Set theory

elective courses

The lecture presents basic notions of advanced set theory (ordinal and cardinal numbers, axioms of set theory) and introduces elements of infinite combinatorics.

A general introduction to set theory: the axioms, well orderings, transfinite induction and recursion, ordinal and cardinal numbers.

Topics in combinatorial set theory: ideals and filters, stationary sets, trees, Delta systems, partition calculus.

W.Just, M.Weese, Discovering modern set theory, I: The basics, II: Set-theoretic tools for every mathematician, Graduate Studies in Mathematics vol. 8 (1996), 18 (1997), American Mathematical Society.

A student:

1. knows Zorn's Lemma and applies it to prove the existence of sets of various properties, including non-principal ultrafilters;

2. knows the notion of a well-ordering and von Neumann's definition of an ordinal. Is familiar with ordinal arithmetic. Knows how to prove statements by transfinite induction and construct sets or sequences by transfinite recursion;

3. knows the notion of a cardinal number, basics of cardinal arithmetic and most important theorems, including Hessenberg's Theorem and the Hausdorff Formula. Is able to express cardinalities of various sets using cardinals.

4. knows the notion of cofinality of a cardinal and the notions of a regular and singular cardinal.

5. knows the notions of a closed unbounded and stationary subset of a regular cardinal. Knows and uses Fodor's Lemma.

6. knows theorems about existence and sizes of families with special combinatorial properties, including almost disjoint and independent families and Delta-systems.

7. knows the notion of a tree and basic theorems about existence of cofinal branches in a tree, including theorems of Koenig and Aronszajn;

8. knows basic partition theorems, including the Ramsey, Erdos-Rado and Erdos-Dushnik-Miller theorems;

9. knows axioms of ZFC theory and some additional set-theoretic axioms, including CH and GCH.