Two Sizes of Infinity

Start with the whole numbers: 1, 2, 3, ... You can never finish counting them, but you can always say which one comes next. An infinity you can line up and count off, one by one, is called countable.

Now take the real numbers between 0 and 1 — every decimal, including endless ones like 0.333... and 0.101101110... There are so many that you cannot line them up at all. This is an uncountable infinity, and it is strictly larger than the countable one. Same word, "infinite," but two different sizes.

Cantor's Diagonal Trick

Why can't you list the reals between 0 and 1? Cantor's diagonal argument (1891) shows it is impossible. Suppose someone hands you a complete list of them, written as decimals.

Now build a new number this way: make its first digit differ from the 1st number's first digit, its second digit differ from the 2nd number's second digit, and so on down the diagonal. The number you just made differs from every entry in at least one place, so it is missing from the list. The list was never complete — and no list ever can be.