Why was this question about axiom of choice and the Wiles's proof of FLT deleted? Namely the question was as follows:

Does the Wiles's proof of Fermat's last theorem rely on axiom of choice?

[I heard that FLT can be proved without AC due to the Shoenfield absoluteness theorem. However, it is not clear whether the Wiles's proof of Fermat's last theorem relies on AC or not. Hence the title question.]

It is the same kind of questions as this one. I think it's unreasonable that the one is deleted while the other is not.

• As Francois said in a comment, the same question had already been closed on math.se. Why on earth did you think it would be acceptable here? – Andy Putman Dec 31 '13 at 4:27
• (and it was also a little dishonest to not link to the previous question on math.se; did you really think people wouldn't notice?) – Andy Putman Dec 31 '13 at 4:28
• @AndyPutman Why on earth did you think it would be acceptable here? There are two reasons. One is that MSE and MO are different sites. The other is that a similar question as mine was accepted in MO. – Makoto Kato Jan 1 '14 at 1:19
• @AndyPutman Oh, there's yet another reason. Anybody who has enough reps can vote to close a question for whatever reasons. So it's a subjective matter whether a question is closed or not. Hence a fact that a question was closed in MSE does not necessarily means that it should be closed in MO. – Makoto Kato Jan 1 '14 at 1:40
• @AndyPutman [did you really think people wouldn't notice?] Of course, not. I knew sombody like you who were watching my posts in MSE would find it out right away. – Makoto Kato Jan 1 '14 at 19:04
• The answers below make clear that the MO post at issue was problematic, and deserved whatever closings it received. But this meta thread is not itself a problem --- I would say this type of question ("why was my question closed") is one of the reasons for meta.MO --- unless I am unaware of a pattern of meta spamming. So I disagree with the down votes and closure votes here on this meta question. – Theo Johnson-Freyd Jan 5 '14 at 4:01
• @Theo: You're just unaware. – Asaf Karagila Jan 5 '14 at 14:09
• @AsafKaragila [You're just unaware.] What's your definition of meta spamming? – Makoto Kato Jan 5 '14 at 18:55
• @AsafKaragila Good to know, and I'm not surprised. It does occasionally happen that meta questions are down-voted for the wrong reasons, so I wanted to make sure that that wasn't happening here. – Theo Johnson-Freyd Jan 5 '14 at 19:42

As François points out in his MO comment on the deleted question, and as noted by Andy above, the same question was asked on MSE - math.stackexchange.com/questions/618147.

Currently the question is closed there (as a duplicate [correction: now as "too broad" after a second closure, as mentioned by Asaf in a comment below]; indeed there was another question which brought up the issue of FLT and AC, and it was given an answer very similar to the response of Andres indicated below). If anyone is interested in this, they probably should have a look at the MSE discussion. There is also a lengthy MSE meta discussion attached to it.

Of course you (Makoto) are welcome to open a discussion about the suitability of this question for MO. If other MO users would like to see this question opened here, then by all means they should weigh in with an answer or comment. But I would like summarize some of the relevant MSE comments (and one of the comments at MO, which some can't see):

• The question received substantial and authoritative response, notably by Andres Caicedo. Paraphrasing what he wrote (I ask the experts to correct me if I garble this in any way), for any universe $V$ that models ZF, FLT holds in $V$ iff it holds in the subuniverse of constructible sets $L$, because the statement of FLT is an arithmetic statement. But FLT is provable in ZFC (NB: without assuming an inaccessible cardinal according to all the experts; see the last discussion that Makoto links to). Therefore it holds in any model of ZFC, such as $L$, and hence it holds in $V$ as already mentioned. Since it holds in every model $V$ of ZF, it is provable in ZF by Gödel's completeness theorem. (This may be even too heavy-handed an explanation, but it's what I came up with.)

• Makoto edited his question to say he was not particularly interested in whether FLT could be proven without AC, but whether AC could be removed from Wiles's proof. I don't know what this could mean if Andres's response (which I thought was excellent) is deemed inadequate by Makoto. At some point Makoto wrote at MSE meta: "That a proof exists does not necessarily mean a human being can ever actually come up with it. Anyway, I'm asking whether AC can be removed from the Wiles's proof. Nobody has answered it yet." Andres responded to that a little later, and couldn't have been clearer: any proof of an arithmetic statement [my emphasis] like FLT, including Wiles's (which without doubt passes inter alia through some heavy machinery, going into the proof of his Modularity Theorem), can be formalized in ZF, and the mechanism for doing that is well-understood by set-theorists.

• In what I thought was a real clincher for this line of argument, MO user Yury cleverly summed things up in his MO comment: "As @FernandoMuro points out, it is extremely unlikely that the proof can be formalized in ZF without any changes. However, one can add just one sentence in the beginning of the proof to make it a proof formalizable in ZF: “all the formulas below are relativized to L (the constructible universe)”." It would be hilarious if Wiles had actually written a paper on FLT with that declaration appearing in section 1! His audience would think he'd gone mental. But note that such a meta-declaration would be in force all throughout the complicated argument, the Modularity Theorem, the whole schmeer in fact, and then just before the final QED which ends the proof of FLT, he brings in the remarks about arithmeticity and Shoenfield Absoluteness or whatnot. So his proof would be a few lines longer.

I think what I personally take away from that is that Andres (and Yury at MO, but this is not visible to users with < 10k rep) gave a very satisfactory mathematical response to the question as asked, as did André Nicolas in a very similar answer to the MSE question cited as duplicate. If Makoto (or anyone else for that matter) wants to ask a separate question about the use of AC in the proof of a (possibly) non-arithmetical statement, such as what might possibly appear as some lemma buried somewhere in Wiles's opus, then that would be an entirely different question. [It would not be reasonable to request that someone scour several hundred pages of Wiles's material just to ferret out such a sublemma -- that's the questioner's duty, and it's the issue underlying the "too broad" reason for the second closure.]

And now a true "meta" comment, since much of the above has technical mathematical content which is arguably not the real concern of MO meta (but which may have been necessary to fully establish the context). The decision of François to delete the question from MO was made no doubt in part because it's been answered already. But possibly also for the very pragmatic reason that, based on prior experience, many users will be completely put out by the prospect of seeing another discussion dragged out, arguing over the meaning of a question in a whole bunch of comment boxes, just as it currently played out at MSE. Personally, I think one gracious thing to do would have been to thank Andres for his terrific answer (please note he was one very cool cat throughout the entire discussion: utterly calm and professional in fact).

I don't understand why you (Makoto) don't do that more often: thank your respondents for their insights and seek to resolve issues with grace and dispatch. It is regrettable, but I'm afraid many here feel it is a waste of time trying to engage with you, because of patterns as well exemplified in the MSE thread and elsewhere, where discussions go on and on and are never resolved to your satisfaction. This is frustrating for people who are busy as it is. Changing people's perceptions about that will be hard, I'm afraid.

• +1000. Thanks for writing this. – Andy Putman Dec 31 '13 at 6:42
• The modularity theorem is $\Sigma^1_2$ (this is an overkill), so Shoenfield absoluteness applies to it as well. – Andrés E. Caicedo Dec 31 '13 at 7:39
• It should be pointed that the question was reopened from being a duplicate, and closed as "too broad" instead (after pointing out that the OP is looking for an answer regarding Wiles' proof). – Asaf Karagila Dec 31 '13 at 8:02
• @ToddTrimble You seem to misunderstand my question. The Wiles's proof involves a stronger result than FLT. Namely Shimura-Taniyama conjecture for semistable elliptic curves. Are you sure that it is provable without AC? – Makoto Kato Dec 31 '13 at 16:27
• You have now changed the question, Makoto, as I explained thoroughly in my answer. But see what Andres wrote in his comment above -- that is an answer to your present question. (I will not discuss the technical mathematics any further here. Take it up at MSE.) – Todd Trimble Dec 31 '13 at 16:44
• @ToddTrimble: You have well-justified our election of you with this post, and many others---Thanks! :-) – Joseph O'Rourke Jan 1 '14 at 2:07
• @MakotoKato Yes, I absolutely think Andres is correct, and I think you can find strong clues in the Wikipedia article on Absoluteness that you yourself linked to, specifically in the subarticle on the Shoenfield Absoluteness theorem, and the results on $\Sigma_2^1$ statements particularly. Andres is much better placed to give details than I am. If you post a respectful question at MSE and don't futz around arguing in comment boxes, I'll bet you could derive benefit from what he'd have to say. – Todd Trimble Jan 1 '14 at 3:38
• @MakotoKato Putting aside possible invidious comparisons between the reception of your question and the question about use of Grothendieck universes in (the backdrop to) Wiles's work, I'd guess the number of upvotes there has partly to do with the fact that there was already a history of grappling on this point. For example, there were discussions on the FOM board about this some years ago. Very often questions that touch on foundational matters (and that aren't somehow tainted by past histories of users) garner a lot of points, and these aspects converged to produce quite a lot of upvotes. – Todd Trimble Jan 1 '14 at 3:47
• Let me add, Todd, that the mention of the completeness theorem is somewhat distracting, as it makes it appear as if Shoenfield's absoluteness is a metamahematical trick. Though it needs some understanding of set theory, the argument is much closer to classical analysis. In fact, what one does is to associate to each $\Sigma^1_2$ set a tree representation. Shoenfield's absoluteness is simply the fact that a $\Sigma^1_2$ statement holds iff the associated tree is illfounded. (Cont.) – Andrés E. Caicedo Jan 5 '14 at 16:41
• My point in mentioning this is that I have seen the criticism that the translation-to-$L$ argument gives us "non-canonical" constructions, while the opposite is in fact true. One can think of the whole thing as showing that we have explicit, canonical choice functions in the case under consideration, so we are not actually making any appeals to the axiom of choice, since choice is provable in the relevant instances. (On the other hand, the proof is definitely non-finitary.) – Andrés E. Caicedo Jan 5 '14 at 16:45
• @AndresCaicedo: Dear Andres, If you're referring to my comments when you speak of "criisicm" about the translation to $L$-argument, maybe I can say that I didn't mean it to be so much of a criticism, than as an explanation as to why this explanation might be unsatisfying to number-theorists. As emerged in the comment thread with Todd at meta.MSE, it might well be that the replacement of Taylor--Wiles patching by a finitistic argument in the Taylor--Wiles argument might in fact be an explicit (and I would guess more elementary and naive) manifesation of the kind of argument you have been ... – Emerton Jan 5 '14 at 19:46
• ... explaining here; if so, I would find that pretty interesting (because it would expand my horizons on this point somewhat, among other things). But when I say that number-theorists would like a "canonical" explanation, what I mean is that they would like an explanation in number-theoretic terms. In this sense, whether one uses choice, diagonal arguments, and so on when conducting patching, or replaces them by finitistic arguments, or by canonical choice functions coming from reduction-to-$L$, one is still unsatisfied: one would like to see the objects being constructed by the patching ... – Emerton Jan 5 '14 at 19:48
• ... process as having a genuine number-theoretic meaning. (Indeed, I believe they do: they should be explained by ideas from $p$-adic Langlands; but this isn't the place to discuss that, I don't think --- I'm just mentioning it to indicate that the "canonical" description I (and maybe others) would like is very far removed from foundational issues and explanations.) So I guess there are two levels of reaction to the appearance of choice/compactness in TW patching --- the first is just the immediate question of whether it can be removed, and here the answer is "yes", by your general ... – Emerton Jan 5 '14 at 19:51
• ... argument, and by the explicit finitistic commutative algebra that was worked out soon after TW appeared. As I said, I would find it interesting to reconcile these. But there is also the more overarching issue of explaining the objects constructed by TW patching in more meaningful number-theoretic terms, and this is something that has to be addressed by number-theory; it is not an issues of foundations. You might then say that it doesn't have much to do with (eliminating) choice, and that would be fair; but I think that the appearance of choice in the initial construction, and the ... – Emerton Jan 5 '14 at 19:53
• ... initial reaction against it, was in part a reflection of a harder-to-articulate sentiment that the TW argument was proving something important, but using objects that appeared somewhat random and meaningless; a state of affairs that is not entirely satisfying. Anyway, sorry for the length of this comment; feel free to ignore it if it's not of interest. Best wishes, – Emerton Jan 5 '14 at 19:55

I would just like to record that I am disappointed about the way this question was handled here (and at MSE). In this respect I seem to be in a minority. (But I always have been with regard to closing and related matters, no matter what the topic.)

I work in the area related to Wiles's argument, and from time to time I find myself thinking about the role of choice in arguments, since it is implicitly used e.g. in the foundations of commutative algebra. I don't find the general argument of "reduction to $L$" very appealing; I would like to think that results in the area are constructive (in some vague sense), relying more-or-less just on Kronecker's dictum about the natural numbers, and not on more abstract foundations. Perhaps this is naive; nevertheless, I think it is real consideration for some number theorists. On the MSE threads, zyx made various remarks that seemed to echo my thoughts on these matters.

Incidentally, I have answered many of the OP's questions on MSE, and never been treated with anything but appreciation and respect.

Added: Here I discuss one concrete example of actual number theorists doing actual work to eliminate an apparently genuine appeal to choice in the Taylor--Wiles argument. It is well-known to those in the field, but may not be to those outside it. I think it would have made a good answer to the OP's question. (Especially because the idea that the Taylor--Wiles construction is non-canonical (which is closely related to it's apparent dependence on choice) is one that continues to impact developments in the field).

(Incidentally, the fact that choice can be removed by relativizing to $L$ is not satisfying in this context: the appeal to choice is related to a dissatisfying non-canonicality and infinitariness in the argument, and it these aspects of the argument, as much as the literal appeal to choice, that made people uneasy. One can argue about whether people should be dissatisfied, but that is an argument with the general culture of algebraic number theory, which I won't try to defend here.)

• Thank you very much for your feedback, Matthew. I think I can understand your reaction to the mathematics, but for various reasons don't think it would be appropriate to pursue this here at meta. I have seen Makoto treat some users with respect and regard. Sometimes he seems to ignore them. I have also seen not a few troublesome situations arise with his questions and comments; some of the history might not be easy to see. I prefer to leave it that and hope for "healing" with regard to his involvement at MO, but I repeat that I appreciate hearing your dissenting opinion on the matter. – Todd Trimble Jan 4 '14 at 6:51
• @ToddTrimble: Dear Todd, Thanks very much for your reply. Best wishes, Matthew – Emerton Jan 4 '14 at 13:47
• I don't think there was anything fundamentally wrong with the question itself. The question was deleted because the OP was using it to circumvent a regular process on MSE. If this process is over and the question will remain deleted there, then there is no problem undeleting here. – François G. Dorais Jan 4 '14 at 16:16
• @Francois: The question is no longer deleted on MSE. – Asaf Karagila Jan 4 '14 at 23:06