(Note: This question is related to my previous mathoverflow question, "Critical Points in $ZF$ without Choice".)

In the *Stanford Encyclopedia of Philosophy* entry "Non-Wellfounded Set Theory" (Section 2.2, "The Foundation Axiom"), one has the following statement (my comments regarding it are in brackets):

The Foundation Axiom ($FA$) may be stated in different ways. Here are some formulations; their equivalence in the presence of the other [$ZF$?] axioms is a standard result of elementary [$ZF$?] set theory [the last two, (4) and (5), are particularly relevant to my previous question]:

(4). For every set $x$, there is an ordinal $\alpha$ such that $x$$\in$$V_{\alpha}$. [seemingly necessary for Asaf's proof in his answer to my previous question]

(5). $V_{[ZF?]}$=$WF$ [the class of well-founded sets].

Question 1: Can the equivalence of (4) and (5) (and their equivalence to $FA$) be proved in $ZF$ alone, without recourse to Choice?

Question 2: Regarding (5) (i.e. $V$=$WF$--my comment excluded), does $V$=$V_{ZF}$? I ask this question because of the following: in the Daghighi, Golshani, Hamkins, and Jerabek paper, "The Role of the Foundation Axiom in the Kunen Inconsistency", they claim (and prove, in $GBC^{-f}$ and in $ZFC^{-f}$) that the Kunen inconsistency (in the following form: "There is no nontrivial $\Sigma_1$-elementary embedding $j$:$WF$$\rightarrow$$WF$.") holds for $WF$. If $V_{ZF}$=$WF$, then it seems that there must be a way to adjust their proof so that it doesn't need Choice (in which case, a major open problem will have seemingly been solved). What, if anything, is wrong with this picture?

theuniverse generated by ...") suggests that you have something else in mind, but I can't imagine what. $\endgroup$ – Andreas Blass Jun 29 '16 at 15:56