3 ms·
His initial explanation of uncountable infinity isn't exactly correct, and I'm afraid it will give people the wrong idea. He says that the real numbers are unco
by baddox 11y ago
His initial explanation of uncountable infinity isn't exactly correct, and I'm afraid it will give people the wrong idea. He says that the real numbers are uncountable, because even between 0 and 1 on the number line there are an infinite number of real numbers. But that is also true of the rational numbers, which are countable! After all, what is the smallest rational number larger than 0?
- baddox 11y agoHe does go on to explain diagonalization, which is of course a better way to demonstrate the uncountability of the reals.
- egonschiele 11y agoIf anyone's curious on how it is possible to list all the rational numbers, it would go something like this: 0, 1, -1, 2, -2, 1/2, -1/2, 3, -3, 1/3, -1/3, 2/3, -2/3, 3/2, -3/2, 4 ... at each step you list the numbers where the numerator and denominator are <= x. For example, if x = 2, we can count 1, -1, 2, -2, 1/2, and -1/2. Obviously it is an infinite list, but you can list them.
- baddox 11y agoMy favorite way to visualize the bijection is https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree.