One way to stratify the large cardinal hierarchy between I3 and I1 is by using second-order elementary embeddings. We may view $j\colon V\_{\lambda+1}\to V\_{\lambda+1}$ as a second-order embedding $j\colon (V\_\lambda,V\_{\lambda+1})\to (V\_\lambda,V\_{\lambda+1})$, and it is known that $\mathrm{I2}(\lambda)$ is equivalent to there is an $\Sigma^1\_1$-elementary embedding $j\colon (V\_\lambda,V\_{\lambda+1})\to (V\_\lambda,V\_{\lambda+1})$, or equivalently, $\Sigma\_1$-elementary embedding over $V\_{\lambda+1}$ to itself.
(A sentence is $\Sigma^1\_1$ if it is of the form $\exists X\subseteq V\_\lambda \phi^{V\_\lambda}(x)$.)


Laver proved that $\Sigma^1\_{2n-1}$-elementary embedding is also $\Sigma^1\_{2n}$-elementary for $n\ge 1$, and if there is a $\Sigma^1\_{2n+1}$-elementary embedding $j\colon V\_\lambda\to V\_\lambda$, then we can find $\lambda'<\lambda$ and a $\Sigma^1\_{2n-1}$-elementary $k\colon V\_{\lambda'}\to V\_{\lambda'}$.


We may turn our view to third-order embeddings $j\colon V\_{\lambda+2}\to V\_{\lambda+2}$, or alternatively, $\Sigma\_n$-elementary embeddings over $V\_{\lambda+2}$. Of course, full elementary embedding $j\colon V\_{\lambda+2}\to V\_{\lambda+2}$ is incompatible with $\mathsf{ZFC}$, but it does not automatically rule out all $\Sigma\_n$-elementary embeddings over $V\_{\lambda+2}$ to itself. On the other hand, coding Woodin's proof into $V\_{\lambda+2}$ seems to exclude $\Sigma\_n$-elementary embeddings over $V\_{\lambda+2}$ for some $n$ (maybe $n\ge 2$?)



> 
> **Question.** Is a $\Sigma\_1$-elementary embedding $j\colon V\_{\lambda+2}\to V\_{\lambda+2}$ compatible with $\mathsf{ZFC}$?
> 
> 
> 



The consistency of $\Sigma_1$-elementary embeddings $j\colon V_{\lambda+2}\to V_{\lambda+2}$ over $\mathsf{ZFC}$
The answer is no, it is not consistent. If $j : V\_{\lambda+2}\to V\_{\lambda+2}$ is $\Sigma\_1$-elementary, then $\mathcal U = \{A\subseteq P(V\_\lambda) : j[V\_\lambda]\in j(A)\}$ is a normal fine ultrafilter concentrating on $\sigma$ such that $\text{ot}(\sigma\cap \lambda) = \lambda.$ Then taking an ultrapower of the universe, we obtain an elementary embedding $j\_\mathcal U : V \to M$ with $j\_{\mathcal U}[V\_\lambda] = [\text{id}]\_{\mathcal U}\in M$ and $j\_{\mathcal U}(\lambda) = \lambda$, contrary to Kunen's inconsistency theorem.


For more information (and in the ZF context), see section 4.3 (Reinhardt ultrafilters) of my joint paper ["Periodicity in the cumulative hierarchy"](https://ems.press/content/serial-article-files/26941) with Farmer Schlutzenberg. The work in this section is for the most part due to Farmer.


