Empty set

The empty set is a set with no elements. Its existence can be proven in Kripke-Platek set theory, even without collection, by applying \(\Delta_0\)-separation with a contradictory formula to an infinite set. The existence of an empty set may seem paradoxical to a beginner to set theory, yet it does not pose any definitional issues and is useful. In particular, the empty set is used in the Von Neumann ordinal system, in which it encodes the number 0. Also, the empty set is used in the formal definition of gap ordinals.

The empty set is denoted \(\varnothing\), \(\emptyset\) or \(\{\}\). When working with ordinals, it may be used interchangeably with \(0\).