Editing Empty set (section)

=== Axiomatic set theory ===
In [[Zermelo set theory]], the existence of the empty set is assured by the [[axiom of empty set]], and its uniqueness follows from the [[axiom of extensionality]]. However, the axiom of empty set can be shown redundant in at least two ways:
*Standard [[first-order logic]] implies, merely from the [[logical axiom]]s, that {{em|something}} exists, and in the language of set theory, that thing must be a set.  Now the existence of the empty set follows easily from the [[axiom of separation]].
*Even using [[free logic]] (which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely the [[axiom of infinity]].