What is Numerability in Set Theory?

In the context of set theory, a set is said to be numerable if its cardinality (i.e., the number of elements it contains) is a countable infinite number. This means that the set can be well-ordered, meaning that it has a total order such that every non-empty subset has a least element.

For example, the set of natural numbers is numerable because it can be well-ordered: we can list out all the natural numbers in a sequence, and each non-empty subset (such as the set of even numbers or the set of multiples of 3) has a least element.

On the other hand, the set of real numbers is not numerable because it cannot be well-ordered. There is no total order of the real numbers that satisfies the above property.

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