On the Use of Impredicative Reasoning to Construct a Class of Partial Models of ZF Within ZF UNPUBLISHED DRAFT

Gödel’s incompleteness results show that if ZF is consistent, it is impossible to construct within ZF itself a single object (set) that represents a complete and precise semantic model of ZF. Nevertheless, it has also become clear since then that many kinds of partial models can be constructed within ZF that reveal interesting characteristics of ZF and related formal systems. We develop here one particular class of partial models of ZF, which rely on an extreme form of impredicative reasoning that nevertheless follow the accepted rules of set theory and first-order logic and are constructible within exactly the same variant of ZF as the one being modelled. Some study of these partial models yields interesting results.