Working in ZfC + Wholeness:

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.


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.

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    $\begingroup$ @Zuhair How can you be sure that the compactness theorem provides the same $V$ as you previously used? I am not even sure that you can apply the compactness theorem for class models internally. $\endgroup$ – Hanul Jeon Feb 23 at 16:07
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    $\begingroup$ @ZuhairAl-Johar I don't understand your compactness idea, since the $j_n$'s do not agree with each other. You can't combine them into one function $F$. $\endgroup$ – Monroe Eskew Feb 23 at 17:20
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    $\begingroup$ As Monroe mentioned, ``$F$ is not surjective'' seems to be in the theory, so $F$ isn't an automorphism. If you construct a model $W$ via compactness satisfying some reasonable finite fragment of the theory of the model $(V,\left<k_n\right>_{n<\omega})$ costructed above (which was wellfounded), together with... $\endgroup$ – Farmer S Feb 23 at 20:21
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    $\begingroup$ ...the statements ``$F:V\to V$ is elementary and non-identity and $\mathrm{range}(k_n)\subseteq\mathrm{range}(F)$'', for each $n$, including separation for $F$ and the sequence of $k$'s, then $W$ will have nonstandard $\omega$, and the interpretation $F^W$ of $F$ could just be $k^W_a$ for one of the nonstandard integers $a$ of $W$. $\endgroup$ – Farmer S Feb 23 at 20:24
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    $\begingroup$ @ZuhairAl-Johar The problem is that, although all standard x will get into the range of F, some new nonstandard sets will appear outside the range. You can add the sentence “F is surjective” to the collection, but what is the interpretation for the finite fragments then? $\endgroup$ – Monroe Eskew Feb 23 at 22:54

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