DescriptionIn 1873, Georg Cantor proved that there is no 1-1 correspondence between the integers and the real numbers. This and subsequent discoveries led him to the beginnings of modern set theory, an area which is fundamental to many branches of mathematics nowadays.Careless dealings with set theory, for example, forming the set of all sets, can lead to paradoxes, such as the one discovered by Bertrand Russell in 1901: Let R = { x | x ∉ x }. Then R ∈ R ⇔ R ∉ R. Such paradoxes led to a foundational crisis of mathematics, and David Hilbert suggested a programme to address these issues. The ultimate goal was to obtain a formalized system consisting of a few axioms, a proof that these axioms are consistent, and that any true mathematical statement could be derived from these axioms.In 1931, Kurt Gödel published two Incompleteness Theorems, which basically destroyed any hope that such a theory could be developed. The first states that for any axiomatic system powerful enough to describe natural numbers, there exists true statements about natural numbers which are not derivable from the axioms of the system. The second theorem, a corollary to the first, states that any such system cannot demonstrate its own consistency. Nevertheless, such axiomatic systems are still important today, but one has to live with the fact that one cannot prove any statement. Perhaps the most famous is the Continuum Hypothesis, which states: There is no set whose cardinality is strictly between that of the integers and that of the real numbers. Work of Gödel (1940) and Paul Cohen (1963) showed that the Continuum Hypothesis is independent of Zermelo-Fraenkel Set Theory with the Axiom of Choice, today the most commonly used axiomatic system for set theory, provided this system is consistent.In this project we will study some of the techniques of Gödel and others developed to prove these statements. The language of Mathematical Logic tends to be very formal, so an important aspect of the project will be to communicate the essential content to a third-year audience. Prerequisites2H core modules.ResourcesFor an introduction to Mathematical Logic, see Introduction to mathematical logic. Another book is A course in mathematical logic. A layman's introduction to Gödel's Incompleteness Theorem is given in Gödel's proof. For an introduction to axiomatic set theory, see Axiomatic Set Theory. |
email: D Schuetz