)\) is the supremum of all \(\pi(\gamma,\pi(\alpha,\beta))\) for \(\gamme <\alpha\).

For successor \(\alpha\), where it is not the successor of a limit ordinal, \(\pi(\alpha+1,0)\) is the first fixed point of \(\gamma = \pi(\alpha,\gamma)\). And \(\pi(\alpha+1,\beta+1)\) is the next fixed point of that after \(\pi(\alpha+1,\beta)\).

The most interesting, but also hardest to explain, case is if \(\alpha\) is the successor of a limit ordinal. In that case, We have to extend the fundamental sequence of that ordinal. After the finite terms, These will just be written like \(\alpha[\beta]\). \(\omega\)th term is always hat ordinal itself, but they refer to different things in operations. To get to \\(\omega+1\)th term, you have to do the same thing to the \(\omega\)th term that you did to get from the nth term to the n+1-th term. So, after \(\omega^2 = \omega^2[\omega]\), there's \(\omega^2[\omega]+1, \(\omega^2[\omega]+2, \), and the limit of that is \(\omega^2[\omega]+\omega = \omega^2[\omega+1]\). Anyway, to get the actuall definition of \(\pi(\alpha+1,\beta)\) in this case, it is the fixed points of \(\gamma = \(\pi([\alpha[\gamma],0)\).

Now I will analyze this notation with normal ordinal notation.

Up to \(\pi(\omega)\)
Because there are no infinites in the first argument, this is easy to understand. It's just like \(\phi(\alpha,\beta)\).