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To find more digits of x, you would only have to see more terms of my sequence.
Now, behold: Your number x is not on my list. That’s because
(a) x is different from r1 because their first digits are different.
(b) x is different from r2 because their second digits are different.
(c) In general, x is different from rn because their n-th digits are different.
So, x is different from every number on my list. My list didn’t include all the numbers in
[0,1) after all. Since you can do this with any list, we conclude that it is impossible to
arrange the numbers in [0,1) in a sequence indexed by the positive integers. So, [0,1) is
uncountable. (End of proof.)
What’s going on is that the real numbers are actually a “bigger” set — a higher order of
infinity — than the integers. You can count the integers, but you can’t count the real
numbers.
Are there other examples of uncountable sets?
Yes, there are uncountably many examples. Since we care about sets of sets (or, to be
less confusing, collections of sets), let’s describe some uncountable collections of sets:
The collection of all intervals in R
The collection of all singletons (one-point subsets) in the
interval S = [0, 60)
The collection of all intervals of the form (-x, x) where x is positive
The Cantor set
The collection of all subsets of R3.
Why does this matter in a probability class?
The additivity axiom applies only to countable (or finite) collections of sets.
That second example above — the collections of singletons in S = [0,60) — is important
to us. In class we constructed a “uniform probability measure” on S, with the probability
measure defined by
P ( [a, b] ) = (b – a)/60
for every interval with b ≥ a. Singleton sets are intervals of the form [a, a], so they all
have measure (a-a)/60 = 0.
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