*Instructor:* Dr. Miklós ERDÉLYI SZABÓ

*Text*: P. Hamburger, A. Hajnal: *Set Theory *.

*Topics covered:*

**Introduction** Notation. Subsets. Empty set. Union,
intersection. Known sets: **N, Z, Q, R.**

Russell's paradox.

Zermelo-Fraenkel axioms: equality, existence, subsets, pair, power sets, infinite sets.

Ordered pair. Cartesian product. Functions. Classes, operations.

**Size of sets** Equivalence of sets, cardinals. Comparison of cardinals, Bernstein's equivalence theorem.

|*A*|< |*P(A)*|.

Addition, multiplication and exponentiation of sets and cardinals.

The axiom of choice, its role in the proofs that the
union of countably many countable sets is countable

and that the two definitions
of convergence are equivalent.
*2 ^{|A|
}=|*P

The monotonicity of

Every nontrivial real interval is of cardinality

An example that exponentiation of cardinals is not monotonic in the strict sense.

For every set

**Counting** Ordered sets, order preserving functions, isomorphisms. Order types.

Well
ordered sets, ordinals.

Segments, segments determined by elements.

Ordinals, ordinal
comparison, it is irreflexive, transitive, trichotomic.

a+1, successor, limit ordinals.

Theorems on transfinite
induction/recursion. Addition and
multiplication of ordinals, rules.

The well ordering theorem. Trichotomy of
cardinal comparison.

Every vector space has a basis. *f(x)=x* is the sum of 2 periodic functions.

In the first uncountable ordinal every countable subset is bounded.

**Exponentiation of Cardinals** The continuum hypothesis.
Sierpinski decomposition of the plane.

König's Theorem, Bernstein-Hausdorff-Tarski Theorem.

Regressive functions and Fodor's theorem.

Silver's Theorem.