A Classical View of the Intuitionistic Continuum
A Classical View of the Intuitionistic Continuum
复制标题
直觉主义连续体的经典观点
DOI：
10.1016/0168-0072(95)00050-x
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
J. Moschovakis
中科院分区：
文献类型：
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作者：
J. Moschovakis
斯蒂芬·科尔·克莱因本质上是一位建构主义者，他对待排中律的不可避免的应用就像二十世纪早期数学家对待选择公理的应用一样。他的工作递归fimctions和基础的直觉数学旨在建立建设性的观点作为一个有用的和可理解的细化经典。数可实现性解释由Kleene [4]为直觉主义数论发展，并由大卫纳尔逊[181]形式化，给出了理论的经典可理解的“外部模型”;这项工作补充了GGdel早期的否定翻译[11]，这表明直觉主义数论“只是明显比经典数论窄，实际上包含了它，尽管有一些偏离的解释。Giidel继续说：“直觉主义似乎只对分析和集合论引入了真正的限制;然而，这些限制是由于拒绝，而不是排除中间的原则，而是不断言定义引入的概念。为了沟通，并认为有效地为这些限制，这是必要的公理化的一个重要组成部分布劳威尔的分析和集理论，一项任务主要落在布劳威尔的学生阿伦Heyting。不幸的是，海廷的形式主义并不适合于与经典系统进行比较，他的分析使布劳威尔对他的“棒定理”的“证明”中一个伪装得很好的循环得以延续（见[2]，在第三版中更正）。由于布劳威尔认为连续统是一种结构，其元素并非都是单独可知的，但关于这一点（正是由于这个原因），人们可以得出一些有力的结论，从古典数学家收缩，克莱因[lo]揭示了公理性质的“酒吧定理”。他开发了一种可用的
Stephen Cole Kleene was at heart a constructivist who treated unavoidable uses of the law of excluded middle the way early twentieth-century mathematicians treated uses of the Axiom of Choice. His work on recursive fimctions and the foundations of intuitionistic mathematics aimed at establishing the constructive viewpoint as a useful and comprehensible refinement of the classical. The number-realizability interpretation developed for intuitionistic number theory by Kleene [4] and formalized by David Nelson [181 gives a classicaily comprehensible “outer model” of the theory; this work complements GGdel’s earlier negative translation[11, which shows that intuitionistic number theory “is only apparently narrower than the classical one, and in truth contains it, albeit with a somewhat deviant interpretation.” Giidel goes on to say,“Intuitionism appears to introduce genuine restrictions only for analysis and set theory; these restrictions, however, are due to the rejection, not of the principle of the excluded middle, but of notions introduced by impredicative definitions.” In order to communicate and argue effectively for these restrictions it was necessary to axiomatize a significant part of Brouwer’s analysis and set theory, a task which fell mainly to Brouwer’s student Arend Heyting. Unfortunately Heyting’s formalism did not lend itself to comparison with the classical systems and his analysis perpetuated a well-disguised circularity in Brouwer’s “proof” of his “bar theorem”(see [2], corrected in the third edition).Attracted by Brouwer’s view of the continuum as a structure whose elements are not all individually knowable, but about which (for this very reason) one can draw some powerful conclusions from which the classical mathematician shrinks, Kleene [lo] uncovered the axiomatic character of the “bar theorem.” He developed a usable