The prevalent interpretation of Gödel’s Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of the underlying mathematical theorem depends on several arbitrary formalisation choices. In this paper I examine the theorem’s dependency regarding Gödel numberings. I introduce deviant numberings, yielding provability predicates satisfying Löb’s conditions, which result in provable consistency sentences. According to the main result of this paper however, these “counterexamples” do not refute the theorem’s prevalent interpretation, since once a natural class of admissible numberings is singled out, invariance is maintained.

中文翻译：

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哥德尔第二定理关于数的不变性

对哥德尔第二定理的普遍解释指出，一个足够充分和一致的理论并不能证明它的一致性。然而，如何证明这种非正式阅读的合理性并不完全清楚，因为基本数学定理的公式取决于几个任意的形式化选择。在本文中，我研究了该定理对哥德尔数的依赖性。我介绍异常编号，产生满足 Löb 条件的可证明谓词，从而产生可证明的一致性句子。然而，根据本文的主要结果，这些“反例”并不能反驳该定理的普遍解释，因为曾经有一个自然类可接受的编号被挑出，保持不变。