Elementary embeddings and replacement

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?

• Oh, you changed the question. I was about to post an answer to the earlier one. – Joel David Hamkins May 13 at 11:36
• @JoelDavidHamkins I’m so sorry! The original question is the one I’m interested in ultimately, but I thought the new question would be more accessible. If you have an answer, though, should I just reinstate that question? – Sam Roberts May 13 at 11:49
• The questions are similar, since you get $\Sigma_n$-elementary embeddings in the old version by using a universal $\Sigma_n$-truth predicate. I'll try to post an answer a bit later. – Joel David Hamkins May 13 at 11:50
• @JoelDavidHamkins Right. But I thought the new question might be more tractable. In fact, the new question is the one I was focusing my attention on when I was thinking this all through. – Sam Roberts May 13 at 11:53
• @AsafKaragila That’s right, the assumption is weaker. But getting an elementary embedding between ranks that isn’t the identity requires a lot of large cardinal strength. To show that we can have the assumption without replacement, we need a bunch of elementary embeddings that aren’t the identity and thus a lot of large cardinal strength. (I was working with extendables and Vopkenka’s principle, e.g. to try and get such a model.) – Sam Roberts May 13 at 15:13

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.
• Thanks, Gabe! That's just what I was looking for. I take it the easiest way to implement your idea is to let $\alpha$ be the limit of the first $\omega$ supercompacts. Then we get that $\beta$ is supercompact in $V_\alpha$ just in case it is supercompact simpliciter and thus that they are the first $\omega$ supercompacts from the perpsective of $V_\alpha$. – Sam Roberts May 14 at 7:06
• Yes, that works. And after all the theory "ZFC + infinitely many supercompacts" is not really much stronger than "ZFC + the existence of an ordinal $\alpha$ that is a limit of $\alpha$-Magidor supercompacts," which already implies the consistency of Zermelo set theory + infinitely many supercompacts. – Gabe Goldberg May 15 at 0:19