Cantor's diagonal argument

A proof technique that defeats a proposed enumeration by constructing an object differing from the list’s diagonal entries. Applied to real numbers, it proves they are uncountable.

Countable set: A set whose elements can be matched one-to-one with the natural numbers. The argument challenges whether the real numbers can be listed in this way.
Natural number: An element of the familiar sequence 0, 1, 2, 3, and onward. Natural numbers index the entries in a proposed enumeration.
Georg Cantor: A German mathematician who founded set theory and developed the arithmetic of infinite cardinalities. Cantor used diagonal reasoning to distinguish the sizes of infinite sets.
Cantor's nested intervals proof: A proof of real-number uncountability using nested closed intervals that avoid successive listed numbers. It reaches the same conclusion without changing decimal digits.
Enumeration: An ordered listing of the elements of a set, often indexed by natural numbers. The proof assumes an enumeration and constructs a number that it cannot contain.
Bijection: A function that is both one-to-one and onto its target set. A set is countable when a bijection with the natural numbers exists.
Cantor's 1874 theorem: Cantor’s proof that the real numbers cannot be placed in one-to-one correspondence with the natural numbers. This earlier result established uncountability before the familiar diagonal presentation.
Rational number: A number expressible as a ratio of two integers with a nonzero denominator. Unlike all real numbers, the rational numbers can be enumerated.
Diagonalization: A method that builds an object differing from each object in a proposed sequence at a selected position. Cantor’s proof is a foundational example of this general method.
Real number: A number representing a point on the continuous number line, including rational and irrational values. The argument’s target is the full set of real numbers, not just a chosen list.

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