
> 
> I [know](https://mathoverflow.net/a/136067/143814) we can't formalize the Kunen inconsistency as an assertion in the first-order language of set theory. But
>  [the wikipedia say](https://en.wikipedia.org/wiki/New_Foundations#Strong_axioms_of_infinity) "**Con([MK](https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory) + The proper class ordinal is a measurable cardinal.)**$\leftrightarrow$ **Con([NFU](https://en.wikipedia.org/wiki/New_Foundations) + Infinity + Large Ordinals + Small Ordinals)**", So I think it's no problem for **NFU**.
> 
> 
> 


**NFU** is a non-well founded set theory. NFU had Universal set **V**, Universal set cardinal and automorphism j, NFU's axiom schema of comprehension is very different from **[MK](https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory)/[TG](https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory)**, They are very important in Kunen's inconsistency theorem. So a interesting question is:


Can we use Kunen's inconsistency theorem in **NFU + AC**, to proof **NFU + AC + Reinhardt/super Reinhardt/totally Reinhardt/limit club Berkeley cardinal** $\to 0=1$?



Can we use Kunen's inconsistency theorem in NFU+AC?
In general, NFU + AC is going to have exactly the same possible large cardinals as ordinary set theory. NFU + AC has the same stratified mathematics as ordinary set theory. There will be no interesting results of this kind, unless they involve unstratified assertions in the language of NFU.


I should be slightly more explicit. NFU + AC has the same strength as bounded Zermelo set theory. However, there will be models of NFU + AC associated with any of the usual levels of strength we are used to.


