I have heard it said that it is possible to use Woodin's iterated collapse forcing to prove that the theory $\textsf{ZF}$+"there exists a non-trivial elementary embedding $j:V_{\lambda+2} \rightarrow V_{\lambda+2}$ for some ordinal $\lambda$" is equiconsistent with the theory $\textsf{ZF}$+"for some ordinal $\lambda$, $V_{\lambda}$ is well-orderable and there is a non-trivial elementary embedding $j:V_{\lambda+2} \rightarrow V_{\lambda+2}$". I was wondering if anyone could tell me where I could read about this kind of forcing?
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$\begingroup$ You might get a partial answer to your question if you email Gabriel Goldberg (google his homepage for that) and ask for is preprint, "On the consistency strength of Reinhardt cardinals", which , according to Rupert Mc'Callum in his arXiv preprint (arXiv:1812.03837v1 [math.LO] 10 Dec 2018), "New Large-Cardinal Axioms an the Ultimate-L Program", one can use to show that "It is not consistent with $ZF$ that there exists an ordinal $\lambda$ and a non-trivial embedding $j$: $V_{\lambda + 2}$ $\prec$ $V_{\lambda +2}$" (Theorem 5.1 in Mc'Callum's preprint) $\endgroup$– Thomas BenjaminJan 17, 2019 at 14:45
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$\begingroup$ Add "elementary" to "non-trivial embedding (so it reads "non-trivial elementary embedding") in my quote of Theorem 5.1 in my previous comment. Also substitute "his" for 'is" in the phrase "ask for is preprint" as well. Thanks. $\endgroup$– Thomas BenjaminJan 17, 2019 at 14:57
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$\begingroup$ Also look through Woodin's papers "Suitable Extender Models" I and II--there might be information on Iterated Collapse Forcing in those papers as well. $\endgroup$– Thomas BenjaminJan 17, 2019 at 15:05
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