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On a Question of Friedman
 Information and Computation
, 1995
"... In this paper we answer a question of Friedman, providing an !separable model M of the fijcalculus. There therefore exists an ffseparable model for any ff 0. The model M permits no nontrivial enrichment as a partial order; neither does it permit an enrichment as a category with an initial ob ..."
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In this paper we answer a question of Friedman, providing an !separable model M of the fijcalculus. There therefore exists an ffseparable model for any ff 0. The model M permits no nontrivial 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 nontrivial 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 modeltheoretic 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....
Functionality, polymorphism, and concurrency: a mathematical investigation of programming paradigms
, 1997
"... ii COPYRIGHT ..."
λcalculus as a foundation for mathematics
 Logic, Meaning and Computation, Synthese Library 305
, 2001
"... Church introduced λcalculus 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. ..."
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Church introduced λcalculus 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 κScott domains. The new version sheds some light on the difference between Russells and BuraliFortis paradoxes, and also sheds some light on why it is consistent to allow nonwellfounded sets in a ZFstyle system.
A kappadenotational semantics for Map Theory in ZFC + SI
 in ZFC+SI, Theoretical Computer Science 179
, 1997
"... 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 settheoretic. MT was original ..."
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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 settheoretic. 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...
Computing in Cantor’s Paradise With λZFC
"... Abstract. Applied mathematicians increasingly use computers to answer mathematical questions. We want to provide them domainspecific languages. The languages should have exact meanings and computational meanings. Some proof assistants can encode exact mathematics and extract programs, but formalizi ..."
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Abstract. Applied mathematicians increasingly use computers to answer mathematical questions. We want to provide them domainspecific languages. The languages should have exact meanings and computational meanings. Some proof assistants can encode exact mathematics and extract programs, but formalizing the required theorems can take years. As an alternative, we develop λZFC, a lambda calculus that contains infinite sets as values, in which to express exact mathematics and gradually change infinite calculations to computable ones. We define it as a conservative extension of set theory, and prove that most contemporary theorems apply directly to λZFC terms. We demonstrate λZFC’s expressiveness by coding up the real numbers, arithmetic and limits. We demonstrate that it makes deriving computational meaning easier by defining a monad in it for expressing limits, and using standard topological theorems to derive a computable replacement.
Dedekind completion as a method for constructing new Scott domains
"... Many operations exist for constructing Scottdomains. This paper presents Dedekind completion as a new operation for constructing such domains and outlines an application of the operation. Dedekind complete Scott domains are of particular interest when modeling versions of λcalculus that allow quan ..."
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Many operations exist for constructing Scottdomains. This paper presents Dedekind completion as a new operation for constructing such domains and outlines an application of the operation. Dedekind complete Scott domains are of particular interest when modeling versions of λcalculus that allow quantification over sets of arbitrary cardinality. Hence, it is of interest when constructing models of powerful specification languages and when using λcalculus as a foundation for mathematics. 1