Seminarium KMMF "Teoria Dwoistości"

By Gödel's incompleteness theorem, for any reasonable (and reasonably strong) theory there exists an undecidable sentence, i.e. a statement in the language of the theory which the theory can neither prove nor disprove. Gödel's undecidable sentences concerned logical phenomena such as provability and consistency, but it gradually became clear that undecidability applies also to statements with a more evident mathematical meaning.For typical theories that attempt to axiomatize "all of mathematics", such as Zermelo-Fraenkel set theory, known undecidable statements tend to concern matters rather distant from the everyday experience of most mathematicians: the continuum hypothesis, the existence of very large cardinal numbers, and so on. Eventually logicians began to ask what sort of axioms are needed to prove basic theorems about more mundane mathematical objects such as e.g. natural and real numbers, continuous functions or separable metric spaces. I will talk about a research programme known as reverse mathematics, which was initiated almost 50 years in an attempt to answer such questions. I will be assuming that the topic is completely alien to most of the seminar audience, and I will attempt to make the talk at least somewhat understandable.