Mathematical structures in Logic - Fall 2011
Deze pagina betreft het vak `Mathematical structures in Logic', gedoceerd aan de Universiteit van Amsterdam van september tot december 2011. De voertaal van het college is Engels.
This page concerns the course `Mathematical structures in Logic', taught at the University of Amsterdam from September-December 2011.
Course description and prerequisites
Aims and Topics
This course aims at providing the basic toolbox for understanding and doing research in a fast-growing branch of modern mathematical logic. This field:
- links logics with several kinds of mathematical objects (topological spaces, partial orders and more in general relational structures, algebras and categories) via semantic interpretations;
- explores the mathematical properties of each of these logics in relation with the properties of the mathematical objects that provide its semantic interpretations;
- explores the logical significance of each kind of mathematical object, as well as the underlying relations between objects of different kinds, via the notion of logical invariance.
Categories
Of the different kinds of mathematical objects mentioned above, categories are the most general, and are especially important because not only they provide the semantic interpretations for specific logics, but also they are the `universe of discourse' and the language where this theory can be properly developed. Therefore the course will pay particular attention to introducing students to categorical reasoning, that is, reasoning in terms of the properties, rather than on the representation of mathematical entities.
Orders
Partial orders are of fundamental importance in logic because they naturally interpret the logical entailment relation. Partial orders also occupy a very special place in category theory: since each partial order is a category, the theory of partial orders reflects category theory very much like in a microcosm/macrocosm relationship: this connection makes partial orders a very useful source of counterexamples in category theory and also underlies the deep relationship between logic and category theory. Moreover, the specialization of the category-theoretic notion of adjunction to partial orders is of crucial importance to logic. All of these themes and results will be presented in the course.
The tutorials
The course alternates lectures and tutorials. The latter will be focused on interesting examples that will be developed in class, or specific concepts that require some interaction to be understood.
Prerequisites
We assume that you are able to understand and produce mathematical definitions and proofs. This includes some very basic background knowledge, e.g. the notions of set, function, relation, graph of a function, and proofs by reductio ad absurdum, contraposition, induction.
Lecturers
Alessandra Palmigiano - e-mail: A.Palmigiano@uva.nl - phone: 525 5360
Vincenzo Ciancia - e-mail: vincenzoml@gmail.com
Time and place
Classes run from the 5th of September to the 20th of October, and from the 31th of October to the 15th of December. There will be no classes during Week 43.
There are two sessions per week, on Monday (9.00-11.00) and Thurdsay (17.00-19.00).
Contents of the lectures and the tutorials
The contents of the lectures and the tutorials can be found at this link.
Homework and grading
To be announced.
Material
Course material
The following book covers the topics related to Order Theory: Introduction to Lattices and Order (Second Edition) - B.A. Davey, H.A. Priestley - Cambridge University Press 2002.
For the topics related to Category Theory, we will refer to the book: Category Theory (Second Edition) - S. Awodey - Oxford University Press 2010.
Another excellent book, which we recommend to those who have a strong mathematical background to deepen their knowledge of category theory beyond the scope of the course, is the classical text by Mac Lane: S. Mac Lane, Categories for the Working Mathematician, Second Edition, Springer 1998.
Further reading
A motivational introduction to category theory, tailored to computer scientists, is the paper "A categorical manifesto" by Joseph A. Goguen, available here.
Depending on the specific background of a student, other books may be more interesting. Should you be willing to learn more about Category Theory, Order Theory, or specific subjects that you hear of during the course, do not hesitate to ask us.