Map Theory: From Well-Foundation to Antifoundation

Map Theory is a powerful extension of type-free lamba-calculus (with only a few term constants added). Due to Klaus Grue, it was designed to be a common foundation for Computer Sciences and for Mathematics. All the primitive notions of first-order logic and set theory, including truth values, connectives and quantifiers, set-membership and set-equality, are interpreted as terms. All the usual set-theoretic constructs, including inductive data-types, get computational interpretations. Now, Grue’s version of Map Theory is founded, in the sense that it only considers mathematical sets or classes which are well-founded with respect to the membership relation. In [19], we have shown that it was possible to design an alternative version which takes non-well-founded sets into account, and allows for co-inductive reasoning over them. This new version opens the way to a direct representation of co-inductive data-types and of circular processes and phenomena in Map Theory. In this article, we give parallel presentations of the two versions of Map Theory and of their relations with ZFC. We also give a flavor of the proofs of their relative consistency with respect to the existence of a strongly inaccessible cardinal. These proofs take place inside the κ-continuous semantics, which is an extension of Scott’s continuous semantics (where κ = ω) to any regular cardinal κ.