The notion of noncommutative set is used for the intuition as the noncommutative analog of a set, as the von Neumann algebras or the ${\rm C}^*$-algebras are for the measurable or topological spaces. But unlike these notions of noncommutative topological or measurable space which are well-defined in the operator algebras framework, the notion of noncommutative set is not (yet) (well-)defined. See the post: [What's a noncommutative set?][1] [1]: http://mathoverflow.net/q/137520/34538