A list can fail even when it never ends
Cantor’s argument turns a question about infinity into a test of whether a list is complete. Suppose someone claims to have listed every real number between 0 and 1, written as an infinite decimal: the first number on line one, the next on line two, and so on. The list may continue forever. The question is whether every number in the interval appears somewhere on it.
Build a number the list cannot contain
Look at the first decimal digit of the first number, the second digit of the second number, and the third digit of the third. Now choose a new digit at each position that differs from the digit you found. For example, if the selected digits are 4, 8, and 1, the new number might begin with 5, 9, and 2.
The resulting decimal disagrees with line one in its first digit, line two in its second, and line three in its third. More generally, it differs from every line at least once, so it cannot equal any number on the list. Yet it is itself a number between 0 and 1. That contradicts the claim that the list contained them all.
To avoid the oddity that some decimals have two spellings—such as 0.5000… and 0.4999…—the construction can choose digits that are never 9, such as 1 and 2. Then the new decimal has an unambiguous representation.
Why this makes one infinity larger
For a finite collection, adding an item eventually makes the list longer. With an infinite collection, “same size” has a more precise meaning: two collections have the same size if their members can be paired off one-to-one. Cantor showed that no such pairing can match the natural numbers—1, 2, 3, and so on—with all the real numbers between 0 and 1. The real numbers are uncountable: no sequence indexed by the natural numbers can list them all.
The punchline is not merely that the interval contains infinitely many decimals. It is that even an endlessly long numbered list has a specific limit: there will always be a real number left out.
