Working in ZfC + [Wholeness](https://en.wikipedia.org/wiki/Wholeness_axiom):


Can we have a countable sequence of non-trivial elementary embeddings of the universe to itself, such that the range of each embedding is a subclass of the range of its successor, and such that the union of all the ranges of those embeddings is the universe itself?


Formally can we have a proper class sequence $j`\mathbb N$ such that:


$\forall n \in \mathbb N : j\_n: V \prec V \land rng(j\_{n+1}) \supset rng(j\_n) \\ \forall x \exists n \in \mathbb N : x \in rng(j\_n)$


Where $\mathbb N$ is the set of all finite von Neumann ordinals.


Note: ZfC means ZFC but with replacement restricted to the pure language of set theory, i.e. it doesn't use the symbol $j$; however, $j$ is allowed in all instances of separation.



Can there exist such a sequence of elementary embeddings of the universe to itself?
It was pointed out by Monroe that if $\lambda$ is a limit and $j:V\_\lambda\to V\_\lambda$ is elementary and $j\_n$ is the $n$th iterate, i.e. $j\_0=j$ and $j\_{n+1}=j\_n(j\_n)$, and $\kappa\_n=\mathrm{crit}(j\_n)$, then since $\lim\_{n<\omega}\kappa\_n=\lambda$, we get every $x\in V\_\lambda$ is in $\mathrm{rng}(j\_n)$ for some $n<\omega$.


Given this, we can arrange the increasing range condition as follows.
Choose a strictly increasing sequence $\left<n\_i\right>\_{i<\omega}$
such that for each $\alpha<\lambda$ there is $i<\omega$ such that $j\_{n\_{i-1}}\circ j\_{n\_{i-2}}\circ\ldots\circ j\_{n\_0}(\alpha)<\mathrm{crit}(j\_{n\_i})$. (Just construct the sequence recursively on $i$, and at stage $i$, make it work for $\alpha=\kappa\_i$.) Now define $k\_i:V\_\lambda\to V\_\lambda$ as the direct limit
\begin{equation}\ldots \circ j\_{n\_{i+2}}\circ j\_{n\_{i+1}}\circ j\_{n\_i},\end{equation}
i.e. for each $x\in V\_\lambda$ set $k\_i(x)=$ the common value of
\begin{equation}j\_{n\_{i+\ell}}\circ j\_{n\_{i+\ell-1}}\circ\ldots\circ j\_{n\_i}(x)\end{equation}
for large $\ell<\omega$, noting that by the choice of the $n\_i$'s, for every $x\in V\_\lambda$, $k\_i(x)\in V\_\lambda$ is well-defined; and $k\_i$ is easily elementary. Note that $k\_{i}=k\_{i+1}\circ j\_{n\_i}$, and therefore $\mathrm{rng}(k\_i)\subseteq\mathrm{rng}(k\_{i+1})$. And note $\lim\_{i<\omega}\mathrm{crit}(k\_i)=\lambda$. So we get the desired conditions.


