Logic Seminar
James Walsh
New York University
New results on incompleteness and ordinal analysis
Abstract: We present an analogue of Gödel’s 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.
Wednesday April 24, 2024 at 4:00 PM in 712 SEO