Koellner: Gödel’s Disjunction
UNIVERSITY SEMINAR ON LOGIC, PROBABILITY, AND GAMES
Peter Koellner (Harvard University)
5:00 pm, Friday, April 7th, 2017
716 Philosophy Hall, Columbia University
Abstract. Gödel’s disjunction asserts that either “the mind cannot be mechanized” or “there are absolutely undecidable statements.” Arguments are examined for and against each disjunct in the context of precise frameworks governing the notions of absolute provability and truth. The focus is on Penrose’s new argument, which interestingly involves type-free truth. In order to reconstruct Penrose’s argument, a system, DKT, is devised for absolute provability and type-free truth. It turns out that in this setting there are actually two versions of the disjunction and its disjuncts. The first, fully general versions end up being (provably) indeterminate. The second, restricted versions end up being (provably) determinate, and so, in this case there is at least an initial prospect of success. However, in this case it will be seen that although the disjunction itself is provable, neither disjunct is provable nor refutable in the framework.