History of Science – The foundations saga: from Euclid to Gödel

With his “Elements” Euclid has coined the one and only acceptable method of conceiving Mathematics: a solid deductive process based on a careful selection of axioms and the Aristotelian rules of inference. During the 19th century, the appearance of various alternative theories has given rise to a new problem: to assess a collection of axioms with respect to its consistency and completeness. This has led to the famous Gödel's incompleteness theorems. In this course we will study the ideas and limitations of axiomatic theories drawing examples from non Euclidean geometries, set theory and number theory.

Course outline
I. Pre – Euclidean mathematics. The Elements.
II. The challenges of non – Euclidean geometries.
III. Cantor’s set theory, Peano’s axioms and related paradoxes.
IV Hilbert’s second problem and Gödel’s theorem.
V. Alternative approaches: Turing machines, halting problem

Course leader

Tefcros Michaelides, Ph.D.

Target group

High School students, 1st and 2nd year University students

Course aim

The course aims at the study of the ideas of axiomatic theories drawing examples from non Euclidean geometries, set theory and number theory