An incompleteness theorem via ordinal analysis

2021/09/20 by Walsh, James · 1 citation
#03F15 #03F40 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2109.09678

Abstract

We present an analogue of Gödel's second incompleteness theorem for systems of second-order arithmetic. Whereas Gödel showed that sufficiently strong theories that are Π⁰₁-sound and Σ⁰₁-definable do not prove their own Π⁰₁-soundness, we prove that sufficiently strong theories that are Π¹₁-sound and Σ¹₁-definable do not prove their own Π¹₁-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.

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• Ordinal Analysis of Well-Ordering Principles, Well Quasi-Orders Closure Properties, and Σ_n-Collection Schema

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