There have been questions on whether proper classes have cardinality (some say yes). However, I have my own axioms about it.
Firstly, we define the 'cardinality' of a proper class as the conglomerate of proper classes which form a bijection with said proper class (in ZFC). $\kappa < \lambda$ ($\kappa$, $\lambda$ cardinals) if there is an injection, but not a surjection, between a proper class with 'cardinality' $\kappa$ and one with 'cardinality' $\lambda$.
The 'cardinality' of the ordinals is called $\Omega$ (absolute infinity).
The 'cardinality' of the surreal numbers is called $\Omega_1$ (absolute one infinity).
The 'cardinality' of the Von Neumann universe is called $ↇ$ (never).
Question: Is the axiom $\Omega < \Omega_1 < ↇ$ independent of ZFC? If not, does ZFC imply it, or does it imply it is not the case? And why or why not?
(Credit to Mathis R.V and maybe also other people for the names and symbols of the 'cardinals', but the meanings described above are probably invented by me)