Cantordiagonalizációs

Cantordiagonalizációs, often shortened to Cantor diagonalization, is a proof technique developed by Georg Cantor. It is most famously used to demonstrate that the set of real numbers is uncountably infinite, meaning it cannot be put into a one-to-one correspondence with the set of natural numbers. The core idea involves constructing a new element that is demonstrably not present in a given countable list of elements, thereby proving the list cannot be exhaustive.

The process begins by assuming a list or enumeration of all elements in a set, for example, all real numbers between 0 and 1, written in decimal form. Cantor's method then constructs a new real number by taking the first digit after the decimal point of the first number in the list, changing it to something else, then taking the second digit of the second number and changing it, and so on. This process creates a new number whose nth decimal digit is different from the nth digit of the nth number in the original list. This newly constructed number, by definition, cannot be present anywhere in the original list because it differs from every number in the list in at least one decimal place.

Therefore, the assumption that a complete, countable list of these numbers could be made must be false. This means the set of real numbers is larger than the set of natural numbers, hence uncountable. The Cantor diagonal argument has profound implications in set theory and is a cornerstone in understanding different sizes of infinity. It has also been adapted to prove other results, such as the uncountability of the power set of any set.