User blog:Syst3ms/Formal pi notation is far weaker than expected

I was a bit too confident about the strength of this, it seems. Eh, happens.

It turns out that the formal definition of pi notation drastically loses in strength past some point. The point at which I noticed the steep drop was \(\psi(\Omega_\omega^2)\).

Instead of equalling \(\pi(L(\pi(\Omega),\pi(\Omega))), it equals \(\pi(\pi(\Omega)+1)\), and the latter is far larger than the former. After that, analysis shows a massive drop in strength, which leads to \(\pi(\Omega,1)=\psi(\Omega_\omega^{\Omega_2})\) instead of the expected \(\psi(\Omega_{\Omega_\omega})\).

So yeah, clearly something isn't working as it should in my formalization. So why did this happen ? This is because the expansion rules break a nice pattern for whatever reason, specifically that \(\pi(L(\pi(\Omega),x))\approx\psi(\Omega_x)\). This is true for finite \(x\) and \(x=\omega\), but for \(x=\omega+1\), we have \(\pi(L(\pi(\Omega),\omega+1))=\psi(\Omega_\omega+\Omega_2)<\psi(\Omega_{\omega+1})).

If we want pi notation to be as strong as we want, we need to have \(\pi(L(\pi(\Omega),\omega+1))=\psi(\Omega_{\omega+1})\).

How to achieve that is still unclear, but I will be investigating... Maybe some other day.