90
$\begingroup$

The Whitney graph isomorphism theorem gives an example of an extraordinary exception: a very general statement holds except for one very specific case.

Another example is the classification theorem for finite simple groups: a very general statement holds except for very few (26) sporadic cases.

I am looking for more of this kind of theorems-with-not-so-many-sporadic-exceptions

(added:) where the exceptions don't come in a row and/or in the beginning - but are scattered truly sporadically.

(A late thanks to Asaf!)

  • 3
    There are many examples coming from Waring's problem: Every sufficiently large number is a sum of at most 7 cubes, for example. But, removing "sufficiently large" requires changing the 7 to a 9.2012-08-24
  • 1
    @Andres: Please allow me to count Waring's problem as *one* further example.2012-08-24
  • 134
    **Theorem:** All natural numbers are larger than $10$, except $0,1,2,3,4,5,6,7,8,9$.2012-08-24
  • 0
    Go ahead. Ellison's and Vaughan-Wooley surveys (both mentioned on the wikipedia page) are excellent.2012-08-24
  • 0
    @Asaf: Your theorem is not of **this kind**. (Forgive me for not having defined precisely what *this kind* actually is - just by examples.)2012-08-24
  • 11
    @Hans: I know it's not. I'm trying to prod the discussion into helping you shape the definition in your head, so you could write it up. If I thought this is something you're looking for, I'd post it as an answer.2012-08-24
  • 0
    @Asaf: If you know what I'm aiming for - could you **please** help me shaping it in my head, so I can write it up?2012-08-24
  • 2
    I just knew that you're not aiming for this. I'm not very good at reading minds in general... I also think that what I said above had to be said.2012-08-24
  • 0
    It would be so great if people were better at reading other people's minds ;-) @Asaf: I did hope you not only guessed what I was *not* aiming for but also in the positive. I was hopeful that my two examples were enough. But alas - they were not. For the moment, I am helpless.2012-08-24
  • 20
    @Asaf We need to add 10 to the list of exceptions, no?2012-08-24
  • 0
    Wouldn't theorems with no exceptions be in this list too? (Such as all integers are divisible by 1)?2012-08-24
  • 1
    Specifically, are we talking about theorems of the form "every $x$ with properties $A$ has property $B$, except for the following explicit list"? Or do you allow "for every $x$ with property $A$, all but finitely many $y$ that satisfy $B$ have property $C$" (where the list of exceptions can depend on $x$).2012-08-24
  • 1
    And what about "every $x$ with $A$ has $B$, except for finitely many" (but we don't necessarily know which, or even how many)?2012-08-24
  • 0
    @Hans: commenting on the answers you upvoted was _completely_ unnecessary. Please don't do this.2012-08-24
  • 5
    Actually all answers to [this question](http://math.stackexchange.com/q/178183/34930) should qualify.2012-08-24
  • 0
    @Qiaochu: I guess you are right. (I found it a not so bad way of bookkeeping: which answers meet the intention of the question. But I understand that you don't like this, and I will not do this anymore.)2012-08-24
  • 2
    @AsafKaragila: Of course your **theorem** is a fallacy. It fails for number **10**.2012-08-24
  • 3
    @ypercube: It was pointed out before. Sadly, MSE do not allow editing comments after some time, and there is no point in deleting and re-posting it again. I am very happy that you were able to find a mistake in my sarcasm, it is *very* important to do that!2012-08-24
  • 1
    @AsafKaragila, Have we not yet learned the perils of sarcasm?2012-08-24
  • 1
    @Steve: Who are you? [Sir Lancelot](http://www.imdb.com/title/tt0071853/quotes?qt=qt0470578)?2012-08-24
  • 2
    Somehow, a lot of answers (even Asaf's marvellous theorem) are of the type “no object is of low complexity, except from a finite number of example”, which is not as satisfying as the examples given by the OP.2012-08-24
  • 2
    Many of the answers are not about sporadic exceptions, but rather the first n cases are exceptions before the theorem actually begins to work.2012-08-24
  • 0
    I don't think my own answer about Heegner numbers is really a good example either. All class numbers are greater than 1 is not a great theorem in the context of this question. All it is saying is that the class number considered as a function f(n) is equal to 1 at these values of n and not equal to 1 at other values of n.2012-08-24
  • 0
    @AsafKaragila, Am not.2012-08-24
  • 0
    Someone *please* make this community wiki!2012-08-26
  • 0
    @SteveD: Uhh it's CW for **two** days now...2012-08-26
  • 0
    Weird, when I answered it wasn't indicated as such...2012-08-26
  • 0
    Interesting question. But, I think you've sort of contradicted yourself. If we take classical logic for granted, and a theorem has any exceptions, then such theorems have instances where they end up false. That implies contradiction, and thus we don't have a theorem in the first place. The statement of these theorems actually hold in *all* cases, e. g. with "all primes are odd except 2" holds in all cases since the cases are 3, 5, 7, 11 ... I'd suggest you write instead something like "theorems which come as easily described by a pattern which admits of a few exceptions."2012-09-18

37 Answers 37