Cantor's Diagonal Argument
Cantor's diagonal argument demonstrates that some infinities are larger than others. Imagine you have a list of all possible decimal numbers between 0 and 1. If you assume this list is complete, you can create a new number by changing the nth digit of the nth number in the list. This new number will differ from every number on the list, proving it’s not included. Thus, you cannot list all decimal numbers, showing that the set of real numbers is "more infinite" than the set of natural numbers, which can be counted.
Additional Insights
-
Cantor's diagonal argument demonstrates that not all infinities are equal, specifically that the set of real numbers is larger than the set of natural numbers. Imagine listing all possible decimal numbers between 0 and 1. Cantor showed that you can always create a new number by changing the nth digit of the nth number on the list, ensuring this new number isn’t on the list. This process proves that no matter how you try to list real numbers, there will always be some missing, indicating that real numbers are uncountably infinite, unlike the countably infinite natural numbers.