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It can't be $5$. And it can't be $4.\overline{9}$ because that equals $5$. It looks like there is no solution... but surely there must be?

user153965
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Tom
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    Why must there be one? – Tobias Kildetoft May 20 '13 at 20:09
  • I don't know, must there be at least a way to describe it? – Tom May 20 '13 at 20:09
  • Tom, you already answered your question. There is no nextdoor neighbor to 5. (That counts for both sides of 5) – imranfat May 20 '13 at 20:10
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    @Tom Not all subsets of the real numbers have a maximum, the set ${x\in \Bbb R: x<5}$ is one such instance. – Git Gud May 20 '13 at 20:10
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    Who says $x$ is a real number. I say $x$ is a number which is less than or equal to $4$ hence the greatest such $x$ is 4. Problem solved ;) – James S. Cook May 20 '13 at 20:11
  • This is one of those places where we confront the nature of infinite sets: there is always "room" between a chosen value of $x$ less than 5 and 5 itself to insert yet another number -- 4.9999, 4.99999, 4.999999, etc. – colormegone May 20 '13 at 20:12
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    If $x<5$ then $x<\frac{x+5}{2}<5$. So there is no greatest such $x$. – Thomas Andrews May 20 '13 at 20:13
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    What do you mean "must there still be a way to describe it?" There are lots of ways to describe things that don't exist. I can describe a moon made of green cheese, but that doesn't mean it exists. – Thomas Andrews May 20 '13 at 20:14
  • I was wondering if there was a clever way of writing it down with limits or something, so that it doesn't describe (or evaluate to) an actual number. But it does not seem so. – Tom May 20 '13 at 20:23
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    You might be interested in Vihart's video, where she briefly discusses surreal numbers and infinitesimals (an alternate number system where numbers actually CAN be infinitely close to $5$ yet larger than any real number $x < 5$): http://www.youtube.com/watch?v=TINfzxSnnIE&t=5m2s – Adriano May 20 '13 at 20:34
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    @Adriano Surreal numbers look interesting. So if $x$ is surreal then we can get closer to 5 than all reals with $\epsilon$ = { 4, 4.5, 4.75, … | 5 }. But we still have the same problem. { $\epsilon$ | 5 }, and so on... – Tom May 20 '13 at 21:10
  • Somehow the Berry paradox comes to mind... – TC1 May 20 '13 at 21:59
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    4 may be the only interesting solution to this problem. – Gary S. Weaver May 21 '13 at 01:53
  • @Adriano while the surreal numbers are interesting, they still do not contain a largest element strictly smaller than $5$ (no ordered field will be able to do so). – Tobias Kildetoft May 21 '13 at 13:28

3 Answers3

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There isn't one. Suppose there were; let's call it $y$, where $y<5$.

Let $\epsilon = 5 -y$, the difference between $y$ and 5. $\epsilon$ is positive, and so $0 < \frac\epsilon2 < \epsilon$, and then $y < y+\frac\epsilon2 < y+\epsilon = 5$, which shows that $y+\frac\epsilon2$ is even closer to 5 than $y$ was.

So there is no number that is closest to 5. Whatever $y$ you pick, however close it is, there is another number that is even closer.

Consider the analogous question: “$x < \infty$; what is the greatest value of $x$?” There is no such $x$.

MJD
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    To answer your analogous question, there is no such $x$ as a value, but $x = \infty-\frac1\infty$ would approach the limit quite nicely. – Gary S. Weaver May 21 '13 at 01:00
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    I brought that up only to illustrate the point that one can make up a set of properties that are not satisfied by any object. – MJD May 21 '13 at 01:12
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    Actually, if you wanted a simple way to describe this number, couldn't you just say $x = 5 - \frac1\infty$ ? – Darrel Hoffman May 21 '13 at 01:30
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    You can say whatever you like, but hardly anyone will understand what you mean. If you want a simple way to describe this particular collection of properties, then Tom's original description, the greatest $x$ that is less than 5, is perfectly clear. Your suggestion just obfuscates that. – MJD May 21 '13 at 01:38
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    @GaryS.Weaver What about $\infty - \frac1{2\infty}$? If that doesn't work, so many things are broken. – Christopher King May 30 '15 at 14:58
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The answer is $4$, assuming the domain of $x$ is $\Bbb Z$.

Christopher King
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If $x<5$ then $2x<x+5$ so $x<\frac{x+5}{2}$. Similarly, $x<5$ means $x+5<5+5=10$ or $\frac{x+5}{2}<5$. So if $x<5$ we have $x<\frac{x+5}{2}<5$, and therefore there is a larger number, $\frac{x+5}{2}$ less than $5$.

Basically, the average of two different numbers must be strictly between those two numbers.

Thomas Andrews
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