Let $\alpha\not= 0$ be such that for every $\beta<\alpha$ there is $\beta<\gamma<\alpha$, where $V\_\gamma$ is an elementary substructure of $V\_\alpha$. In other words, $V\_\alpha$ is a limit of its $V\_\beta$ elementary substructures. Then it is a simple result that $V\_\alpha$ models replacement.


My question: Let $\alpha\not= 0$ be such that for every $\beta<\alpha$ there is $\beta<\gamma<\alpha$ and an elementary embedding from $V\_\gamma$ to $V\_\alpha$. Does it follow that $V\_\alpha$ models replacement?



Elementary embeddings and replacement
If $\alpha$ is a limit of $2^\alpha$-supercompact cardinals,
then by the Magidor characterization of supercompactness, for each $2^\alpha$-supercompact cardinal $\kappa < \alpha$, for some $\gamma < \kappa$, there is an elementary embedding $j : V\_{\gamma}\to V\_\alpha$ with critical point arbitrarily large below $\kappa$. Thus for each $\beta < \gamma$, there is an elementary embedding $j : V\_{\gamma}\to V\_\alpha$ such that $\gamma$ is between $\beta$ and the next $\alpha$-supercompact cardinal, which yields the property you asked about.


But if $\alpha$ is the least cardinal that is a limit of $\alpha$-supercompact cardinals, then $V\_\alpha$ does not model replacement: from the perspective of $V\_\alpha$, there are $\omega$-many supercompact cardinals that are cofinal in the ordinals. The reason is that the $2^{\alpha}$-supercompacts of $V$ are precisely the supercompacts of $V\_\alpha$. Clearly the forwards implication holds, but conversely if $\delta$ is supercompact in $V\_\alpha$, then $\delta$ is supercompact up to a cardinal $\kappa < \alpha$ that is $2^\alpha$-supercompact, and as a consequence, $\delta$ itself is $2^\alpha$-supercompact.


The optimal hypothesis is the existence of an ordinal $\alpha$ that is a limit of $\alpha$-Magidor supercompact cardinals, where a cardinal $\kappa$ is $\alpha$-Magidor supercompact if for some $\gamma < \kappa$, there is an elementary embedding $j : V\_\gamma\to V\_\alpha$ such that $\kappa = j(\text{crit}(j))$. Let $\alpha$ be the least ordinal as in your question. Let $\beta$ be the supremum of the $\alpha$-Magidor supercompact cardinals. If $\beta < \alpha$, then fix an elementary embedding $j : V\_\gamma\to V\_\alpha$ with $\beta < \gamma < \alpha$, and note that $j(\text{crit}(j)) \leq \beta < \gamma$ and hence $j$ witnesses that $\text{crit}(j)$ is huge, contrary to the minimality of $\alpha$. So $\alpha = \beta$ and hence $\alpha$ is a limit of $\alpha$-Magidor supercompact cardinals.


