This course is an introduction to set theory, focusing on foundational issues but with an eye also on the study of combinatorial properties of infinite objects.
We will start by motivating and introducing ZFC. Then we will develop the basic theory of the ordinals and cardinals in this theory, and will prove some classical theorems of combinatorial flavour. Possible topics may include cardinal arithmetic, Aronszajn trees, infinite Ramsey theory and/or some results on determinacy of games. Time permitting, I will briefly discuss large cardinal axioms, the independence phenomenon, and the problem of finding natural extensions of ZFC.
One of the goals of the course is to engage a working mathematician into looking at the foundations of the mathematical building.
There are no prerequisites for this course, except for a reasonable level of mathematical maturity. Having been exposed to a course in mathematical logic would be desirable but not necessary. I will in fact give brief introductions to the relevant notions from logic.
Naive set theory: Sets as foundational framework for mathematics. Paradoxes.
Axiomatic set theory: ZFC.
Ordinals and cardinals. Transfinite recursion and induction. The cumulative hierarchy.
Countable and uncountable sets.
The Axiom of Choice.
Basic cardinal artihmetic.
Some combinatorial set theory: Aronszajn trees, infinite Ramsey theory.
Determinacy of infinite games.
Large cardinal axioms: Weakly compact, measurable, and beyond.
Natural axioms for mathematics: Extending ZFC.