Inconsistency of Reinhardt cardinals in ZF+DC

Question

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) 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?

Answer 1

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 $\kappa$.

Answer 2

EDIT: Tidying this up to make it appear less "passive-aggressively" defensive on the good advice of a friend...

[paragraph deleted]

I should just clarify, I said that I've been waiting on a peer-reviewer's report for 19 months and it sounds like I'm complaining, but I'm not, these are times of COVID, and any time an expert volunteers to check your work the correct answer is always "Thank you, I really appreciate what you're doing". In addition, regarding P versus NP, Henry Towsner was most kind to volunteer his time and effort to give feedback on an earlier version, and I think he might have been a bit frustrated about perceived lack of clarity in the write-up, so I'm sorry about that, however I've addressed the concerns he raised, the current version is up there on Arxiv, I have not been made aware of why it is wrong, Annals of Mathematics have said "the editors will have a look and get back to you" but no more than that. And now, back to Ultimate-$L$ and inconsistency of Reinhardt.

Okay. I agree the version of the Ultimate-$L$ paper up there right now needs a bit of a re-write. Here's how the whole thing works. I'm going to open with a discussion of how I came up with the $\alpha$-tremendous and $\alpha$-enormous cardinals and why I think they're consistent.

First of all, check out this one which is recently accepted for publication.

https://arxiv.org/abs/1403.8058

You'll find me explaining there in some detail why I think belief in large cardinals up to a supercompact cardinal is justified (roughly speaking). It's a bit of a delicate matter whether I want to say that a prediction of consistency of a supercompact is epistemically warranted, I'll write a bit about that some other time. But, to the extent that we can speak of "intrinsic justifications" in set theory in the sense defined in that paper, yeah I am roughly saying I think a supercompact cardinal should be seen as intrinsically justified. I'm a little bit more nuanced in exactly how I state the three main theses of the paper. I also cite Victoria Marshall, I think if you take the same line of thought a bit further with reference to Victoria Marshall then it's pretty fair to say that $n$-huge cardinals are justified. But Victoria Marshall went all the way up to the point of inconsistency with the axiom of choice. So what went wrong.

So let's talk about what Victoria Marshall does when she motivates $n$-huge cardinals. She has a fixed sequence of length $n$ of "sub-universes" of the Universe $V$, right, and the idea is that you start with a formula $\phi$ and a parameter from your universe $V$ and you reflect downwards, going through your sequence of sub-universes, roughly speaking, and there's an elementary embedding which guides how your parameter gets reflected downward, and you're saying reflection like that always works, and that's equivalent to the ordinal height of your smallest sub-universe being an $n$-huge cardinal, or something. I'm being very vague here but it's all presented in detail in Victoria Marshall's "Higher Order Reflection Principles" which you can find on JSTOR, and I will be more precise when we get up to my own arguments, I explaining here how to motivate the $\alpha$-tremendous and $\alpha$-enormous cardinals.

So you see, when Victoria Marshall goes all the way up to $j:V_{\lambda+2} \rightarrow V_{\lambda+2}$, here's the problem. I'll be showing you shortly how you motivate the $\alpha$-tremendous cardinals assuming that Victoria Marshall's motivation of $n$-hugeness is okay, and then those ones rank-reflect I3 as shown in Section 2, yes I do believe the version of Section 2 that is currently up there is okay, let me know if anyone can see issues there. So if $\alpha$-tremendous cardinals are okay then I3 cardinals are okay. You're all right with $j:V_{\lambda+1} \rightarrow V_{\lambda+1}$ as well, because if you've got an I3 embedding which you use to reflect parameters in $V_{\lambda}$ then you're okay with saying it can reflect parameters in $V_{\lambda+1}$ as well because there's only one possible choice for how your parameter can be reflected. But when you go to $V_{\lambda+2}$ that's no longer the case, and that's why that much reflection is dodgy. You understand, at this point I am simply talking about vague philosophical arguments which I will write up properly in some future paper. Now we're up to how I came up with the $\alpha$-tremendous and $\alpha$-enormous cardinals and why I think they're well-motivated.

So for $\alpha$-tremendous you're assuming $\alpha$ is a limit ordinal greater than zero, and you've got a sequence of cardinals of length $\alpha$ less than your cardinal $\kappa$, and for any finite subsequence you can do reflection going down that sequence similar to what Victoria Marshall does. That's roughly the idea. The definition that's up there is the correct definition, and the arguments of Section 2 showing that you get something between I3 and I2 consistency-strength-wise are correct. Okay, let's go to $\alpha$-enormous.

In the definition that's up there, Definition 1.3, after I say "member of the $\kappa$th iterated HOD of $V_{\kappa}$", after that insert "and $k(S) \subseteq S$". Apart from that, definition that's up there is correct. I'm sure you don't want to hear any more philosophical waffle about why you should believe it's consistent, let's get up to some actual maths. If you can show it's inconsistent, you've killed it. I will defend Section 2 if anyone can poke holes in it, I think the version up there is fine, and let us totally ignore Section 3 and Section 4. My task is to expound on Section 5 and Section 6, and I'll do Section 6 first.

"It will follow from the results of this section together with known results about the Ultimate-L Conjecture that Ultimate-L so defined does not in fact depend on the choice of the sequence." Scratch that sentence, it might be dodgy. We'll do without it. We'll say this is a possibly non-unique Ultimate-$L$, which can be chosen to be $\Sigma_2$-definable without parameters if you want it to be.

"The necessary elementary embedding within the model can be constructed using arguments of Section 3." That was written when we had a different definition of $\alpha$-enormous. Now with the current definition of $\alpha$-enormous, don't even need to go back to Section 3, you've got the elementary embeddings you need in the inner model, it's fine.

"any supercompact cardinal is necessary hyper-enormous, and all necessary elementary embeddings for witnessing this do descend to the model Ultimate-$L$". And you've also got to note that all of the elementary embeddings you need for Magidor's characterisation of supercompactness would be available down there too.

"Well-known generic absoluteness results". What I'm talking about here is Theorem 14 of this one. It's in Section 3, on page 21.

http://logic.harvard.edu/koellner/QAU_reprint.pdf

Okay then. That argument which is up there on Arxiv, with all those modifications and clarifications. So yeah, what's wrong with it? I am happy to discuss. I guess if it falls down it will be the bit where I say I can find a universally Baire set which does what I want.

And proof of Theorem 5.5. Okay I just re-read it. Well, [un-named expert] said it's a probem that $e_{X,n}$ depends on $n$, or at least that's how I understood his point. What he said was "yes, you can do inverse limit reflection but it's not uniform" (roughly). Ah okay, got it, the elementary embeddings $e_{X,n}$ would have to fit together in the right way wouldn't they. No, I don't think I need that. It's all about the way my choice sets $C_{\alpha}$ fit together, I can get from one to the other using some member of a fixed family of elementary embeddings. That's really all I need. That's enough for saying "the function mapping $n$ to the chosen member of the equivalence class of $X$ is in fact eventually constant". But [un-named expert] thought there was a problem. I thought I saw why he thought there was a problem and now I don't see it, I think it's fine. I will defend the proof of Theorem 5.5 on request.

EDIT: See, here's the thing. Your parameter X is made up of $\omega$ many pieces from $V_{\kappa_0}$, $V_{\kappa_1} \setminus V_{\kappa_0}$, $V_{\kappa_2} \setminus V_{\kappa_1}$, $\ldots$, and yeah if you reflect those down with the different embeddings $e_{X,n}$, then sure a different embedding will give you a different reflected piece of your parameter. But you'll eventually be up the point where you're in the "tail" which all your different choice functions are going to leave unchanged. It's like, when I'm past a certain point, I reflect it down, use my choice function, then lift it back up again, and when you're past a certain point there's a certain "tail" of your countable set of ordinals which the choice function isn't going to touch. When $n$ is sufficiently big then $e_{X,n}$ is not going to modify the bit that's not in the tail, and the tail itself doesn't get changed by the choice function, so you're just reflecting down and lifting back up again. That's the trick. That's why I can say it's eventually constant. This is the bit that [un-named expert] convinced me I ought to clarify.