Artemov (2019, The provability of consistency) offered the notion of constructive truth and falsity of arithmetical sentences in the spirit of Brouwer–Heyting–Kolmogorov semantics and its formalization, the logic of proofs. In this paper, we provide a complete description of constructive truth and falsity for Friedman’s constant fragment of Peano arithmetic. For this purpose, we generalize the constructive falsity to |$n$|-constructive falsity in Peano arithmetic where |$n$| is any positive natural number. Based on this generalization, we also analyse the logical status of well-known Gödelean sentences: consistency assertions for extensions of PA, the local reflection principles, the ‘constructive’ liar sentences and Rosser sentences. Finally, we discuss ‘extremely’ independent sentences in the sense that they are classically true but neither constructively true nor |$n$|-constructively false for any |$n$|⁠.

中文翻译：

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Peano算术中的构造真与假

Artemov（2019，一致性的可证明性）提供了建构性真相和算术句子虚假性的概念，本着Brouwer-Heyting-Kolmogorov语义及其形式化，证明逻辑的精神。在本文中，我们对弗里德曼的Peano算术常数片段的构造真与假提供了完整的描述。为此，我们将构造性虚假概括为| $ n $ | Peano算术中的构造破坏性，其中| $ n $ | 是任何正自然数。在此概括的基础上，我们还分析了著名的哥德林句子的逻辑状态：PA扩展的一致性断言，本地反映原则，“建设性”骗子句和罗瑟句子。最后，我们从“极端”独立的句子的角度来讨论它们，它们在经典意义上是正确的，但既不是建设性的，也不是| $ n $ | -对任何| $ n $ |⁠都构造错误。