I would like to suggest another argument for Kunen's inconsistency result, and I wonder to know if the argument is correct. I am also interested to see, if the proof is correct, which part of the argument uses AC.


Theorem.
There is no non-trivial elementary embedding $j: V \rightarrow V.$


Proof. Assume on the contrary that there is a non-trivial elementary embedding $j: V \rightarrow V$ with $\kappa=crit(j).$ Iterate it to get $\langle \langle M\_n: n\leq \omega \rangle, \langle j\_{n,m}: n\leq m\leq \omega \rangle \rangle$ where $\langle M\_{\omega}, \langle j\_{n, \omega}: n\leq \omega \rangle \rangle =dirlim \langle \langle M\_n: n<\omega \rangle, \langle j\_{n,m}: n\leq m< \omega \rangle \rangle.$ Note that $j=j\_{0,1}$ and that for all $n<\omega, M\_n=V.$ Let $\kappa\_n = j\_{0, n}(\kappa), n\leq \omega.$ Then $\kappa\_{\omega}=sup\_{n<\omega}\kappa\_n.$ Also we have $M\_{\omega}=V$ and we get a contradiction, since $\kappa\_{\omega}$ must be regular in $M\_{\omega}=V.$



More on Kunen's inconsistency result
This is an interesting idea, but I don't believe that your
argument succeeds.


Notice first that if it were correct, then it would also refute
the existence of [$I\_1$ rank-into-rank cardinals](http://cantorsattic.info/Rank_1_into_rank_1), that is, $j:V\_{\lambda+1}\to V\_{\lambda+1}$, since
$\lambda=\kappa\_\omega$ and the singularity of this cardinal is
revealed in $V\_{\lambda+1}$.


Some people might object that you haven't said what you mean by iterating $j$, but this is not actually a problem, for there is a natural way to do this. Any class embedding $j:V\to M$ can be iterated, since the
definition of $j$ can be extended to any proper class $A$ by
defining $j(A)=\bigcup\_\alpha j(A\cap V\_\alpha)$. This way, we get
$j(j)$, which will be an elementary embedding from $M$ to $j(M)$.
Thus, if we start with $j:V\to V$, we may indeed iterate
$j$ to form a system $V\to V\to V\to\cdots$ as you describe.


The problem is that you claimed that the direct limit of this system is $V$, but I
don't see why this should be true; this is a gap in
your proof. It may seem reasonable to suppose that the
direct limit of a system of structures, all of which are the same,
is again that same structure, but in fact there are counterexamples to
this general principle. (For a quick example, embed the discrete order
$\mathbb{Z}$ into itself by stretching by a factor of two; the
direct limit of the iteration of this process is the dense order
$\mathbb{Q}$.)


Indeed, let me argue directly that the direct limit is definitely
not $V$. The reason is that the direct limit model has only
$\lambda$ many subsets of $\lambda$, where
$\lambda=\kappa\_\omega$. This is simply because every subset of
$\lambda$ in the direct limit is born at some stage $M\_n$ as a
subset of $\kappa\_n$, having the form $j\_{n,\omega}(A)$ for some
$A\subset\kappa\_n$. But there are only $\lambda$ many $A$ like
that, and so the limit model is missing most subsets of $\lambda$.


Note that the limit of the iteration is well-founded, by
essentially the usual Kunen argument. Namely, if it were
ill-founded, then we may consider the least $\gamma$ with
$j\_{0,\omega}(\gamma)$ in the ill-founded part of the limit. By
pushing $\gamma$ into $M\_n$, we may conclude on the one hand that
$j\_{0,n}(\gamma)$ must be least sent into the ill-founded part,
but on the other hand we may also drop down to smaller ordinal
born at that stage still in the ill-founded part, making a
contradiction. (To formalize this precisely, one may avoid the
complications caused by $j$ being a proper class simply by
chopping off $j$ to the set $j\upharpoonright V\_\alpha$, such that
the iteration whose iteration is ill-founded.) This argument relies fundamentally on the fact that the iteration of $j$ in $M\_0$ produces the same limit as the iteration of $j(j)$ in $M\_1$. 


Finally, if on the other hand you had formed your iteration by using the same
embedding $j\_{n,n+1}=j$ at each stage, then this is clearly
ill-founded, since the threads starting with $\kappa$ at each
stage form a descending sequence of ordinals in the limit.


