diagonalization argument

mathematics

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July 23, 2026

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The term "diagonalization argument" in mathematics primarily refers to a proof technique developed by Georg Cantor. This method was crucial in demonstrating Cantor's theorem, which states that the cardinality (size) of any set is strictly less than the cardinality of its power set (the set of all its subsets).

Cantor's diagonalization argument was instrumental in proving that not all infinities are equal. It showed that the set of real numbers is a larger infinity than the set of integers. This concept challenged the prevailing view that there was only one size of infinity. The argument involves constructing a new number that differs from every number in a proposed list of numbers, thereby proving the list is incomplete and the set is larger than initially assumed.12

In a different mathematical context, "diagonalization" also refers to the process of transforming a matrix into a form where all non-zero elements are on the main diagonal.34

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Cantor’s theorem
…a version of his so-called diagonalization argument, which he had earlier used to prove that the cardinality of the rational numbers is the same as the cardinality of the integers by putting them into a one-to-one correspondence. The notion that, in the case of infinite sets, the size of a…
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…a version of his so-called diagonalization argument, which he had earlier used to prove that the cardinality of the rational numbers is the same as the cardinality of the integers by putting them into a one-to-one correspondence. The notion that, in the case of infinite sets, the size of a…
Read More