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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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    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.2011-12-24

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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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    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.2011-12-24