Logic Seminar: New results on incompleteness and ordinal analysis, by James Walsh
April 24, 2024
4:00 PM - 4:50 PM
James Walsh (New York University): New results on incompleteness and ordinal analysis
We present an analogue of Gödels second incompleteness theorem. Whereas Gödel showed that sufficiently strong theories that are $\Pi^0_1$-sound and $\Sigma^0_1$-definable do not prove their own $\Pi^0_1$-soundness, we prove that sufficiently strong theories that are $\Pi^1_1$-sound and $\Sigma^1_1$-definable do not prove their own $\Pi^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.
We will then turn to characterizations of ordinal analysis itself. One of the main goals of ordinal analysis is measuring the strength of theories by calculating their proof-theoretic ordinals. But in what sense do proof-theoretic ordinals measure the strength of theories? What is the attendant notion of strength? We provide some abstract answers to this question.
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Date posted
May 1, 2024
Date updated
May 1, 2024
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