UNDERSTANDING GÖDEL'S INCOMPLETENESS THEOREM

Professor McCarthy studies logic, the philosophy of mathematics, and the foundations of semantics. In previous work, he has explored a number of generalizations of Gödel’s incompleteness theorem. Gödel’s theorem, which appeared in 1931, states that for each formal axiomatic system of a certain not-very-special sort there exist propositions that are undecidable within the system; that is, sentences in the language of the system that are neither provable nor refutable. Because the systems in question include those regularly used to provide an axiomatic foundation for classical mathematics, Gödel’s result has often been taken to indicate an intrinsic limitation of the use of the formal, axiomatic method in mathematics. Professor McCarthy has shown that an analog of Gödel’s phenomenon also arises in more general models of mathematical reasoning, incorporating various nondemonstrative inference structures. During his Center appointment, he will work on a book that extends these results and puts them into a general framework for the analysis of inductive inference. The book will also explore parallel generalizations of a number of more recent proofs of Gödel’s theorem.