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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?

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  • $\begingroup$ Oh, you changed the question. I was about to post an answer to the earlier one. $\endgroup$ May 13, 2021 at 11:36
  • $\begingroup$ @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? $\endgroup$ May 13, 2021 at 11:49
  • $\begingroup$ 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. $\endgroup$ May 13, 2021 at 11:50
  • $\begingroup$ @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. $\endgroup$ May 13, 2021 at 11:53
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    $\begingroup$ @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.) $\endgroup$ May 13, 2021 at 15:13

1 Answer 1

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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.

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  • $\begingroup$ 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 $V_\alpha$. $\endgroup$ May 14, 2021 at 7:06
  • $\begingroup$ 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. $\endgroup$ May 15, 2021 at 0:19
  • $\begingroup$ Exactly! And it makes obtaining the relevant limit trivial. $\endgroup$ May 15, 2021 at 4:49

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