When faced with a mathematical statement, the usual mathematician's response is to try to prove it, or prove that it is false. Surprisingly, there are some statements for which neither proof nor refutation is possible using the commonly accepted foundations of mathematics. Gödel (1931) showed that such statements exist, but the statements constructed in his proof were "artificial". In 1963 Paul
Cohen introduced a powerful method for proving that natural mathematical statements are neither provable nor refutable. In the 50 years since Cohen's work, a large number of natural-sounding questions from a wide variety of areas of mathematics have been shown to be unanswerable.
In this talk we will explain precisely what is meant when we say that a statement cannot be proved or refuted. We will not describe the technical machinery used to prove that a question is unanswerable; instead, we will survey some (hopefully surprising!) examples of undecidable statements.
Our examples will be drawn from core areas of "ordinary" mathematics, such as algebra, analysis, and topology. No background in mathematical logic will be required.
Free pizza will be available at 5:45 pm in the lounge of the sixth floor of Bahen Centre. Please join us at the Pour Girl Pub on College (191 College Street) after the seminar; we shall muster at 7:15 pm in the lounge, and walk to the venue.
**note later time**