In a note appended to the translation of “On consistency and completeness” (1967), Godel reexamined the problem of the unprovability of consistency. Godel here focuses on an alternative means of expressing the consistency of a formal system, in terms of what would now be called a ‘reflection principle’, roughly, the assertion that a formula of a certain class is provable in the system only if it is true. Godel suggests that it is this alternative means of expressing consistency that we should be interested in from a foundational point of view, and he gives a result that shows certain reflection principles to be underivable in extensions of elementary number theory under conditions significantly weaker than the Hilbert-Bernays derivability conditions. In this paper I shall discuss the background to Godel's result and the foundational significance he claims for it. Along the way, I shall present a new proof of the result which places it in an even more general context than the one considered by Godel in the 1960s.

中文翻译：

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哥德尔第三不完备性定理

在“论一致性和完整性”（1967）的译文所附的注释中，哥德尔重新审视了一致性的不可证明性问题。哥德尔在这里着重于表达形式系统一致性的另一种方法，根据现在称为“反射原理”的术语，粗略地，断言某个类的公式仅在系统中是可证明的真的。哥德尔建议，从基础的角度来看，我们应该对这种表达一致性的替代方法感兴趣，并且他给出的结果表明，某些反射原理在基本数论的扩展中在明显弱于希尔伯特的条件下是不可推论的-Bernays 可推导条件。在本文中，我将讨论哥德尔的背景 的结果以及他声称的基础意义。在此过程中，我将提出一个新的结果证明，将其置于比哥德尔在 1960 年代所考虑的更广泛的背景中。