The category of inner models (Q1868169)

Inner models are proper classes that are transitive models of ZFC. The author proposes to ``explore some aspects of the family of inner models from a category-theoretical perspective.'' The objects are the inner models that are parametrically \(\Sigma_{n}\)-definable and and the morphisms are the \(\Sigma_{n}\)-definable elementary mappings between them. (\(\Sigma_{1}\)-definability for both is the working assumption in this paper.) The author begins with two limit constructions in the category that are derived from the iteration of a non-trivial elementary embedding \(\pi\colon V\rightarrow M\), where \(V\) is the class of all sets. One is a well-founded direct limit, the other is an ill-founded direct limit. A combination of the two constructions leads to tree-like systems of inner models, indexed by \(\omega^{<\omega}\), in which some branches have well-founded limits and others have ill-founded limits. Since branches correspond to real numbers, every real can be associated with a well-founded or ill-founded limit through the tree of models. This is formalized in the notion of an embedding normal form, ENF, for a set of reals and its strengthened version, an embedding normal form with witnesses or ENFW. The existence of an ENFW for a set of reals \(A\) implies the determinacy of \(A\). Induced embeddings are defined and used to code information into diagrams of inner models. Finally the author indicates how ENFs for projective sets of reals can be built and used to prove the Martin-Steel theorem on projective determinacy.