While reading this morning’s post by a number theorist known to the world as *Persiflage*, the author of this blog decided to entertain the possibility that *Mathematics without Apologies* is a *real book* written by a *fictional author*. It’s the sort of thought that comes naturally to the author (of this blog) during his first extended visit to Yale since the years immediately following his PhD, when he used to come to New Haven to pursue his interest in comparative literature. (Naturally that was not his primary object of pursuit; but it was an interest he could always be sure would be gratified.) As of this morning, *Persiflage* has only got as far as the beginning of Chapter 5; had the author of this blog been consulted, he would have advised *Persiflage* to start with Chapter 7, which provides hints as to how the author became convinced of his own fictionality, even before his earliest visits to Yale.

*Persiflage* wants to convince his readers that the fictional author of *Mathematics without Apologies*, whom he describes as “a chimera,” holds real opinions. Out of respect for the genuine cultural critics with whom the author (of MWA) interacted, at Yale and elsewhere, the author (of MWA) cultivated the habit of assuring the reader that, whatever kind of fictional character he might be, he does not mistake himself for a scholar (as *Persiflage* was careful to note, though he seems to have found the disclaimer disingenuous). To illustrate how this works, the author (of this blog) turned straight to the article *Fiction* on the *Stanford Encyclopedia of Philosophy* in search of an answer to the question: do fictional characters have real opinions? If that’s what you want to know, don’t bother looking: you will not find the answer there. (You will find out how philosophers have addressed the tricky question of whether Casaubon in *Middlemarch* was a scholar, or more precisely the slightly less tricky question of whether “[Sherlock] Holmes is cleverer than any actual detective.”)

Maybe the answer is in one or more of the references listed in the bibliography. A true scholar would have refrained from comment until he or she had read all of these references, and continued reading until the relevant literature was exhausted; the author of MWA, in contrast, would have checked a few of the references that looked most promising; but the author of this blog has to finish quickly in order to get to this morning’s lectures at Yale, and therefore will offer a fictional opinion after having looked at *nothing* beyond what is written in the *Stanford Encyclopedia* article, which incidentally needs to be cited as follows:

Kroon, Fred and Voltolini, Alberto, “Fiction”,

The Stanford Encyclopedia of Philosophy(Fall 2011 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2011/entries/fiction/>.

This morning the safest opinion seems to be that the author of MWA is a *Meinongian object*; to quote the *Stanford Encyclopedia*:

Meinong (1904) thought that over and above the concrete entities that exist spatiotemporally and the ideal or abstract entities that exist non-spatiotemporally, there are entities that neither exist spatiotemporally nor exist non-spatiotemporally: these are the paradigmatic Meinongian objects that lack any kind of being.

Some readers may find it helpful to think about mathematical objects in that way; and *Persiflage* may find it helpful to think about MWA’s author as Meinongian, in order to find an ontological slot in which to file his opinions. I (the author of this blog) advise *Persiflage* to assume that both participants in the number theory dialogues reflect the author’s opinions and maybe then to check the *Stanford Encyclopedia* article’s bibliography to determine whether or not Meinongian objects are capable of ambivalence.

I (the author of this blog) do want to make one thing clear: under no circumstances, even Meinongian, would the author of *Mathematics without Apologies *ever want to leave the impression that Vladimir Voevodsky is any kind of “villain,” as *Persiflage* clearly thinks the author of MWA “definitely” thinks. Both authors (of this blog and of MWA) are sympathetic to Voevodsky’s motivations although neither one shares them. If Chapter 3 needs a villain, it’s the *Powerful Beings*, or at least those among them that exist spatiotemporally.

Jon AwbreyNothing of nonbeing comes to be,

nor does being cease to exist;

the boundary between these two

is seen by men who see reality.

— Krishna,

Bhagavad Gita, 2.16LikeLike

mathematicswithoutapologiesPost authorOn the other hand, here is a literal translation from http://www.bhagavad-gita.org:

In the unreal there is no duration and in the real there is no cessation; indeed the conclusion between the two has been analyzed by knowers of the truth.And another one from http://prabhupadabooks.com/bg/2/16?d=1:

Those who are seers of the truth have concluded that of the nonexistent there is no endurance, and of the existent there is no cessation. This seers have concluded by studying the nature of both.The original Sanskrit can be seen on these pages, with a phrase-by-phrase comparison with the translation. And so one can see that

satah(sorry I’m missing the diacriticals), here translated as “real” or as “being,” can also mean “of the eternal” and half a dozen other things. All of which just serves to illustrate the correctness of the verse in question.LikeLike

Jon AwbreyThanks for all that. My snippet was from the B.S. Miller rendering, and I did get an inkling while reading it of an Eleatic influence on the translator. Your recent mention of Arjuna sent me reeling back to some readings and writings I was immersed in 20 years ago. A few days’ digging turned up hard and soft copies of a WinWord mutilation of MacWord document that unfortunately lost all the graphics and half the formatting, but LibreOffice was able to export a MediaWiki text that I could paste up on one of my wikis. Traveling coming up so it may be another couple weeks before I can LaTeX what needs to be LaTeXed, but here is the link for future reference:

☸ Inquiry Driven Systems : Fields Of Inquiry

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Jacob LuriePossibly off topic… I strongly disagree with this sentiment:

“If someone told me today that Voevodsky had discovered an inconsistency in ZFC, I would care slightly less than if someone told me the Collatz problem had been solved, and care much less than if someone (trustworthy) told me that a serious error had been found in the proof of cyclic base change.”

I don’t really think mathematics needs foundations in order to function, or that those foundations need to be based on ZFC. But I -do- think that the axioms of ZFC consist of statements about sets that seem obviously true. So if Voevodsky or anyone else found an inconsistency in ZFC, it would mean that there was something seriously wrong with our intuition about sets. If someone discovered, say, that innocuous-looking impredicative definitions can lead to paradoxes, then huge swaths of mathematics would be due for a rethink.

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Tom LeinsterI don’t think an inconsistency in ZFC would “mean that there was something seriously wrong with our intuition about sets”, at least if “our” is interpreted as the large majority of mathematicians who couldn’t state the ZFC axioms if their life depended on it. Even the starting point of ZFC, that elements of sets are always sets, conflicts with most mathematicians’ intuitions: who truthfully thinks of e = 2.718…, an element of the set of real numbers, as being a set?

(I made this argument at more length here. And, incidentally, do you really find the full strength of Replacement to be obviously true?)

On the other hand, I agree with the spirit of your comment, in the sense that I agree with it if “ZFC” is replaced by a different set theory that better models what mathematicians actually do. In particular, if someone found an inconsistency in the Elementary Theory of the Category of Sets — which amounts to a bunch of statements about sets that mathematicians really do use every day — then that would really be spectacularly consequential.

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galoisrepresentationsDear Jacob, There are two points I am trying to convey here. The first is that the exact formulation of foundations has very little relevance to the way that I personally do mathematics, and that I find it hard to imagine that any modification of the way we conceive them [foundations] will have any effect (philosophically or practically) on how I think about the Fontaine-Mazur conjecture.

The second (related) point is that I find it inconceivable that anything will be discovered that

wouldrequire huge swaths of mathematics to be re-thought. Imagine a world in which Russell’s paradox was not discoved until 1950. Would that have made any difference to the progress of number theory in that period (or, more generally, to the vast majority of mathematics)? Perhaps you might argue that the context today is different, and that after 100 years of study, any issue with ZFC would necessarily have far greater consequences than the observation of Russell. I would certainly be willing to consider accepting that position, never having thought at all about ZFC. I would then simply modify my opinion as follows: if anyone claimed to have discovered an inconsistency in ZFC, It wouldn’t bother me at all, because I would assume they were wrong.LikeLike

Jacob Lurie@GaloisRepresentations

I agree with your first point, and with this statement:

“The second (related) point is that I find it inconceivable that anything will be discovered that would require huge swaths of mathematics to be re-thought.”

But for me, agreement with this statement is almost synonymous with a belief that ZFC is consistent. The possibility that ZFC might be inconsistent seems like much less of a worry than, say, the possibility that there might be an error in the proof of some important theorem in number theory or algebraic geometry. But I think that’s because it’s orders of magnitude less likely to occur: inconceivably unlikely, perhaps. If it were to occur, I think it would likely have a much larger impact on mathematical practice (of course, that might depend on the nature of the contradiction).

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Jacob Lurie@Tom,

“I don’t think an inconsistency in ZFC would ‘mean that there was something seriously wrong with our intuition about sets’, at least if ‘our’ is interpreted as the large majority of mathematicians who couldn’t state the ZFC axioms if their life depended on it.”

I disagree with this. Mathematicians don’t need to have studied ZFC to have intuitions about sets. Rather, I’d say that the axioms of set theory were arrived at by codifying those intuitions.

“Even the starting point of ZFC, that elements of sets are always sets, conflicts with most mathematicians’ intuitions: who truthfully thinks of e = 2.718…, an element of the set of real numbers, as being a set?”

The axioms of ZFC can be rephrased as statements about the category of sets and functions, so the question of whether ZFC is consistent isn’t tied to the conception of sets as things whose elements are also sets, or to the way that familiar mathematical objects such as e are encoded in set theory textbooks.

“And, incidentally, do you really find the full strength of Replacement to be obviously true?”

Yes, in the sense that it codifies an idea that I think most mathematicians wouldn’t really worry about using in practice. For example, I suspect that most mathematicians studying a Banach space B would be happy to consider the sequence of Banach spaces B -> B** -> B**** -> B****** -> … which embeds each term into its double dual, and to contemplate the direct limit. So I’d consider the fact that this construction can’t be carried out in ETCS (without additional axioms) to represent a mismatch with mathematical practice.

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Tom LeinsterJacob, I agree with the general principle that you can have an intuition about some mathematical object without having studied it or knowing axioms for it. I just don’t think most mathematicians have much intuition for sets

as construed by ZFC.To clarify, I’ll use the term “everyday sets” for sets as routinely manipulated and intuitively understood by most mathematicians. Of course, that’s a vague description, but hopefully not so vague as to be meaningless. And I’ll use the term “tree-sets” for sets as axiomatized by ZFC (because sets under ZFC are indeed trees of a sort).

Someone who claims that ZFC successfully models what mathematicians do would presumably claim that everyday sets and tree-sets are the same thing. I’d say not. For instance, it’s a mathematical fact that in ZFC, elements of sets are always sets. And it’s a sociological fact that as far as most mathematicians are concerned, elements (such as

e) of sets (such asR) are often not sets. That’s one basic difference between everyday sets and ZFC’s tree-sets.Incidentally, all this is entirely separate from the issue of how mathematical objects are encoded in set theory textbooks, or indeed anywhere else.

Agreed, and I wasn’t saying otherwise. I was mostly objecting to the idea that mathematicians’ very real intuition about everday sets is the same thing as intuition about ZFC’s tree-sets.

Here’s my position. An inconsistency in ZFC would, of course, be interesting. But an inconsistency in ETCS, a weaker system, would be

devastating, since those axioms encode aspects of ordinary mathematical practice.The difference between these two systems is replacement. It’s a fair critique of ETCS that it excludes replacement, and that replacement is sometimes wanted. But of course, you can easily add replacement to ETCS if you want.

Simple instances of replacement like your Banach one seem very natural. But the reason why I asked you about the “full strength” of replacement is that there are much more complex instances that could be mentioned. The least disruptive kind of inconsistency in ZFC would be one that used very complex instances of replacement — least disruptive, as so much of mathematics can be done without replacement at all.

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David Roberts@Jacob

do you really mean an arbitrary Banach space B? That’s a pretty big ask. I could imagine this construction being used (or useful!) in the case of Banach spaces that turn up in functional analysis, say separable ones, whereby this colimit may be constructed by other means. Given that functional analysis has a reasonably low consistency strength, one doesn’t need to pull out the full Replacement nuke for the (logical) mosquito of PDE theory (for instance).

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Richard SéguinI read the whole book, and did not get the impression that Voevodsky was being portrayed as a villain.

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volochIt’s parodies all the way down.

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mathematicswithoutapologiesPost authorNot the part about true scholars, at least.

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galoisrepresentationsYou’re overthinking it:

michael harris(small letters) is not aMeinongian object, he’s astraw man. That said, I expect to have completed the book by the time I arrive in Luminy, where this discussion can be continued, hopefully with the addition of a bottle of white wine from Cassis.LikeLike

mathematicswithoutapologiesPost authorA straw

person, please, made of fictional straw, like the Scarecrow in theWIzard of Oz.More than one bottle of wine will probably be necessary.

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DT@Jacob,

“But I -do- think that the axioms of ZFC consist of statements about sets that seem obviously true. So if Voevodsky or anyone else found an inconsistency in ZFC, it would mean that there was something seriously wrong with our intuition about sets.”

Do you think the same way about ZFC + large cardinals?

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Jacob LurieNo. If someone were to prove that measurable cardinals (say) were inconsistent with ZFC, I think it would be a surprising development to set theorists, but I don’t think it would suggest that there was anything wrong with the way most mathematicians think about sets.

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mathematicswithoutapologiesPost authorI stumbled on this quotation by John Burgess, from his recent

Rigor and Structure, while trying to determine what the fictional author of a book like MWA would plausibly think about the Axiom of Replacement:“So long as AC remained somewhat controversial, theorems depending on it were sometimes starred, like sports records set by athletes on steroids. Such starring of choice-dependent results has lapsed, but anything depending on assumptions beyond ZFC would still today be literally or metaphorically starred. Assumptions going beyond ZFC used in proofs must be acknowledged as hypotheses in the theorems proved; assumptions included in ZFC do not require special mention. Such is the policy (or so I have it third-hand) of the prestigious

Annals of Mathematics, for instance.”MWA quotes the set-theoretic section 1.2.15 of Jacob’s Annals of Math Studies book

Higher Topos Theory, in which he states his assumptions regarding strongly inaccessible cardinals. This is roughly consistent with Burgess’s hearsay information (the assumption is not literally restated as a hypothesis in every theorem ofHTT) and it is also consistent with the claim in MWA that the various approaches in the literature to defining infinity-categories can be seen as avatars of the theory of infinity-categories mathematicians need.So the fictional author of MWA must be agnostic with regard to the consistency of ZFC, with or without inaccessible cardinals, but strongly believes that mathematicians would find an acceptable replacement for Replacement, if the latter ever proved problematic in hitherto unexplored regions of intuition.

And just a note on the absurdity of the book trade: MWA has consistently been near the top of the amazon.co.uk ranking for books in Philosophy of Mathematics — right now it’s at #4 — whereas Burgess’s book, a genuine work of philosophy written by an authentic philosopher of mathematics (and which both I and the fictional author of MWA will have to read), stands only at #60.

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Jacob Lurie“but strongly believes that mathematicians would find an acceptable replacement for Replacement, if the latter ever proved problematic in hitherto unexplored regions of intuition.”

I agree with this, but not with the sentiment that such a development would be irrelevant to working mathematicians.

For comparison: mathematicians were working with infinite series for centuries before we had formal definitions of limits and convergence. I’m sure that they understood well that certain manipulations of infinite series are fine but that others can lead to contradictions. But an untrained intuition can’t be trusted to tell the difference, so precise definitions are good to have. The existence of divergent series isn’t regarded as an annoyance foisted upon us by pedantic logicians: instead, we regard awareness of potential pitfalls as an important component of a mathematics education.

If someone found an inconsistency within ZFC, I think the situation would be analogous. It would mean that there was something unreliable about our existing intuitions about sets. I think it would then be important to retrain those intuitions, so that we know exactly what is allowed and what isn’t. I read Persiflage’s original statement to imply that “real mathematics” takes place so firmly on one side that being able to pinpoint the line exactly is irrelevant in practice.

I disagree with this, because I think a lot of good ideas have come out of understanding exactly where the line is and walking very close to the edge without crossing it. Russell’s paradox obviously crosses into the realm of illegality, but essentially the same idea appears in Cantor’s proof of the uncountability of the real numbers (a fact which we use all the time) and in the proof of Godel’s incompleteness theorem. It’s easy to spot the illegality of the manipulation

1 = 1 + (-1 +1) + … = (1 + -1) + (1 + -1) + … = 0,

but Barry Mazur used the same idea in a different context to prove the topological Schoenflies conjecture.

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Max@Lurie

//…Russell’s paradox obviously crosses into the realm of illegality, but essentially the same idea appears in Cantor’s proof of the uncountability of the real numbers (a fact which we use all the time) and in the proof of Godel’s incompleteness theorem…

I think Russell’s paradox, as a constructive proof of inconsistency, very different from any metatheorems with formal proofs. This is concrete explicit paradox.

… but essentially the same idea appears in Cantor’s proof of…

If you meant the diagonal arguments, I do not think so:

From assumptions about existence of the actual infinity(actual existence of enumeration : N->N^N)

do not follow diagonal contradiction alike

f(m)(m)=f(m)(m)+1,

this is only one more impredicative definition for infinity (assumed in metatheory)

f(m)(m)=infty.

Infinity that exists in actually infinite table(graph of surjection N->N^N).

//… 1 = 1 + (-1 +1) + … = (1 + -1) + (1 + -1) + … = 0,

Looks like inconsistency of associativity/substitution.

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Yemon ChoiThis is meant to be a reply to David Roberts’s comment at June 5 2015.

David, I don’t see why requiring B to be separable gains anything, since the bidual B^{(2)} will often be non-separable anyway, and then we are taking B^{(4)}, and so on.

That said, I don’t know of any places where this particular colimit has proved useful, although the related simplicial (cosimplicial?) object coming from this (co?)monad was once something I meant to look into for other reasons.

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David Roberts@Yemon

>I don’t see why requiring B to be separable gains anything, since the bidual B^{(2)} will often be non-separable anyway,

and thus I betray the fact I’m not an analyst, by a long shot.

But here is a better way to put it: if the colimit didn’t exist, due to the failure of Replacement, and we needed it, then we could simply work in the category of ind-objects. Unlike the category of finite groups, where even though not all limits exist, but there is a handy larger category in which they do, we might be stuck with the fact the category of sets (by which I mean ZFC – Replacement sets!) without all colimits. Why not, then just pass to the colimit-completion?

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mathematicswithoutapologiesPost authorThis is mainly addressed to Persiflage, but anyone should feel free to answer.

Colin McLarty went to a lot of trouble to develop the basic structures of Grothendieck geometry at the level of finite order arithmetic. I don’t know what that means, and as far as I can tell he has not yet succeeded in proving Fermat’s Last Theorem without Grothendieck universes, though I may well be wrong and it does seem to be widely believed that it’s possible.

Suppose it could be proved somehow (I don’t know whether or not this makes sense) that the full Langlands correspondence could not be proved with ZFC alone, but required inaccessible cardinals. Or maybe any finitely enumerable collection of geometric Galois representations could be proved within ZFC to be automorphic, but no finite proof of the full Langlands correspondence could be constructed within ZFC. (Maybe a logician would see what I am suggesting is obviously impossible; but Manin once speculated in print, long ago, that FLT was undecidable.) I guess number theorists would not hesitate to add the missing axiom and would not worry about possible inconsistency with ZFC.

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galoisrepresentationsThe philosophical equivalent of proving FLT in first order logic would be to re-write the

Tractatuswithout using the lettere— it is a purely intellectual and aesthetic (Oulipo style) endeavor.As to your question, I would answer it as follows. There are a class of problems (certainly including Fermat’s Last Theorem, and at least some versions of Langlands reciprocity) which, if they are

false, are certainly disprovable in ZFC (by exhibiting an counter-example). Hence any argument that shows these results are independent of ZFC should be thought of as constituting a proof that these theorems are true. ZFC was designed (as Jacob said) to codify our intuitions; letting “decidability in ZFC” be the final arbiter of truth would be letting the tail wag the dog. Independence results of this form (if they exist) should be thought of as demonstrating the failure of ZFC to fully capture the arithmetic of the integers. (Matt E has made very similar remarks to me on a number of occasions). Of course, such independence results would be amazingly surprising, but they wouldn’t keep me up at night.LikeLike

mathematicswithoutapologiesPost authorI didn’t say anything about independence, I suggested a scenario in which any proof would provably require inaccessible cardinals. Your alternatives, I believe, would then be either to look for counterexamples or to try to construct a proof using more than ZFC. The former, I think, would not be a very effective use of your talents.

I would also like to be able to talk about codifying intuitions, which is why I’m trying to provoke philosophers to provide their seal of approval for that kind of talk. Maybe they can prove it’s impossible to talk consistently about intuitions. Would we then be in trouble?

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David RobertsMcLarty’s work, briefly, establishes that derived functor cohomology of toposes (e.g. sheaves on a scheme) does not require strong logical foundations. In particular, one doesn’t need all of ZFC. I believe McLarty maintains–based on a sketch of analysis of MacIntyre of the proof of FLT and comments by people like Brian Conrad–that the use of universes implicit in Wiles’ proof is really only related to the SGA treatment of cohomology. Similarly, as the Stacks project shows, one can get a lot done without using universes at all, in a more nuts and bolts way.

I would be happy to say that McLarty’s work eliminates universes from Wiles’ proof, though of course the end goal is to show FLT is a theorem of arithmetic alone.

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VoynichRegarding universes and the original proof of FLT, the comments here (http://mathoverflow.net/questions/35746/inaccessible-cardinals-and-andrew-wiless-proof), especially those of BCnrd, clear up the situation pretty well.

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DT@mathematicswithoutapologies,

I hope an independence result like that would be closely related to or even a consequence of a more interesting quantitative result; a Goodstein theorem to match the Gentzen theorem. Gentzen proved you can’t use the Peano axioms to prove that the set of finite rooted trees can be well-ordered — boring! But it roughly means that when you perform an induction on finite rooted trees, you’re likely to create quantities that are superexponentially large. Maybe that’s easier to sell to someone who doesn’t care about axioms.

Still hard to sell, though. The fact that that kind of induction argument is rare in mainstream math (say, the Langlands program) feels like a clue to me, at least I would like to hear a good explanation.

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Matthew EmertonAs far as I understand, proving that the Langlands correspondence can’t be shown in ZFC alone is the same as proving that ZFC + (not-Langlands) is consistent, which means that we could find a model (presumably rather exotic) of ZFC in which Langlands if false.

To show that the Langlands correspondence is independent of ZFC is to show that both ZFC + Langlands and ZFC + (not-Langlands) are consistent. Showing that ZFC + Langlands is consistent means that we can find a model of ZFC in which Langlands is true.

Assuming for a moment that the full Langlands correspondence is true, then it seems that in the scenario of the first paragraph, if it existed, the integers as defined internally in the model of ZFC under consideration would be a non-standard model of PA, in which some non-standard solutions to a Diophantine equation, or maybe a Diophantine equation with non-standard coefficients, was causing Langlands to fail.

My intuition for non-standard things is not solid enough to be exactly sure how to think about the scenario of the second paragraph. E.g. would allowing non-standard integers, or other non-standard constructions, arising from some non-standard model of ZFC, be able to make the Langlands correspondence become true, if it wasn’t already true in the standard world? I would guess not, just because it is pretty finitary in nature — in which case, the scenario of the second paragraph would imply the truth of the usual Langlands correspondence (i.e. for the standard model of PA).

(One overall difficulty of trying to think about this hypothetical situation is captured by the sentiment of this post on MO.)

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mathematicswithoutapologiesPost authorI don’t know that Maass forms and their generalizations have a lot to do with Diophantine equations, and they are also part of the Langlands program. But I can lower the price to make my point clearer. Suppose it could be proved (don’t ask me how) that no complete proof of Langlands functoriality in ZFC alone was possible in fewer than 10^{10^10} symbols, or however many you like, but that a 400-page reasonably comprehensible proof was available if we allow inaccessible cardinals. Would number theorists petition for a rule change?

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JSEYes, and we’d win.

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