The set of real numbers in the closed interval [0, 1] is an uncountable set.

Explanation:
To understand why the set of real numbers in the closed interval [0, 1] is uncountable, we need to understand what it means for a set to be countable or uncountable.

A set is countable if its elements can be put into a one-to-one correspondence with the natural numbers (1, 2, 3, ...). In other words, a set is countable if we can list its elements in a sequence.

On the other hand, a set is uncountable if its elements cannot be put into a one-to-one correspondence with the natural numbers. In other words, a set is uncountable if we cannot list all its elements in a sequence.

To determine whether the set of real numbers in the closed interval [0, 1] is countable or uncountable, we can use a proof by contradiction known as Cantor's diagonal argument.

Cantor's diagonal argument starts by assuming that the set of real numbers in the closed interval [0, 1] is countable. This means that we can list all the real numbers in the interval in a sequence.

Next, we construct a new number by taking the digits in the diagonal of the sequence and changing each digit to a different digit. This new number will be different from every number in the original sequence because it will differ from each number in at least one digit.

Now, we have a new number that is not in the original sequence, which contradicts the assumption that the set of real numbers in the closed interval [0, 1] is countable. Therefore, the set of real numbers in the closed interval [0, 1] must be uncountable.

In conclusion, the set of real numbers in the closed interval [0, 1] is an uncountable set.