User blog comment:Meowzz/Question: 0=1 Cardinal/@comment-26428969-20180422163440

"0=1" is usually just used to mean inconsistency/contradiction/absurdity

On the other hand, ZF (or rather NBG (without Choice/LoS), which is more convenient to talk about Reinhardts for language restriction reasons) is not known to be inconsistent with existence of a Reinhardt, though it us suspected to be

Ultimate L / the Omega conjecture are a bit too complicated to explain here, so I'm just going to mention some relevant consequences of the Omega conjecture and the related HOD conjecture

1) under some precise (forcing-related) meaning of "large cardinal axiom", the LCAs are linearly/totally ordered by consistency strength. however there is no maximal elements in that order

2) if ZFC proves the HOD conjecture (closely related to the Omega conjecture) then ZF/NBG alone can prove there are no Reinhardts, and so that LCA is inconsistent with ZF/NBG