The student is expected to develop a critical eye with regard to mathematical reasoning: is there one objective Logic for Mathematics? Are there more than one? He will also be encouraged to evaluate the significance of mathematical theorems about Logic. More generally, he will learn how mathematical methods can be applied to domains which are less stable than those concerning mathematical objects.
Course contents summary
At the beginning of the course, a few lectures will be devoted to the Intuitionistic Philosophy of Mathematics, i.e. an approach to do Mathematics constructively without infinitistic assumptions. Then, Heyting's formalization of Intuitionistic reaoning is introduced followed by an analysis of both algebraic and Kripke-type semantics. By way of a conclusion, the existence of a close connection between the two different semantical systems is established, thereby showing that there is a uniform and accurate representation of the Intuitionist's notion of meaning in a formal language.
1) M. Fitting, Intuitionistic Logic, Model Theory and Forcing, NothHolland 1969.
2) H. Rasiowa & R. Sikorski, The Mathematics of Metamathematics, Warsavia 1963,
3) Notes for Students.
There will be lectures, exercises will be assigned and corrected in class. The strategy will be to go from the concrete to the abstract, to further confirming examples and to reflect on the significance of the work.
Assessment methods and criteria
The final mark will depend upon the student's performance in an oral exam and the quality of his participation in the comunalwork in class.