The cardinality of a set is a measure of how many elements it contains. Two sets have the same cardinality if there is a one-to-one correspondence between them, meaning that every element of one set can be paired with a unique element of the other set and vice versa.

The interval [0, 1] of real numbers is the set of all real numbers between 0 and 1, including 0 and 1 themselves. Each real number in this interval can be written as an infinite decimal expansion, such as 0.123456789…, where each digit is between 0 and 91.

The cardinality of the interval [0, 1] of real numbers is uncountable, meaning that it is larger than the cardinality of any countable set, such as the natural numbers or the rational numbers. One way to show this is by using a technique called Cantor’s diagonal argument2, which proves that there is no way to list all the elements of [0, 1] without missing some.

Another way to show that the interval [0, 1] of real numbers has uncountable cardinality is by using the fact that any open interval (a, b) of real numbers has the same cardinality as (0, 1)3. This means that we can find a one-to-one correspondence between (0, 1) and any other open interval, such as (0, ∞) or (-∞, ∞). For example, one possible correspondence between (0, 1) and (0, ∞) is given by the function f(x) = -ln(x), which maps every number in (0, 1) to a unique number in (0, ∞) and vice versa4. Since we know that the set of all real numbers (-∞, ∞) has uncountable cardinality, it follows that (0, 1) and [0, 1] also have uncountable cardinality.

Therefore, the answer to your question is that the cardinality of the interval [0, 1] of real numbers is uncountable.