Abstract:This paper is an investigation of the relationship between Gödel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector that the relation is well-founded. We provide an alternative proof of this fact that uses Gödel's second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any , if the rank of is , then is the th admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of is .

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中文翻译：

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不完整和跳转层次结构

摘要：本文研究了哥德尔的第二个不完全性定理与跳跃层次的充分根据之间的关系。从Spector的经典定理可以得出，这种关系是有充分根据的。我们使用Gödel的第二个不完全性定理代替可容许序数理论，为这一事实提供了另一种证明。然后，我们从该结果中得出第二个不完全性定理的语义版本，最初是由于Mummert和Simpson造成的。最后，我们转向在这种有根据的关系中计算实数等级。我们证明，对于任何，如果的等级为，则为第t个可允许的序数。因此，假设有适当的大基数假设，则圆锥上的 是。

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