The supra-natural part of the IFF is called the IFF metashell. It consists of only three levels.
- The iff level is located at the very top of the IFF architecture just above the type level. It contains the tiny (5 terms) IFF-IFF namespace, which provides the terminology for the fundamental ideas of set and function. The IFF-IFF namespace represents a (directed) graph of abstract sets and functions.
- The type level is located just below the iff level and just above the meta level. It contains the medium-sized (∼ 450 terms) IFF-TYPE namespace, which provides typing terminology for the IFF-META namespace. The IFF-TYPE namespace represents a finitely-complete category of abstract sets and functions, and introduces the categorical representation of subsets and set membership for generalized elements.
- The meta level is located just below the type level and just above the natural part of the IFF. It contains the large-sized IFF-META namespace, which services the natural part consisting of the generic metalevels.The IFF-META namespace, following the adjunctive style of axiomatization, represents a topos of Cantorian feartureless abstract sets and functions.