Cantor's Diagonal Argument¶

Theorem¶

Cantor's diagonal argument shows that there is no bijection between $\mathbb{N}$ and the interval $[0,1]$ .

Interpretation¶

If a list claimed to contain every binary expansion in $[0,1]$ , one can construct a new sequence by flipping the diagonal entries. This new sequence differs from the $i$ th listed sequence in its $i$ th digit, so it cannot be on the list. Strictly, a few numbers have two binary expansions ( $0.0111 \dots =$ $0.0111\ldots=0.1000\ldots$ $0.1000 \dots$ ), which is why each 0-1 sequence corresponds to a unique number, but not quite conversely. Every number has at most two expansions, so a list of all numbers in $[0,1]$ would still give a list of all 0-1 sequences, and the conclusion stands.

A list of decimal expansions r1, r2, r3 and so on with the diagonal digits highlighted; changing each diagonal digit builds a new number that differs from every number in the list.

The decimal illustration highlights diagonal digits $3,7,1,0,3$ , which can be changed to $4,8,2,1,4$ , giving a new number with prefix $0.48214\ldots$ . At every subsequent position, choose either $1$ or $2$ so that the chosen digit differs from the corresponding diagonal digit. The resulting expansion is neither eventually all $0$ nor eventually all $9$ , so it avoids the ambiguity of two decimal representations. It therefore represents a number different from every $r_i$ in the proposed list.

Key Results¶

The interval $[0,1]$ is uncountable.
The real numbers have strictly larger cardinality than the natural numbers.

Examples¶

From a list $s_1,s_2,s_3,\ldots$ , define $s^*$ by making its $i$ th digit different from the $i$ th digit of $s_i$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 1

Aliases: diagonal argument, Cantor's diagonalization, Cantor diagonalization, diagonalization argument, Cantor's diagonal method Symbols: natural numbers

Last updated: 20 October 2026 03:02