# Can there exist such a sequence of elementary embeddings of the universe to itself?

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 $$\left_{i<\omega}$$ 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 $$$$\ldots \circ j_{n_{i+2}}\circ j_{n_{i+1}}\circ j_{n_i},$$$$ i.e. for each $$x\in V_\lambda$$ set $$k_i(x)=$$ the common value of $$$$j_{n_{i+\ell}}\circ j_{n_{i+\ell-1}}\circ\ldots\circ j_{n_i}(x)$$$$ for large $$\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.
• @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. – Hanul Jeon Feb 23 at 16:07
• @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 $F$. – Monroe Eskew Feb 23 at 17:20
• 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... – Farmer S Feb 23 at 20:21
• ...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 $W$. – Farmer S Feb 23 at 20:24