User blog:Bubby3/Analysis of BMS v2.2

Here is an analysis of BM version 2.2

Everything is the same until (0,0,0)(1,1,1)(2,1,0). However, ordinals after that are different.

\(\psi(\Omega_\omega \Omega)\) to \(\psi(\Omega_\omega \Omega + \Omega_omega)\)
This might seem mysterious, as normally (0,0,0)(1,1,1)(2,1,0)(1,1,1) would have a level much higher than that. However, in this version, we just cut off the (1,1,1).
 * (0,0,0)(1,1,1)(2,1,0) has level \(\psi(\Omega_\omega \Omega)\)
 * (0,0,0)(1,1,1)(2,1,0)(1,1,1) has level \(\psi(\Omega_\omega \Omega) + 1\)
 * (0,0,0)(1,1,1)(2,1,0)(1,1,1)(2,0,0) has level \(\psi(\Omega_\omega \Omega) + \omega\)
 * (0,0,0)(1,1,1)(2,1,0)(1,1,1)(2,0,0)(3,1,1)(4,1,0) has level \(\psi(\Omega_\omega \Omega) 2\)
 * (0,0,0)(1,1,1)(2,1,0)(1,1,1)(2,0,0)(3,1,1)(4,1,0)(3,1,1) has level \(\psi(\Omega_\omega \Omega) 2 + 1\)
 * (0,0,0)(1,1,1)(2,1,0)(1,1,1)(2,0,0)(3,1,1)(4,1,0)(3,1,1)(4,0,0)(5,1,1)(6,1,0) has level \(\psi(\Omega_\omega \Omega) 3\)
 * (0,0,0)(1,1,1)(2,1,0)(1,1,1)(2,1,0) has level \(\psi(\Omega_\omega \Omega) \omega\)
 * (0,0,0)(1,1,1)(2,1,0)(1,1,1)(2,1,0)(1,1,1)(2,0,0)(3,1,1)(4,1,0)(3,1,1)(4,1,0) has level \(\psi(\Omega_\omega \Omega) \omega 2\)
 * (0,0,0)(1,1,1)(2,1,0)(1,1,1)(2,1,0)(1,1,1)(2,1,0) has level \(\psi(\Omega_\omega \Omega) \omega^2\)
 * (0,0,0)(1,1,1)(2,1,0)(2,0,0) has level \(\psi(\Omega_\omega \Omega) \omega^\omega\)
 * (0,0,0)(1,1,1)(2,1,0)(2,0,0)(3,1,1)(4,1,0) has level \(\psi(\Omega_\omega \Omega) ^ 2\)
 * (0,0,0)(1,1,1)(2,1,0)(2,1,0) has level \(\psi(\Omega_\omega \Omega + 1)\)
 * (0,0,0)(1,1,1)(2,1,0)(2,1,0)(2,1,0) has level \(\psi(\Omega_\omega \Omega + \Omega)\)
 * (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,1,0) has level \(\psi(\Omega_\omega \Omega + \psi_1(\Omega_\omega \Omega))\)
 * (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,1,0)(3,2,1) has level \(\psi(\Omega_\omega \Omega + \psi_1(\Omega_\omega \Omega)) + 1\)
 * (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,1,0)(3,2,1)(4,1,0) has level\(\psi(\Omega_\omega \Omega + \psi_1(\Omega_\omega \Omega) + 1)\)
 * (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,1,0)(5,2,1)(6,1,0) has level\(\psi(\Omega_\omega \Omega + \psi_1(\Omega_\omega \Omega) 2)\)
 * (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,2,0) has level \(\psi(\Omega_\omega \Omega + \psi_1(\Omega_\omega \Omega) \omega)\)
 * (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,2,0)(4,2,0) has level \(\psi(\Omega_\omega \Omega + \psi_1(\Omega_\omega \Omega + 1))\)
 * (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,2,0)(4,2,0)(4,2,0) has level \(\psi(\Omega_\omega \Omega + \Omega_2)\)
 * (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,2,0)(5,3,1)(6,1,0) has level \(\psi(\Omega_\omega \Omega + \psi_2(\Omega_\omega \Omega))\)
 * (0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,2,0)(5,3,1)(6,3,0)(6,3,0)(6,3,0) has level \(\psi(\Omega_\omega \Omega + \Omega_3)\)

\(\psi(\Omega_\omega \Omega + \Omega_omega)\) to \(\psi(\Omega_\omega \Omega_2)\)

 * (0,0,0)(1,1,1)(2,1,1) has level \(\psi(\Omega_\omega \Omega + \Omega_omega)\)
 * (0,0,0)(1,1,1)(2,1,1)(1,1,1) has level \(\psi(\Omega_\omega \Omega + \Omega_omega 2)\)
 * (0,0,0)(1,1,1)(2,1,1)(1,1,1)(2,1,0) has level \(\psi(\Omega_\omega \Omega 2\)
 * (0,0,0)(1,1,1)(2,1,1)(1,1,1)(2,1,1) has level \(\psi(\Omega_\omega \Omega 2 +\Omega_\omega)\)
 * (0,0,0)(1,1,1)(2,1,1)(2,1,1) has level \(\psi(\Omega_\omega \Omega^2 + \Omega_\omega)\)
 * (0,0,0)(1,1,1)(2,1,1)(3,1,1) has level \(\psi(\Omega_\omega \Omega^\Omega + \Omega_\omega)\)
 * (0,0,0)(1,1,1)(2,2,0) has level \(\psi(\Omega_\omega psi_1(0))\)
 * (0,0,0)(1,1,1)(2,2,0)(3,1,1) has level \(\psi(\Omega_\omega psi_1(0)) + 1\)
 * (0,0,0)(1,1,1)(2,2,0)(3,1,1)(4,2,0) has level \(\psi(\Omega_\omega psi_1(0)) + \omega\)
 * (0,0,0)(1,1,1)(2,2,0)(3,2,0) has level \(\psi(\Omega_\omega psi_1(0)) + \omega^2\)
 * (0,0,0)(1,1,1)(2,2,0)(3,3,0) has level \(\psi(\Omega_\omega psi_1(0)) + \omega^\omega\)
 * (0,0,0)(1,1,1)(2,2,0)(3,3,1) has level \(\psi(\Omega_\omega psi_1(0)) + \psi(0)\)
 * (0,0,0)(1,1,1)(2,2,0)(3,3,1)(4,3,1) has level \(\psi(\Omega_\omega psi_1(0)) + \psi(1)\)