Set Theory  SET

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(A)|, 2a>a  for every cardinal a.
The monotonicity of ab.
Every nontrivial real interval is of cardinality c.
An example that exponentiation of cardinals is not monotonic in the strict sense.
For every set  A of cardinals there is a cardinal  b such that  b>a holds for every element a of A.

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.