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arXiv · 2409.05973

A classification of incompleteness statements

Henry Towsner·James Walsh
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Abstract

For which choices of $X,Y,Z\in\{\Sigma^1_1,\Pi^1_1\}$ does no sufficiently strong $X$-sound and $Y$-definable extension theory prove its own $Z$-soundness? We give a complete answer, thereby delimiting the generalizations of G\"odel's second incompleteness theorem that hold within second-order arithmetic.

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Henry Towsner, James Walsh. 2024-09-09. A classification of incompleteness statements. https://doi.org/10.4153/s000843952510091x

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