diagonaalargument

A diagonaalargument, often translated as diagonal argument, is a proof technique used in mathematics and logic. It is most famously associated with Georg Cantor's proof of the uncountability of the real numbers. The core idea of a diagonaalargument is to assume that a certain set of objects can be enumerated or listed, and then to construct a new object that is demonstrably not in that list. This contradiction proves that the original assumption of enumerability must be false.

In Cantor's original proof, he considered an arbitrary list of all real numbers between 0 and 1. He then constructed a new real number by taking the first digit after the decimal point of the first number in the list and changing it, then taking the second digit of the second number and changing it, and so on, along the diagonal of the list. The resulting number differs from every number on the list in at least one decimal place, meaning it cannot be present in the list. Since the list was assumed to contain all real numbers in that interval, and we constructed a number that is not in it, the assumption that all real numbers in that interval can be listed must be false. This demonstrates that the set of real numbers is "uncountable."

The diagonaalargument is a powerful tool for proving the existence of infinite sets that are larger than others. It has been applied in various areas of mathematics, including set theory and computability theory, to establish limitations on what can be computed or enumerated. Its elegance lies in its directness and the clear contradiction it establishes.