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In this paper we answer a question of Friedman, providing an !-separable model M of the fij-calculus. There therefore exists an ff-separable model for any ff 0. The model M permits no non-trivial enrichment as a partial order; neither does it permit an enrichment as a category with an initial object. The open term model embeds in M: by way of contrast we provide a model which cannot embed in any non-trivial model separating all pairs of distinct elements. 1 Introduction Separability is a recurring topic in the -calculus. It is usually defined syntactically; there is also an interesting model-theoretic definition. Say that a subset A of an applicative structure (X; \Delta) is separable if any function f : A ! X is realised by some f in X , by which is meant, that for all a in A, f(a) = f \Delta a. This idea first appears in work of Flagg and Myhill [FM]. They termed the concept "discreteness," employing a topological analogy; we prefer to extend the usual -calculus terminology....

Church introduced -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 -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 grue@diku.dk Contents 1 Introduction 3 1.1 Differences with Churchs approach . . . . . . . . . . . . . . . . . 4 1.2 Inclusion of non-functions . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Set abstraction versus -abstraction . . . . . . . . . . . . . . . . 5 1.4 Selection of well-behaved maps ....

Map theory, or MT for short, has been designed as an \integrated" foundation for mathematics, logic and computer science. By this we mean that most primitives and tools are designed from the beginning to bear the three intended meanings: logical, computational, and set-theoretic. MT was originally introduced in [17]. It is based on -calculus instead of logic and sets, and it fullls Church's original aim of introducing -calculus. In particular, it embodies all of ZFC set theory, including classical propositional and classical rst order predicate calculus. MT also embodies the unrestricted, untyped lambda calculus including unrestricted abstraction and unrestricted use of the xed point operator. MT is an equational theory. We present here a semantic proof of the consistency of map theory within ZFC + SI, where SI asserts the existence of an inaccessible cardinal. The proof is in the spirit of denotational semantics and relies on mathematical tools which reect faithful...