Self-reference with negative types

Abstract

The universe of sets, V, is usually seen as an entity structured in successive levels, each level being made up of objects and collections of objects belonging to the previous levels. This process of obtaining sets and axioms for set theory can be seen in Scott [74] and Shoenfield [77].The approach we want to take differs from the previous one very strongly: the seeds from which we want to generate our universe of classes are to be the one-variable predicates (given by one-free-variable formulas) of the formal language we shall be using. In other words, any one-variable predicate of the language is to be represented as a class in our universe. In this sense, we view our theory as being about a self-referential language, a language whose predicates refer to objects which are predicates of the language itself.We want, in short, a system such that: (i) any predicate may be represented by an object to be studied by the theory itself; (ii) the axioms for the theory may be derived from the general principle that we are dealing with a language that aims at describing its own predicates; and (iii) the theory should be strong enough to derive ZFC and suggest answers to the existence of large cardinals and to the continuum hypothesis.An objection to such a project arises immediately: in view of the Russell-Zermelo paradox, how is it possible to have all predicates of the language as elements of the universe? This objection will be easy to deal with: we shall provide our language with a type structure to avoid paradox.