λ-calculus as a foundation of mathematics

Church introduced λ-calculus in the beginning of the thirties as a foundation of mathematics and map theory from around 1992 fulfilled that primary aim. The present paper presents a new version of map theory whose axioms are simpler and better motivated than those of the original version from 1992. The paper focuses on the semantics of map theory and explains this semantics on basis of κ-Scott domains. The new version sheds some light on the difference between Russells and Burali-Fortis paradoxes, and also sheds some light on why it is consistent to allow non-well-founded sets in a ZF-style system. ∗DIKU, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen, Denmark, E-mail [email protected]