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This is maybe a soft question, I am not sure yet.

Anyway, I am delivering a 8 (+ 4 supervisions) hour course on 'basic set theory' for undergraduates : set notation, bijections, functions, count-ability, Schroeder-Bernstein theorem [basically extremely naive set theory].

Now, there are loads of ways to show $ \mathbb R$ is uncountable (or equivalenetly some interval of $ \mathbb R$ is uncountable) - perfect sets, diagnol argument, real numbers of (0,1) in binary, Schoreder-Bernstein with the power set of $ \mathbb N$ , ...

HOWEVER!

I - course mates included - found it mind blowing getting a theorem proving the rationals and irrationals were not equinumerous [barely 8 days into our first term at University]. I want to provide some heuristics supporting the claim - I read somewhere about throwing a 10 sided die and letting the faces produce some real in (0,1); for instance

throw 1: 9

throw 2: 0

throw 3: 5

...

Yields the number 0.905 ...

This at least supports the claim that we should expect to get more irrational numbers after throwing the die in an intuitive way - but infinity isn't intuitive! A similar argument might go:

Between 0 and 1 there is 1/2 , between 0 and 1/2 there is a 1/4, ... - surely we can find more rational numbers than natural numbers? EEEErrr uh oh! No we cannot.

Do you have some plausible heuristic to back up uncountability arguments for irrationals?

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  • $\begingroup$ Your die throwing illustration is good. I'd stick with that. At some point your students will need to confront the fact that infinity is weird. Also, the size of the irrationals is something we really can't get a good hold of. Remember that ZFC (i.e. standard set theory) does not pin down how "big" the irrationals are. $\endgroup$
    – Bill Cook
    Commented Dec 24, 2011 at 22:07
  • $\begingroup$ You might be able to go straight to the definition: take a number line, draw separate points representing the integers, then draw a whole thick line representing R. It's "obvious" that the thick line has more stuff in it than the separated points. (It might take more work to convince people that Z is the same size as Q. Clearly Q is at most Z^2 because every rational is in the form a/b, though, and seeing that Z^2 is the same size as Z isn't /too/ mind-blowing.) $\endgroup$
    – Lopsy
    Commented Dec 24, 2011 at 22:13
  • $\begingroup$ Yes, that sounds less incorrect (not that there is a correct way to deliver this material). I'll use Lopsy's argument first, then prove the claim using diagonal argument, then do the dice comparison. Do you know any questions that students may have after the lecture of such topics? $\endgroup$
    – Adam
    Commented Dec 24, 2011 at 22:34
  • $\begingroup$ If you look closely at all "loads of ways to show that $\mathbb R$ is uncountable" you will discover many (all?) of them boil down to diagonal argument. $\endgroup$
    – sdcvvc
    Commented Dec 24, 2011 at 23:43

1 Answer 1

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Heuristically speaking, we expect a set to be countable if each of its elements can be specified by a finite amount of data, and uncountable otherwise. This is because the set of finite sequences of elements from some countable set (say $\mathbb{N}$) is countable (exercise), but the set of infinite sequences of elements from some infinite set is uncountable (exercise). So

  • $\mathbb{Q}$ ought to be countable because rational numbers can be specified using a denominator and a numerator, but
  • $\mathbb{R}$ ought to be uncountable because real numbers have infinitely many digits which need not be related.

Of course to back up these heuristics requires standard proofs. There's a point at which one should stop looking for heuristics in terms of what one already understands and accept that a genuinely unfamiliar mathematical phenomenon is occurring that demands to be understood on its own terms.

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  • $\begingroup$ Isn't just paraphrasing the statement "given a finite/countable alphabet, the set of all finite sequences using said alphabet is countable. While this is a great way to back up if a set is, or is not, countable prior to exploring more, I definitely think it is non-intuitive to students two weeks into Higher education! It perhaps could be useful - it is a shame there are no guinea pigs to be found on stackexhange. $\endgroup$
    – Adam
    Commented Dec 24, 2011 at 22:31
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    $\begingroup$ It would be hard to improve on this answer. @Adam: The suggested heuristic is very good, but the last paragraph is probably even more important. $\endgroup$
    – Brian M. Scott
    Commented Dec 24, 2011 at 22:37

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