# Largest counter example to a mathematical conjecture **URL:** \ **Category:** Factual Questions\ **Created:** [November 19, 2009, 2:02am UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106 "2009-11-19T02:02:15Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 20, 2009, 1:36am UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/21 "2009-11-20T01:36:04Z")
Clearly, any number can be trivially given compact notation in _some_ system. Just say “Alright, that number? I’m going to represent it as ‘R’ from now on. There; just one letter to express it, in the particular notation system I just made up”. It’s only when you start asking more of the notation system than simply “Has a short name for this particular number” that there’s any kind of non-trivial design problem. If you don’t impose any further requirements/constraints, then it’s easy. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 20, 2009, 1:47am UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/22 "2009-11-20T01:47:11Z")
> [@](#): > > Is there any such thing as a number that is finite but “inexpressibly huge”? That is, it is so large that all human attempts at notation of it are hopelessly inadequate? Of course, but I obviously can’t give you an example. Consider, though: There is a finite amount of matter in the observable Universe, which can be arranged in a finite number of ways. Even if I could consider every possible arrangement of matter in the observable Universe to be a representation of a number, that’s still a finite list of numbers that can be represented. Any finite list of numbers must contain some largest element, so any number larger than that would meet your criterion. And it’s no fair saying that you could just use some different notation to make the same arrangement of the Universe represent a different number, since you need to specify your notation somehow, and whatever you’re using to specify the notation must itself be part of the observable Universe. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 20, 2009, 1:51am UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/23 "2009-11-20T01:51:44Z")
Well, much like in the thread linked below, let me just note there is a difference between “For every notation system, there is some number which lacks a short name within it” and “There is some number which lacks a short name within any notation system”. The former is obviously true and the latter is obviously false. But to say “You need to specify your notation system somehow” is to require an infinite regress… in what system should I specify my notation system? And in what system should I specify _that_ system? And so on. (I spoke a bit about the “Wittgensteinian” resolution of this issue [here](http://boards.straightdope.com/sdmb/showpost.php?p=11462324&postcount=27)) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 20, 2009, 2:01am UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/24 "2009-11-20T02:01:22Z")
> [@Indistinguishable](#): > > But to say “You need to specify your notation system somehow” is to require an infinite regress… in what system should I specify my notation system? And in what system should I specify _that_ system? And so on. Which is to say, if you really stick to your guns with the principle “You need to specify your notation system somehow” (and don’t just off the bat pick some particular notation system as privileged and not in need of specification), then _all_ descriptions become infinite: one must augment the notation N\_0 of a number with some notation N\_1 of the notation system in which N\_0 is to be interpreted, which must itself be augmented with the notation N\_2 of the notation system in which N\_1 is to be interpreted, ad infinitum. (And, of course, in the end this still doesn’t clear up ambiguity over exactly how all these notations are to be interpreted, as two people could still go on interpreting every link in the chain differently. Which just goes to show that this kind of requirement accomplishes nothing and isn’t the right way to think about what counts as notation.) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 20, 2009, 2:10am UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/25 "2009-11-20T02:10:47Z")
(If one does pick some particular notation system as privileged, then, of course, there are numbers which cannot be given short names within it, no matter what, even with such techniques as auxiliary definitions of new languages or what have you given in the privileged notation system. But this is just “In every particular notation system, there are numbers which lack short names” again. It’s not as though those numbers are intrinsically unwieldy to express in themselves; they’re just unwieldy to express relative to _that_ notation system. To the extent that privileging that system over others is arbitrary, so is the predicate of being easily expressed within it as opposed to within others.) ---
**Author:** ![Super\_Kapowzler](https://avatars.discourse-cdn.com/v4/letter/s/a698b9/32.png) [@Super\_Kapowzler](https://boards.straightdope.com/u/Super_Kapowzler)\ **Post date:** [November 20, 2009, 2:47am UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/26 "2009-11-20T02:47:24Z")
The number of thy answer shall be three. No more, no less. Four would be right out. As would be 2. ---
**Author:** ![Uncertain](https://avatars.discourse-cdn.com/v4/letter/u/6a8cbe/32.png) [@Uncertain](https://boards.straightdope.com/u/Uncertain)\ **Post date:** [November 20, 2009, 3:04am UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/27 "2009-11-20T03:04:38Z")
> [@Indistinguishable](#): > > [QUOTE=Uncertain] > > Yes, but there is a difference between a proposition that fails only for very large numbers and a proposition whose truth or falsehood has been settled by the discovery of a very large counterexample. What do you mean by this as relates to the conjecture I put forth? [/QUOTE] That your “conjecture” is the first, but not the second. Nobody (and certainly not the mathematical community) thought that this proposition might be true and then settled the question by discovering a large counterexample. The falsehood of the proposition follows from Peano’s axioms in just a few steps. In fact, if “conjecture” is to mean something more specific than “proposition” (as it is intended to), your “conjecture” is not, and was never, an conjecture at all. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 20, 2009, 3:09am UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/28 "2009-11-20T03:09:00Z")
I’ll grant you that no one ever thought the conjecture was true [well, I recall once as a very young child confidently stating something along such lines to my sister (though with an embarrassingly far more modest upper bound)…]. But, I say, the only reason everyone knows it to be false is because they all readily see how to discover a very large counterexample. So as far as I’m concerned, it’s both the first _and_ the second. 🙂 (And if we settle Goldbach’s conjecture by finding a counterexample, it will be both the first and the second as well) ---
**Author:** ![Uncertain](https://avatars.discourse-cdn.com/v4/letter/u/6a8cbe/32.png) [@Uncertain](https://boards.straightdope.com/u/Uncertain)\ **Post date:** [November 20, 2009, 3:46am UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/29 "2009-11-20T03:46:20Z")
> [@Indistinguishable](#): > > I’ll grant you that no one ever thought the conjecture was true Or even entertained the notion that it might be true. So it’s not a conjecture, and is disqualified on that ground alone. > [@](#): > > But, I say, the only reason everyone knows it to be false is because they all readily see how to discover a very large counterexample. So as far as I’m concerned, it’s both the first _and_ the second. 🙂 This is an attempt to rob a legitimate question of its meaning. And I can dismiss all claims of the form “all numbers are less than or equal to _n_”, without finding a counterexample of any of them. > [@](#): > > (And if we settle Goldbach’s conjecture by finding a counterexample, it will be both the first and the second as well) Sure, and that might become the answer to the OP. But your pseudo-conjecture is not a candidate. ---
**Author:** ![Sage\_Rat](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/sage_rat/32/399_2.png) [@Sage\_Rat](https://boards.straightdope.com/u/Sage_Rat)\ **Post date:** [November 20, 2009, 4:11am UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/30 "2009-11-20T04:11:19Z")
> [@Chronos](#): > > [Graham’s number](http://mathworld.wolfram.com/GrahamsNumber.html) is far, far larger. In fact, with Graham’s number, you don’t even really need the “serious mathematical context” qualifier, since it’s so big that non-mathematicians can’t even conceive of the mental gymnastics that would be needed to come up with a bigger number. (Graham’s Number) + 1 ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 20, 2009, 7:00am UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/31 "2009-11-20T07:00:02Z")
I don’t really care about whatever argument we seem to have going. However, out of curiosity: > [@Uncertain](#): > > And I can dismiss all claims of the form “all numbers are less than or equal to _n_”, without finding a counterexample of any of them. I’m sure you can, but I imagine the first idea that occurs to most people is “Well, there’s always n+1” (see above), which certainly is finding a counterexample. [And, of course, there are myriad ways to reword this sort of thing]. Still, what would your approach be? ---
**Author:** ![Uncertain](https://avatars.discourse-cdn.com/v4/letter/u/6a8cbe/32.png) [@Uncertain](https://boards.straightdope.com/u/Uncertain)\ **Post date:** [November 20, 2009, 3:17pm UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/32 "2009-11-20T15:17:21Z")
> [@Indistinguishable](#): > > I don’t really care about whatever argument we seem to have going. OK, good. I hate those pointless acrimonious arguments too. > [@](#): > > I’m sure you can, but I imagine the first idea that occurs to most people is “Well, there’s always n+1” (see above), which certainly is finding a counterexample. [And, of course, there are myriad ways to reword this sort of thing]. Still, what would your approach be? I may not have been clear. I didn’t mean that I could dismiss _any_ such proposition. I meant that I can, right now, simultaneously dismiss _every_ such proposition. The reasoning is obvious enough. It could be done by noting that every number has a successor, and for every number’s successor is bigger than that number. And that’s really all you’re doing in your example, but in a narrower way. Consider the following: Proposition: There is no number larger than 1,000,000,000!. “Counterexample”: (1,000,000,000!) + 1 Proposition: There is no number larger than the billionth prime “Counterexample”: The billionth prime plus one Proposition: There is no number larger than the first number for which Goldbach’s conjecture fails, if there is one “Counterexample”: (The first number for which Goldbach’s conjecture fails) + 1 In none of these cases have we found a counterexample in the sense of the OP. ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [November 20, 2009, 5:27pm UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/33 "2009-11-20T17:27:05Z")
> [@Sage\_Rat](#): > > (Graham’s Number) + 1 Or, indeed, (Graham’s Number)[sup](Graham’s Number)[/sup]. 🙂 ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [November 20, 2009, 5:34pm UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/34 "2009-11-20T17:34:35Z")
> [@Colophon](#): > > Or, indeed, (Graham’s Number)[sup](Graham’s Number)[/sup]. > > 🙂 Raised a Graham’s Number of times. :eek: ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 20, 2009, 7:03pm UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/35 "2009-11-20T19:03:56Z")
> [@Exapno\_Mapcase](#): > > Raised a Graham’s Number of times. :eek: Graham’s number -\> Graham’s number -\> Graham’s number. Those who are interested in discussing large numbers should check out my old thread [On large numbers](http://boards.straightdope.com/sdmb/showthread.php?t=149305). ---
**Author:** ![Arnold\_Winkelried](https://avatars.discourse-cdn.com/v4/letter/a/3d9bf3/32.png) [@Arnold\_Winkelried](https://boards.straightdope.com/u/Arnold_Winkelried)\ **Post date:** [November 20, 2009, 7:57pm UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/36 "2009-11-20T19:57:53Z")
Or, to steal an idea from [xkcd](http://www.xkcd.com/207/): calling the [Ackermann function](http://en.wikipedia.org/wiki/Ackermann_function) with [Graham’s number](http://en.wikipedia.org/wiki/Graham%27s_number). A(g64,g64) example of how quickly the Ackermann function grows (from the Wikipedia page) A(4,3) = (approximately) 10^(10^19727.78) ---
**Author:** ![MilTan](https://avatars.discourse-cdn.com/v4/letter/m/71e660/32.png) [@MilTan](https://boards.straightdope.com/u/MilTan)\ **Post date:** [November 20, 2009, 10:06pm UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/37 "2009-11-20T22:06:35Z")
> [@ultrafilter](#): > > Graham’s number -\> Graham’s number -\> Graham’s number. > > Those who are interested in discussing large numbers should check out my old thread [On large numbers](http://boards.straightdope.com/sdmb/showthread.php?t=149305). Oh yeah? (Graham’s number -\> Graham’s number -\> Graham’s number) -\> (Graham’s number -\> Graham’s number -\> Graham’s number) -\> (Graham’s number -\> Graham’s number -\> Graham’s number) So there 😛 ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 20, 2009, 10:45pm UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/38 "2009-11-20T22:45:30Z")
Bah! The largest number specifiable in [your fixed language of choice] in less than [your upper bound of choice] characters. Boy, it’d take you [longer than your upper bound of choice] to beat _that_! ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [November 20, 2009, 10:52pm UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/39 "2009-11-20T22:52:33Z")
> [@Indistinguishable](#): > > Bah! The largest number specifiable in [your fixed language of choice] in less than [your upper bound of choice] characters. Boy, it’d take you [longer than your upper bound of choice] to beat _that_! - Graham’s Number ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 20, 2009, 10:54pm UTC](https://boards.straightdope.com/t/largest-counter-example-to-a-mathematical-conjecture/518106/40 "2009-11-20T22:54:15Z")
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