A note on the formalism-freeness of Gödel's L

(Logic, )

n his Princeton Bicentennial Lecture, Gödel set out the challenge of conducting a "Turing analysis" of the informal mathematical notions of definability and provability: i.e., finding a formalization of such notions that is natural and stable under formalism variations. In this talk, I shall examines Kennedy's project to establish the class of constructible sets, L, as a formalism-free characterization of definability in set theory, analogous to the role Turing-computable sets play for computability. One key component in her project is a confluence result in the paper Inner Models from Extended Logics by Kennedy, Magidor, and Väänänen (the KMV paper), stating that if L is constructed using definability in generalized logics extending first-order, then for a large class of logics, the resulting inner model is still L. Kennedy takes this result to be a successful test of L's formalism invariance.

On the contrary, I claim that Kennedy_s sense of "changes in formalisms" rests on a confused analogy: the confluence result in the KMV paper is best understood on a par with examining Turing machines equipped with oracles, rather than genuinely different formalizations. To support my claim, I shall provide a characterization of the class of inner models obtainable from generalized quantifier logics, which also partially answers a question posed in the KMV paper. Using my characterization, I will argue by way of reductio that if one accepts Kennedy's argument, then one is also forced to accept that every inner model of the form L_X_ is formalism-free, thus trivializing the concept.

Chair: Larissa Bolte

Zeit: 14:00-14:30, 11. September 2021 (Samstag)

Ort: SR 1.006

Zesheng Chen

(University of California, Irvine, United States)

Zesheng Chen (preferred name: Jason) is currently a third-year PhD student in the Department of Logic and Philosophy of Science at University of California, Irvine. His research focuses on generalized notions of effectivity, especially on how formalizations of such notions interact with set theory and philosophy. He graduated with a B.A. in Linguistics/Philosophy from University of California, where he also minored in Middle Eastern Languages. When he was younger, fitter, and more follicly blessed, he was the captain of the Shenzhen U15 soccer team, which is the youth team of his hometown in China.

In his spare time, he likes to photograph the stars, learn languages, and pour latte art. He maintains an account on the Chinese Q&A website Zhihu where he writes about set theory and philosophy for his 15k readers.

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