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What is the best way to explain that $\Bbb{R}$ is not countable assuming that the audience is formed of people who are not mathematicians?

I ask this because these days I'm in a debate with someone about this. I have presented him the Cantor diagonal argument, but it seems he doesn't understand the proof. I'm about to give up, since I think that he does not have the mathematical knowledge necessary in order to understand the problem, but before I do that I wanted to know if you had any such 'debate' how would you proceed?

Gibarian
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Beni Bogosel
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  • Does he know what countability is? And what does he understand/is he willing to accept? – Keaton Mar 12 '13 at 19:50
  • If I remember correctly, there is a pretty good video on numberphile. Also, does he understand the concept of countability? – Stefan Mar 12 '13 at 19:52
  • http://www.youtube.com/watch?v=elvOZm0d4H0 that's the numberfile video, great explanation. – fosho Mar 12 '13 at 19:54
  • Might be of interest: http://math.stackexchange.com/questions/160192/ . – dtldarek Mar 12 '13 at 20:00
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    Another video. At least the guy in the movie appears to have understood it better at the end. – user642796 Mar 12 '13 at 20:06
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    @Asaf: It also unfortunately reinforces the stereotype that math students are more interested in math than sex. – user642796 Mar 12 '13 at 20:18
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    Don't. Just... just don't. – Qiaochu Yuan Mar 12 '13 at 20:25
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    @Asaf: Yet this inconstancy is such / As you too shall adore; / I could not love thee, Dear, so much, / loved I not Forcing more. – Brian M. Scott Mar 12 '13 at 20:33
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    @Asaf: I thought that your answer was perfectly reasonable. I’ve done it a little differently, but not much. (And of course with mixed success.) – Brian M. Scott Mar 12 '13 at 20:38
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    The best comment/advice is, of course, Qiaochu's: just don't. I tried this once with my girlfriend and it didn't end well: I married her. Just don't. And if someone ever reads something about Cantor, cardinals and stuff and asks you about this then play it deaf, begin talking about politics, Santa Claus, religion or any other highly discussable issue and avoid getting into that. – DonAntonio Mar 12 '13 at 22:21
  • @DonAntonio I don't think that is a good advice. If I am unable to explain a topic to a genuinely interested person, then there is something wrong with my understanding (or) I am not motivating it in the right spirit. –  Mar 12 '13 at 22:55
  • The key word being "genuinely", @Marvis. Good luck with that. Anyway, I think some subjects in mathematics are not meant for non-mathematicias and, if pushed against the wall for whatever reason, one can try to make some things as easy to diggest as possible, but this would be, imo, as trying to describe the wonders of Bach, Schumann or Led Zeppelin to a deaf person. – DonAntonio Mar 12 '13 at 23:00

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You can play the devil's game with him. ( I have added this link since I could not find a better link explaining this. If someone, finds a better link feel free to edit.)

  • Haha I like it. I've never seen that before. Simplistic, but fun. – Keaton Mar 12 '13 at 20:06
  • @AsafKaragila As a first step, a non-mathematician is not really interested in a rigorous proof for the uncountability of real numbers. I believe, he only needs an appreciation of what uncountability means. Once he gains this appreciation and if he is further interested, he will automatically make efforts to understand and appreciate the proof. –  Mar 12 '13 at 20:12
  • In the link it is explained why $\mathbb{N},\mathbb{Z}$ and $\mathbb{Q}$ are countable. It is only mentioned that $\mathbb{R}$ is uncountable, with a reference to the Wikipedia article on Cantor's diagonal argument. – Martin Brandenburg Mar 12 '13 at 20:26