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A lambda calculus for real analysis
, 2005
"... Abstract Stone Duality is a revolutionary theory that works directly with computable continuous functions, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, but the reasoni ..."
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Abstract Stone Duality is a revolutionary theory that works directly with computable continuous functions, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, but the reasoning looks remarkably like a sanitised form of that in classical topology. This paper is an introduction to ASD for the general mathematician, and applies it to elementary real analysis. It culminates in the Intermediate Value Theorem, i.e. the solution of equations fx = 0 for continuous f: R → R. As is well known from both numerical and constructive considerations, the equation cannot be solved if f “hovers ” near 0, whilst tangential solutions will never be found. In ASD, both of these failures and the general method of finding solutions of the equation when they exist are explained by the new concept of “overtness”. The zeroes are captured, not as a set, but by highertype operators � and ♦ that remain (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than sets of points leads to
The Mathematical Import Of Zermelo's WellOrdering Theorem
 Bull. Symbolic Logic
, 1997
"... this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusi ..."
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this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his wellknown paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :
A FIXED POINT THEOREM FOR THE INFINITEDIMENSIONAL SIMPLEX
, 2006
"... Abstract. We define the infinite dimensional simplex to be the closure of the convex hull of the standard basis vectors in R ∞ , and prove that this space has the fixed point property: any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that ..."
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Abstract. We define the infinite dimensional simplex to be the closure of the convex hull of the standard basis vectors in R ∞ , and prove that this space has the fixed point property: any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner’s lemma. The fixed point theorem is shown to imply Schauder’s fixed point theorem on infinitedimensional compact convex subsets of normed spaces. 1.
SPERNER AND KKMTYPE THEOREMS ON TREES AND CYCLES
, 909
"... Abstract. In this paper we prove a new combinatorial theorem for labellings of trees, and show that it is equivalent to a KKMtype theorem for finite covers of trees and to discrete and continuous fixed point theorems on finite trees. This is in analogy with the equivalence of the classical Sperner’ ..."
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Abstract. In this paper we prove a new combinatorial theorem for labellings of trees, and show that it is equivalent to a KKMtype theorem for finite covers of trees and to discrete and continuous fixed point theorems on finite trees. This is in analogy with the equivalence of the classical Sperner’s lemma, KKM lemma, and the Brouwer fixed point theorem on simplices. Furthermore, we use these ideas to develop new KKM and fixed point theorems for infinite covers and infinite trees. Finally, we extend the KKM theorem on trees to an entirely new KKM theorem for cycles, and discuss interesting social consequences, including an application in voting theory. 1.
Université Catholique de Louvain The Brouwer Fixed Point Theorem for Intervals 1 Toshihiko Watanabe
"... Summary. The aim is to prove, using Mizar System, the following simplest version of the Brouwer Fixed Point Theorem [2]. For every continuous mapping f: � → � of the topological unit interval � there exists a point x such that f(x) = x (see e.g. [9], [3]). ..."
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Summary. The aim is to prove, using Mizar System, the following simplest version of the Brouwer Fixed Point Theorem [2]. For every continuous mapping f: � → � of the topological unit interval � there exists a point x such that f(x) = x (see e.g. [9], [3]).