Extending Martín Escardó’s effectful forcing technique, we give a new proof of a well-known result: Brouwer’s monotone bar theorem holds for any bar that can be realized by a functional of type (ℕ→ℕ)→ℕ in Gödel’s System T. Effectful forcing is an elementary alternative to standard sheaf-theoretic forcing arguments, using ideas from programming languages, including computational effects, monads, the algebra interpretation of call-by-name λ-calculus, and logical relations. Our argument proceeds by interpreting System T programs as well-founded dialogue trees whose nodes branch on a query to an oracle of type ℕ→ℕ, lifted to higher type along a call-by-name translation. To connect this interpretation to the bar theorem, we then show that Brouwer’s famous “mental constructions” of barhood constitute an invariant form of these dialogue trees in which queries to the oracle are made maximally and in order.

中文翻译：

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高阶函数和 Brouwer 的论文

扩展马丁·埃斯卡多的有效强迫技术，我们给出了一个众所周知的结果的新证明：Brouwer 的单调条定理适用于任何可以通过哥德尔的 (ℕ→ℕ)→ℕ 类型的泛函来实现的条系统 T. 有效强制是标准层理论强制参数的基本替代方案，使用编程语言的思想，包括计算效果、单子、按名称调用 λ 演算的代数解释和逻辑关系。我们的论证是通过解释来进行的系统 T程序作为有充分根据的对话树，其节点在查询时分支到类型为ℕ→ℕ的oracle，并通过按名称翻译提升到更高的类型。为了将这种解释与 bar 定理联系起来，我们随后证明了 Brouwer 著名的 barhood“心理结构”构成了这些对话树的不变形式，其中对预言的查询被最大限度地有序地进行。