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J. D. Hamkins proved in "The foundation axiom and elementary 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. 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-AF+Reinhardt and TG-AF+Reinhardt?

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    $\begingroup$ He means no ordinals are moved. $\endgroup$
    – Monroe Eskew
    Commented Oct 2, 2019 at 8:53
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    $\begingroup$ This post is not written in good mathematical English: "to proof" and "no any" are always inappropriate, and "mean for" is inappropriate here. I can not understand question 2 at all. It might help to apply GoogleTranslate after writing the post in some other language, or to compare and combine computer-translated results with the English text proposed so far. $\endgroup$
    – user44143
    Commented Oct 2, 2019 at 15:13
  • $\begingroup$ the title is not that clear? $\endgroup$
    – Zuhair Al-Johar
    Commented Oct 2, 2019 at 16:52
  • $\begingroup$ @MattF. Sorry, I'm not Living in English country. I'm delete "any". "no critical points" just the motivation for my problem. $\endgroup$
    – Ember Edison
    Commented Oct 3, 2019 at 10:23
  • $\begingroup$ @ZuhairAl-Johar, if the title question and question 2 are clear to you, you can edit the post, and the edits would likely be approved. $\endgroup$
    – user44143
    Commented Oct 3, 2019 at 11:51

1 Answer 1

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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.

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