> This is similar to saying that, since the set of integers is infinite, it must contain Pi.
Not exactly. Pi has no chance of occurring in an infinite set of integers.
Not exactly. Pi has no chance of occurring in an infinite set of integers.
This, of course, is impossible.
Proof: let X[n] be the set of numbers with non-zero probability of being produced at the n'th trial. X[n] must be countable, since sum(X[n]) = 1 and the sum of any uncountably infinite set of non-zero numbers must be infinite.
Let X = union(X[n], n=0...infinity). X is countable, being the countable union of countable sets. The reals are uncountable. Thus, most real numbers will NOT eventually be produced.
(It's true, pi in particular could be in X, but the vast majority of numbers could not be in X.)
However, I don't see how the set of numbers found in an infinite number of trials would be countable.
Edit: I googled some about this and can say I definitely learned some math today..
Count the first element from set 1, first element from set 2, second element set 1, second element set 2, first element set 3, third element set 1, etc. Eventually you will count every element from every set.
You can't do that for the real numbers.