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Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and "[A] ham sandwich is better than nothing" in a mathematical tone.

According to Darling, the former is equivalent to "The set of all things that are better than eternal happiness is "All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members.

However, there are very many things that 'have no members', in the set-theoretical sense—namely, all non-sets.

It is perfectly clear why these things have no members, for they are not sets.

Similarly, the product of the elements of the empty set should be considered to be one (see empty product), since one is the identity element for multiplication.

A derangement is a permutation of a set without fixed points.

In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced.

In fact, it is a strict initial object: only the empty set has a function to the empty set.

The intersection of two disjoint sets (two sets that contain no elements in common) is the null set.

For example: = The null set provides a foundation for building a formal theory of numbers.

By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for minimum and infimum.

In any topological space X, the empty set is open by definition, as is X.