图灵和克莱因之间
Betwixt Turing and Kleene
摘要：
图灵著名的“机器”模型构成了第一个直观上令人信服的实数计算框架。Kleene的计算方案S1-S9扩展了图灵的方法，并提供了一个框架，用于计算任何有限型的对象。已经提出了各种研究计划，其中高阶对象，如实数上的函数，被表示/编码为实数，以便使它们服从图灵框架。这是一个自然的问题，是否有任何显着差异之间的Kleene方法或图灵方法通过代码。连续函数被很好地研究在这方面，我们研究有界变差的函数，其中有至多可数多个点的不连续。一个中心结果是若尔当分解定理，即$[0, 1]$上有界变差函数等于两个单调函数的差。我们表明，对于这个定理和相关结果，Kleene方法和图灵方法通过代码之间的差异是巨大的，在完全二阶算术很容易出现在Kleene方法中，在Kleene的量词$\exists^3$的幌子。
英文摘要：
Turing's famous 'machine' model constitutes the first intuitively convincing framework for computing with real numbers. Kleene's computation schemes S1-S9 extend Turing's approach and provide a framework for computing with objects of any finite type. Various research programs have been proposed in which higher-order objects, like functions on the real numbers, are represented/coded as real numbers, so as to make them amenable to the Turing framework. It is then a natural question whether there is any significant difference between the Kleene approach or the Turing-approach-via-codes. Continuous functions being well-studied in this context, we study functions of bounded variation, which have at most countably many points of discontinuity. A central result is the Jordan decomposition theorem that a function of bounded variation on $[0, 1]$ equals the difference of two monotone functions. We show that for this theorem and related results, the difference between the Kleene approach and the Turing-approach-via-codes is huge, in that full second-order arithmetic readily comes to the fore in Kleenes approach, in the guise of Kleene's quantifier $\exists^3$.
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