## Lambda calculus as a foundation of mathematics

Klaus Grue

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### Abstract

Church introduced lambda-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 $\kappa$-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.

Klaus Grue, June 22, 2011