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$ , ...
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?