The IFF Metastack

Overview

IFF Architecture

μ : metashell

ν : natural part

ο : object level

The IFF core axiomatization contains many terms, partitioned according to whether the term is a core concept, a diagram term, a (co)limit term or an exponent term. The IFF core axiomatizates finite limits of specific shapes at the type level and general finite limits at all lower levels. Finite limits are connected with Dedekind’s abstract definition of finiteness (Lawvere and Rosebrugh, 2003). Following the principle of conceptual warrant, terminology has been placed in the IFF core only when it is needed in the lower or more peripheral namespaces. All metalevel ontologies import and use, either directly or indirectly, the core meta-ontology. This includes the all meta-ontologies in the structural component of the IFF axiomatization.

The IFF core axiomatization is in adjunctive form. The generic level of the core plays a central role in the IFF axiomatization — this is the most referenced namespace in the natural part of the IFF. The kernel and finite limits namespace axiomatizations of the core rely heavily upon the metastack partial ordersfor the basic collections, respectively. The finite colimits namespace is a categorical dual to the finite limits namespace. The axiomatization for exponents and curry begins at the meta level of the metashell.

Figure 1: IFF Metastack Structural Framework

The collection of kernel namespaces in the IFF core, called the IFF metastack (Figure 1), represents a chain of toposes

Set  =  ‹Set₁ ⊂ Set₂ ⊂ Set₃ ⊂ … ⊂ Set_n ⊂ … ›

which anchors the entire IFF architecture. The argument for the metastack structure is based upon the Cantorian expansion.

Furthermore, the vertical structuring of the IFF metastack between metalevels n and n+1 (and between metalevel n and the IFF-META namespace of the IFF metashell) requires axiomatization for the metastack partial orders at metalevel n+2 (and the IFF-TYPE namespace of the IFF metashell).