J. D. Hamkins proved in ["The foundation axiom and elementary self-embeddings of the universe"](http://jdh.hamkins.org/foundation-axiom-and-self-embeddings-of-the-universe/) that, working in $ZFGC^− +BAFA$, there are nontrivial automorphisms and elementary embeddings of the universe **V** into itself.


But, he say "so (BAFA) there are no Reinhardt cardinals here to be found. The embeddings provided by BAFA have no critical points." in this [post](https://mathoverflow.net/a/200544/143814). What does he mean for **"no critical points"**?


(1) Can we use Kunen's inconsistency theorem to proof $ZFGC^- + BAFA + Reinhardt \to 0=1$?


(2) Can Kunen inconsistency make sure there is no "useful" anti-foundational axiom in **[MK](https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory)-AF+Reinhardt** and **[TG](https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory)-AF+Reinhardt**?



Is there no anti-foundational theory exists Reinhardt and hold Global Choice?
In general, anti-foundation axioms have nothing to do with issues of large cardinal consistency strength. You can take your universe of set theory with foundation and convert it into a universe with your favorite antifoundation axiom and no large cardinals will be created or destroyed. This process is reversible: large cardinals have nothing to do with anti-foundation axioms.


