What is concrete incompleteness

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This talk is part of Lectures on Logic and Foundations of Mathematcs at Wuhan University starting from September 2020, organized by Wuhan University in China. • Logic and Foundations of Mathematcs: Lecture 22 • Title: What is concrete incompleteness?Speaker: Prof. Colin McLarty (Case Western Reserve University, USA) • Time: 2021-12-22, 9:00-11:30 Beijng time (UTC+8) • Interlocutor: Prof. Harvey Friedman (The Ohio State University, USA) • Organizer: School of Philosophy, Wuhan University, China • Abstract: • David Hilbert made a brilliant mistake. It was a mistake only a visionary genius and great mathematician could make. Hilbert believed: 1) There would be no limit to mathematical knowledge, and 2) Cantor’s transfinite set theory would be a powerful engine for that progress, and 3) formal axiomatics would be a decisive, productive organizing tool. Those were not mistakes. But he also believed we could justify infinitary mathematics by finite methods, since formal axiomatics makes it all a question about axioms, proofs, and theorems which are just finite strings of symbols. Henri Poincaré recognized the brilliance of the idea, and Hilbert’s success at using it in geometry, but argued that it could never express all of geometry, let alone all of mathematics. Kurt Gödel’s Incompleteness Theorems showed Hilbert was wrong to believe these questions about finite strings of symbols can all be answered by ordinary finitary methods. Notably, Gödel showed the consistency of Peano Arithmetic can be expressed in arithmetic, but then the Peano axioms cannot prove or refute it. This talk asks: How large is the gap between these Gödel sentences in logic, on hand, and the kind of things mathematicians are normally interested in on the other? The gap is not absolute, and Harvey Friedman finds many bridges between Gödel sentences and mainstream mathematics. Harvey Friedman, Albert Visser, and others prove more general and more systematic forms of Gödel’s Second Incompleteness Theorem. I discuss how this research may make the study of Incompleteness entirely continuous with ordinary concrete mathematical questions. • Slide: http://yongcheng.whu.edu.cn/webPageCo...

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