Update: Rupert has withdrawn his claim. See the final bullet point below.
Rupert McCallum has posted a new paper to the mathematics arXiv
Rupert McCallum, The choiceless cardinals are inconsistent, mathematics arXiv 2017: 1712.09678.
He is claiming to establish the Kunen inconsistency in ZF, without the axiom of choice, which is a long-standing open question. In particular, this would refute the Reinhardt cardinals in ZF and all the stronger ZF large cardinals that have been studied.
If correct, this result will constitute a central advance in large cardinal set theory.
I am making this post to provide a place to discuss the proof and any questions that people might have about it. Please feel free to post comments with questions or answers to other questions that have been posted. I will plan to periodically summarize things in the main body of this post as the discussion proceeds.
- My first question concerns lemma 0.4, where he claims that $j’\upharpoonright V_{\lambda+2}^N$ is a definable class in $N$. He needs this to get the embedding into $N$, but I don’t see why the embedding should be definable here.
- I wrote to Rupert about this concern, and he replied that it may be an issue, and that he intends to post a new version of his paper, where he may retreat to the weaker claim refuting only the super-Reinhardt cardinals.
- The updated draft is now available. Follow the link above. It will become also available on the arXiv later this week.
- The second January 2 draft has a new section claiming again the original refutation of Reinhardt cardinals.
- New draft January 3. Rupert has reportedly been in communication with Matteo Viale about his result.
- Rupert has announced (Jan 3) that he is going to take a week or so to produce a careful rewrite.
- He has made available his new draft, January 7. It will also be posted on the arXiv.
- January 8: In light of the issues identified on this blog, especially the issue mentioned by Gabe, Rupert has sent me an email stating (and asking me to post here) that he is planning to think it through over the next couple of weeks and will then make some kind of statement about whether he thinks he can save the argument. For the moment, therefore, it seems that we should consider the proof to be on hold.
- January 24: After consideration, Rupert has withdrawn the claim, sending me the following message:
“Gabriel has very kindly given me extensive feedback on many different drafts. I attach the latest version which he commented on [January 24 draft above]. He has identified the flaw, namely that on page 3 I claim that $\exists n \forall Y \in W_n \psi(Y)$ if and only if $\forall Y \in U \psi(Y)$. This claim is not justified, and this means that there is no way that is apparent to me to rescue the proof of Lemma 1.2. Gabriel has directed me to a paper of Laver which does indeed show that my mapping e is an elementary embedding but which does not give the stronger claim that I want.
…So, I withdraw my claim. It is possible that this method of proof can work somehow, but some new insight is needed to make it work.”
-Rupert McCallum, January 24, 2018
Dear Joel, many thanks for starting this discussion. Hugh Woodin has convinced me that the current proof of Lemma 0.4 is wrong. I think I may be able to save it but for now I am retreating to just claiming a proof that super-Reinhardt cardinals are inconsistent. That version should be up there in the next 24 hours and I suggest we discuss whether I have established that first before going on to consider whether I can get stronger results.
OK, thanks for posting here! We’ll look forward to your new version. Meanwhile, Happy New Year!
Sorry arxiv.org is saying it will be up there on Wednesday at 1 a.m. GMT, but I’ll email it to Joel now.
I can post it here, if you like. Let me know. Or take your time; we can wait.
I think it should be fine to put it here until Wednesday. After that arxiv’s license requires that it should not be distributed elsewhere on the internet, so we’d better take it down then.
Great! I posted the revised version above. (I haven’t heard of such a restrictive license at the arXiv before.)
You, as the author, are free to do what you like with writings you hold the copyright in. The standard arXiv license means that *others* can’t take the arXiv version and post it elsewhere.
Okay great. We may as well keep it up here then if it makes life easier.
What is a super-Reinhardt cardinal?
For every $\alpha$ there is $j\colon V\to V$, non-trivial elementary embedding with critical point $\kappa$ such that $j(\kappa) > \alpha$.
In other words, a Reinhardt cardinal with an arbitrarily large target.
I guess I should probably clarify I want $j_n$ to be an elementary embedding which takes $\kappa’_i$ to $\kappa_i$ for $0\leq i\leq n$ and $\kappa_i$ to $\kappa_i+n+1$ for $0\leq i$. That’s probably not all that clear from the current write-up.
I mean $\kappa_{i+n+1}$ of course
Also there is a typo on the second line of page 2 where I say $Y$ instead of $Y’$.
I don’t understand the proof of the first lemma of the current draft. How is the embedding $j’$ constructed? Is the whole top of page 2 supposed to be a construction of $j’$? Or is $j’$ a fixed embedding that can be made to work for all $X’$?
Let me also point out one issue.
Whatever $j’$ is, it seems it must equal the restriction to $V_{\lambda+1}$ of the map $X’\mapsto X”$ that is implicitly defined on the top of page 2 (so essentially this can serve as the definition of $j’$). Running the construction starting with $X’ = \kappa_0’$, one would therefore expect $X” = \kappa_1’$, since $j'(\kappa_0′) = \kappa_1’$ is part of the property claimed to hold of $\langle \kappa_n’: n < \omega\rangle$. Note that $X = e(X') = e(\kappa_0')$ is just $\kappa_0'$ again, since all embeddings involved have critical point at least $\kappa_0'$ and $X$ is the pointwise image of some "glueing" of these embeddings on $\{\alpha : \alpha < \kappa_0'\}$. But then $X''$ is just $j(X)^U = \kappa_0'^U = \kappa_0'$, which looks wrong.
I think if one continues this sort of analysis, one will see that the map $j'$ given by the restriction of $X'\mapsto X''$ is not an elementary embedding at all. (It seems to move $\kappa_0' + 1$.) But I suspect I am misunderstanding the proof, so I don't want to continue too far down this path. Given the length of the current draft, I think it would make sense to expand the proof to include a more detailed construction of $j'$ and a proof that the construction works.
By the way, one can use Woodin's iterated collapse forcing starting with $j : V_{\lambda+2}\to V_{\lambda+2}$ in the ground model to force choice up to $\lambda$ while lifting $j$, so a $j: V_{\lambda+2}\to V_{\lambda+2}$ is equiconsistent with a $j: V_{\lambda+2}\to V_{\lambda+2}$ such that $V_\lambda$ is wellorderable. This means we do not need to focus on the issues in the second section right now.
Hi Gabriel, $e(\kappa’_0)$ is meant to be $\kappa_0$. I think this is the way it comes out but I could be wrong, I will check more carefully tomorrow. I will write up an explanation of what $j’$ is meant to be tomorrow morning.
To get the embedding $j’$, glue together $j_1^{-1}(j \mid V_{\kappa_0})$, and $j_2^{-1}(j \mid V_{\kappa_1})$, and so on. It is meant to work for all $X$.
This is basically the same complaint as before, rephrased for the new version of the paper. You claim on the bottom of page 3 that for $X’\subseteq V_{\bar \lambda+1}$, the embedding $e: V_{\lambda’ + 1}\to V_{\lambda + 1}$ is an elementary embedding $(V_{\bar \lambda’ + 1}, X’)\to (V_{\bar \lambda + 1}, X)$, where I think $X$ is defined to be $e[X’]$. This is not true. For example, take $X’$ to be the critical point of $e$.
Sorry about the $\bar \lambda$s. I meant: you claim that for $X’\subseteq V_{\lambda’ + 1}$, the embedding $e$ is elementarty $(V_{\lambda’ + 1},X’) \to (V_{\lambda+1},X)$.
Thank you. I think I may need a requirement that $X’\notin V_\lambda’+1$ then. Matteo Viale has very kindly been looking carefully over the latest draft and is not saying it is definitely wrong but he recommends that I should state that I am not currently in a position to state the truth of my theorem and need about one week to re-organise the write-up of the proof. This is probably good advice. I will take one week to do a re-write. In the meantime I am happy to continue to respond to any questions or comments about the current draft.
Regarding the proof of Theorem 2.1, why can one assume $\lambda$ is of the form stated above and in particular it has countable cofinality. See my related question asked in mathoverflow:
https://mathoverflow.net/questions/237662/the-axiom-i-0-in-the-absence-of-ac
Dear Mohammad, Suppose that there is an elementary embedding $j:V_{\lambda+2} \rightarrow V_{\lambda+2}$ for some ordinal $\lambda$. If we let $\lambda’$ be the least fixed point of $j$ above the critical point of $j$, we have $j:V_{\lambda’+2} \rightarrow V_{\lambda’+2}$.
I believe that the answer to your question on MathOverflow largely depends on whether it is possible to establish the Kunen inconsistency in ZF.
Thanks,,
I don’t see why $V_{\lambda+2}$ of N is defined.
You’re talking about the earlier draft? Gabriel has said Section 2 of that draft was unnecessary anyway. I was not claiming that $V_{\lambda+2}$ is a set in $N$, I was claiming it is a definable subclass.
I was referring to “January 3 draft” where $N$ is defined as $N=(HOD(j|V_{\lambda})^{H(\Theta)}$, where even it was not clear to me what $N$ is in fact.
Now you have changed the paper, and I have a possibly trivial question regarding the first three lines of the proof of Lemma 2.1. Why $\kappa$ has the reflection property $R$. How can you obtain the $\alpha$ for it?
Now it is clear to me how to prove it. Surry for such a trivial question.
Let $\kappa$ be such that there is a non-trivial elementary embedding $j:V_{\lambda+2}\rightarrow V_{\lambda+2}$ with critical point $\kappa$ where $\lambda$ is the first fixed point of $j$ above $\kappa$. Let $X \in V_{\lambda+2}$. The mapping $j$ has critical point $\kappa$ and is an elementary embedding from $(V_{\lambda+1},X)$ into $(V_{\lambda+1},X’)$ where $X’:=j(X)$. So if we let $\kappa_0:=\kappa$ and $\kappa_n:=j^{n}(\kappa)$, then there exists an ordinal $\alpha$ less than $\kappa_1$ such that there is an elementary embedding $j$ with critical sequence $\langle \alpha, \kappa_1, \kappa_2, \ldots \rangle$ and for all $X \in V_{\lambda+2}$ there is an $X’ \in V_{\lambda+2}$ such that $j$ is elementary from $(V_{\lambda+1},X)$ into $(V_{\lambda+1},X’)$. (Specifically, you can take $\alpha:=\kappa$ and then the property will hold.) Now because $j$ is elementary from $V_{\lambda+2}$ to $V_{\lambda+2}$ you can reflect this downwards, reflecting $\kappa_1$ to $\kappa_0$, $\kappa_2$ to $\kappa_1$ and so on.
Oh okay good you figured it out. 🙂
One issue I noticed in the January 7 draft, is that in lemma 1.2, he has the embedding $j_X$, but the map $X\mapsto j_X$ would seem to require the axiom of choice, since there could be many such embeddings.
Yes, that is true. Since I am working in $\textsf{ZF}$ I am not strictly entitled to say that a map $X \mapsto j_{X}$ exists, I believe that the argument does not use that assumption but it is convenient to use that notation. I could re-write the argument without using that notation but it would be more cumbersome. Thanks for picking up on this point.
In the second paragraph of lemma 1.2 (January 7 draft) he appears to be using DC in order to build the sequence of $j_n$’s. Is he working in ZF+DC, instead of just ZF as stated?
That is a good question, and I agree that I should have said something about that point, but remembering that we are allowed to use the assumption that $V_{\lambda}$ is well-ordered and that each $j_{n}$ can be built using an $\omega$-sequence of elements of $V_{\lambda}$, I think I can avoid using that assumption.
Avoid assuming DC, I mean.
The final paragraph is not written up very well at the moment, unfortunately. I start with a particular choice of $X$, and then construct my mapping $j’_{X}$ which I want to be the right elementary embedding for the set $X”$ downstairs, and then I suddenly change my mind about what $X$ is. The way to do it should be this. Construct my mappings $j_n$ and the mapping $e$ which are independent of $X$ or $X”$. Now choose a set $X”$, I want to find an elementary embedding that works for this set $X”$. Use the mapping $e$ to transport the set $X”$ upstairs to a set which we call $X$, find the corresponding mapping $j_{X}$, then construct my corresponding mapping $j’_{X}$ downstairs which will be the right mapping for the set $X”$. That is the way that it should have been written up. And the notation $j_{X}$ is merely a notational convenience, which is not meant to imply that I am actually in a position to assert the existence of the function $X \mapsto j_{X}$ given that I am not assuming the axiom of choice. And as Joel has pointed out, in my statement of my first lemma I should say that I need to assume either $DC$ or else that $V_{\lambda}$ is well-ordered, and if I can prove it on the assumption that $V_{\lambda}$ is well-ordered then that’s enough. I cannot currently see any other problems with the write-up. I apologise if it is difficult to understand at the moment, but I hope this version will be enough to make the argument clear, or at least clarify where it goes wrong if it does go wrong.
The revision does not substantially address the issue I raised. In the notation of the current draft, the problem is that $e$ may not be an elementary embedding $(V_{\lambda’+1},X ‘ ‘)\to (V_{\lambda+1},X)$, as claimed on page 4. A new counterexample to this claim that is not avoided by your modification is $X’ ‘ = V_{\lambda’+1}$, for which the corresponding $X$ is $V_\lambda\cup e[V_{\lambda’+1}]$, when it should be $V_{\lambda+1}$.
I don’t think an ad hoc modification is going to fix this. If extending $e$ is essential to your argument, which it seems to be, then I think this is a major problem.
Yes, all right. Well, that is a good criticism, and it is not obvious to me right now how to fix it. Thank you for your feedback, and I will think it through.
Okay so thanks Gabriel, do you suppose that if, instead of starting with $X”$ and lifting it up to $X$, I start with $X$, reflect it down to $X”$, and then I don’t change the value of $X$, I just keep it as is and then claim that $j$ is elementary from $(V_{\lambda’+1},X”)$ into $(V_{\lambda+1},X)$ because $X”$ arose from reflection of $X$? Is there any chance that that claim might be true and then in that way the argument can be saved? You could maybe start with some formula that is true in $(V_{\lambda+1},X)$ and argue that it must still be true in $V_{\lambda’+1},X”)$.
So given a value of $X”$, I pick some $X$ that gets reflected down to that $X”$, and then construct the corresponding $j’_{X}$ and say that this witnesses the reflection property that I want for $X”$. Can that work?
No, that is just another way of re-phrasing the problem that you raised, how do I pick the right $X$?
I think that you have identified an important gap in the argument. I do not know whether it is possible to fill this gap. I thank you for drawing this to my attention and will think about it more.
I thank Gabriel for drawing attention to this important gap in my argument, I think that the statement that I need to be true, namely that for any $X”$ we can find the right $X$, could conceivably be true given the assumptions we start with but maybe quite a lot harder to prove than I thought? But I have to admit that right now I really don’t know. I thank everyone who has taken the trouble to examine it so far.
Let $X” \in V_{\lambda’+2}$. We want to find the $X \in V_{\lambda+2}$ that makes the mapping $e$ elementary from $(V_{\lambda’+1},X”)$ into $(V_{\lambda+1},X)$. Remember that $U$ is the range of the mapping $e$ and that $A^U$ is what $A$ gets “reflected” to downstairs. Let $X=\{Y \in V_{\lambda+1}:Y^U \in X”\}$. This is how you go upstairs from $X”$ to $X$. Does this work? Or maybe we are at the point where people can no longer follow my line of thought and I need to do another re-write.
I would be interested to find out what paper of Laver’s Gabriel directed Rupert Mc’Callum to and why Laver’s paper shows that Rupert’s mapping $e$ is an elementary embedding but does not give the stronger claim that Rupert wants. Thanks in advance for that information.