用于实际分析的λ演算

IF 0.3 Q4 LOGIC
P. Taylor
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引用次数: 52

摘要

摘要Stone对偶是一种新的拓扑范式,它直接描述了可计算的连续函数,而不需要使用集合论、无限格理论或离散计算的先验理论。微积分中的每个表达式都表示连续函数和程序,其推理看起来非常像经典拓扑中连续函数和程序的简化形式。这是一般数学家对ASD的介绍,并应用于初等实数分析。这种语言被应用于中间值定理:连续函数在实线上的方程的解。众所周知,从数值和建设性的考虑来看,如果函数“徘徊”在0附近,则方程无法解,而切向解将永远找不到。在自闭症谱系障碍中,这两种失败,以及当它们存在时寻找方程解的一般方法,都可以用公开性的新概念来解释。这些零不是作为集合捕获的,而是由更高类型的模态操作符捕获的。与映射的browwer度不同,它们是自然定义的,并且(Scott)连续跨越参数方程的奇点。用连续函数来表示拓扑,而不是使用点的集合,导致了对开放和封闭概念的处理,这些概念非常接近格-(或德摩根-)对偶,而没有在直觉主义方法中发现的双重否定。在这里,紧凑性的二重性是公开性。场所理论中的会合和连接是不对称有限和无限的,而在ASD中它们具有明显和紧凑的索引。公开性取代了韵律属性,如总有界性和基数条件,如具有可数的密集子集。它还与构造分析中的定位性和递归理论中的递归枚举性有关。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A lambda calculus for real analysis
Abstract Stone Duality is a new paradigm for general topology in which computable continuous functions are described directly, 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, and the reasoning looks remarkably like a sanitised form of that in classical topology. This is an introduction to ASD for the general mathematician, with application to elementary real analysis. This language is applied to the Intermediate Value Theorem: the solution of equations for continuous functions on the real line. As is well known from both numerical and constructive considerations, the equation cannot be solved if the function "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 higher-type modal operators. Unlike the Brouwer degree of a mapping, these are naturally defined and (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than using sets of points leads to treatments of open and closed concepts that are very closely lattice- (or de Morgan-) dual, without the double negations that are found in intuitionistic approaches. In this, the dual of compactness is overtness. Whereas meets and joins in locale theory are asymmetrically finite and infinite, they have overt and compact indices in ASD. Overtness replaces metrical properties such as total boundedness, and cardinality conditions such as having a countable dense subset. It is also related to locatedness in constructive analysis and recursive enumerability in recursion theory.
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来源期刊
CiteScore
0.80
自引率
0.00%
发文量
2
审稿时长
35 weeks
期刊介绍: "Journal of Logic and Analysis" publishes papers of high quality involving interaction between ideas or techniques from mathematical logic and other areas of mathematics (especially - but not limited to - pure and applied analysis). The journal welcomes papers in nonstandard analysis and related areas of applied model theory; papers involving interplay between mathematics and logic (including foundational aspects of such interplay); mathematical papers using or developing analytical methods having connections to any area of mathematical logic. "Journal of Logic and Analysis" is intended to be a natural home for papers with an essential interaction between mathematical logic and other areas of mathematics, rather than for papers purely in logic or analysis.
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