User blog:Syst3ms/A dump of my (on hold) shot at well-defining pi definition

I'll be straightforward : I don't know how to actually formalize stuff, so I won't bother saying this is formal, since it is extremely far from it. I'm just dumping this here so that it doesn't get lost in some Discord channel somewhere. This is definitely better than "transfinite FSes", however.

"""Definition""" :
\(\Omega_0 = 1 \\ \pi_n(0,0) = \Omega_n \\ n \ge 1: L(\Omega_n,\alpha) = \alpha \\ m+1 > n: \pi_n(\Omega_{m+1},\alpha) = \pi_n(\pi_m(\Omega_{m+1},0),\alpha) \\ \text{cof}(\alpha) = \Omega_{n+1}: L(\pi_n(\alpha,0),0) = 0 \\ \text{cof}(\alpha) = \Omega_{n+1}: L(\pi_n(\alpha,\beta+1),0) = \pi_n(\alpha,\beta)+1 \\ \text{cof}(\alpha) = \Omega_{n+1}: L(\pi_n(\alpha,\beta),\gamma+1) = \pi_n(L(\alpha,L(\pi_n(\alpha,\beta),\gamma)),0) \\ \beta \in \text{Lim}: L(\pi_n(\alpha,\beta),\gamma) = \pi_n(\alpha,L(\beta,\gamma)) \\ \beta \in \text{Lim}: L(\pi_n(L(\alpha,\beta),\gamma),\delta) = \pi_n(L(\alpha,L(\beta,\delta)),\gamma) \\ L(\pi_n(0,\alpha+1),0) = 0 \\ L(\pi_n(0,\alpha+1),\beta+1) = L(\pi_n(0,\alpha+1),\beta)+\pi_n(0,\alpha) \\ \alpha \not\in \text{Lim}: L(\pi_n(\alpha+1,0),0) = 0 \\ \alpha \not\in \text{Lim}: L(\pi_n(\alpha+1,\beta+1),0) = \pi_n(\alpha+1,\beta)+1 \\ \alpha \not\in \text{Lim}: L(\pi_n(\alpha+1,\beta),\gamma+1) = \pi_n(\alpha,L(\pi_n(\alpha+1,\beta),\gamma)) \\ \alpha \in \text{Lim}: L(\pi_n(\alpha,0),\beta) = \pi_n(L(\alpha,\beta),0) \\ \alpha \in \text{Lim}: L(\pi_n(\alpha,\beta+1),\gamma) = \pi_n(L(\alpha,\gamma),\pi_n(\alpha,\beta)+1) \\ \alpha \in \text{Lim}: L(\pi_n(\alpha+1,0),0) = 0 \\ \alpha \in \text{Lim}: L(\pi_n(\alpha+1,\beta+1),0) = \pi_n(\alpha+1,\beta)+1 \\ \alpha \in \text{Lim}: L(\pi_n(\alpha+1,\beta),\gamma+1) = \pi_n(L(\alpha,L(\pi_n(\alpha+1,\beta),\gamma)),0)\)

I'd continue this one day, if school weren't such a hassle.