Set Theory, 88-202, 2018a

* Database of previous exams.

Hebrew lecture notes on basic Set Theory:

Lecture Topics
Lecture #1 Partial orders. The Rasiowa-Sikorski Lemma. Every countable dense linearly ordered set without endpoints is order-isomorphic to the rational line.
Lecture #2 Transitive sets. Well-orders are rigid. The class of ordinals is well-ordered.
Lecture #3 Every two well-ordered sets are compatible. Finite ordinals. Every well-ordered set is isomorphic to a unique ordinal. Ordinals arithmetic: sums and multiplications.
Lecture #4 Transfinite induction, and applications to ordinal arithmetic. Ordinal power.
Lecture #5 Hartogs' number. Definition by recursion. The axiom of choice. Zermelo's well-ordering principle. Hausdorff's maximality principle. Zorn's lemma. Hartogs' comparability of cardinals. Tychonoff's theorem.
Lecture #6 The axioms of ZFC. The effect of AC on infinite graph theory: the Shelah-Soifer graph, and the Komjath-Galvin graph.
Lecture #7 ZF implies that Hartogs' number is indeed a set. Cardinals, and cardinal arithmetic. Cantor's theorem.
Lecture #8 kxk=k for every infinite cardinal k, and applications to basic cardinal arithmetic. Successor and limit cardinals. The Aleph function. Cofinality of partially ordered-sets. Regular and singular cardinals. Every successor cardinal is regular.
Lecture #9 Analysis of the Aleph function: cofinality, surjectivity, fixed points. Some more cardinal arithmetic, including Konig's lemma and Hausdorff's formula.
Lecture #10 Cardinal arithmetic assuming GCH. Strong limit cardinals and the Beth function. Ideals and filters. The nonstationary ideal is sigma-additive. Normal functions. Closed and unbounded sets.
Lecture #11 CUB is the dual of NS. Diagonal union. The nonstationary ideal is normal. Fodor's lemma. Ulam's matrix. Every stationary subset of w1 may be partitioned into w1 many pairwise disjoint sets.
Lecture #12 Pigeonhole principles for cardinals, a pigeonhole principle for ultrafilters, Rasmey's theorems - the infinite implies the finite, Sierpinski's anti-Ramsey theorem, and a few infinite-dimensional colorings.
Lecture #13 Delta-systems, free set lemma, Almost disjoint sets, Hausdorff's formula, the Bukovsky-Hechler theorem, Tarski's theorem.
Lecture #14 Set-theoretic implementation of the classic system of numbers: natural numbers, integers, rational numbers, the real line (Dedekind cuts). Order-theoretic uniqueness of the real line. Cardinality of the real line.


p.s.
The corresponding Math-Wiki page is in here.