As I'm just a layperson I don't understand the technicalities involved, but does the paper *New Large Cardinal Axioms and the Ultimate-L Program*, by Rupert McCallum (arXiv:[1812.03837](https://arxiv.org/abs/1812.03837)) prove the inconsistency of a Reinhardt cardinal? If so can someone explain how given the results of the paper one proves Reinhardt cardinals to be inconsistent in ZF+DC?



Inconsistency of Reinhardt cardinals in ZF+DC
Before explaining the results, there are some facts that should be noted:


1. Arxiv papers are not peer-certified scientific papers. It is very possible, and in fact very common, for false theorems to appear on Arxiv. They are endless papers detailing proofs of the inconsistency of $ZFC$ for instance.
2. Rupert McCallum has published, on arxiv, a number of papers containing theorems whose proofs were later discovered to be false, such as his proof of the inconsistency of super Reinhardt cardinals.
3. The main results contained in the paper are almost certainly **false**. This is because the problems related to $V=\text{Ultimate-}L$ are some of the most important problems related to set theory. If they were solved, every set theorist would be discussing the solution. The results would certainly not be relegated to a relatively unknown paper on arxiv.


Now, as to your main question. The paper **claims** to prove the inconsistency of the existence of a cardinal $\lambda$ and some non-trivial elementary embedding $j: V\_{\lambda+3}\prec V\_{\lambda+3}$ with $ZF+DC$. If this is true, then this also means the Reinhardt cardinals are inconsistent.


This is because, if $j: V\prec V$ is a non-trivial elementary embedding with critical point $\kappa$, and $\lambda=\text{sup}\{\kappa,j(\kappa),j^2(\kappa)...\}$, then $j(\lambda)=\lambda$. Therefore, $j\restriction V\_{\lambda+3}: V\_{\lambda+3}\prec V\_{\lambda+3}$ is a non-trivial elementary embedding with critical point $\kappa$.


