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

In the von Neumann construction of the ordinals, 0 is defined as the empty set, and the successor of an ordinal is defined as , such that the Peano axioms of arithmetic are satisfied.

Many possible properties of sets are vacuously true for the empty set. The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Norwegian and Danish alphabets (and not related in any way to the Greek letter Φ).; it is not available in HTML/Unicode.

Null set was once a common synonym for "empty set", but is now a technical term in measure theory and describes a set that is not necessarily empty. In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements; therefore there can be only one set with no elements.

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.