Instructor: Dr. Péter KOMJÁTH

Text: P. Hamburger, A. Hajnal:  Set Theory and handouts.

Topics covered:
Notation. Subsets. Empty set. Union, intersection.
Known set: N, Z, Q, R      Zermelo-Fraenkel axioms.    Equality. Existence. Subsets.
Pair. Power sets. Infinite sets. Ordered pair. Cartesian product. Functions. The empty function.
D_f, R_f are sets. Classes, operations.   Symmetric difference.
Equivalence, cardinals. a £b, a<b for cardinals. Properties, a£b, b£a imply a=b.
|A|< |P(A)|, Russell's paradox.
a+b, ab for cardinals, properties. S_i a_i
General Cartesian product. P_i a_i     a^b, ^BA.
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. If  a  is an infinite cardinal then a³À₀ and a+À₀=a. 2^|A| =|P(A)|,2^a>a  for every cardinal  a.
The monotonicity of a^b.
Every nontrivial real interval is of cardinality c.
 c=2^À0. 1 · 2 · 3  · · · = c. c=c²=c³= · · ·=c^À0.
There are c continuous real functions. The cardinals 2^c, 2^2c, ....
For every set  A of cardinals there is a cardinal  b such that  b>a holds for every aÎ A.
Ordered sets, the axiomatic set theory definition. Order preserving functions, isomorphisms. Order types.
Well ordered sets, examples, ordinals.
If (A,<) is well ordered, f:(A,<)€®(A,<) is order preserving, then f(x)³ x holds for every x\in A.
(A,<) is well ordered iff there is no infinite decreasing sequence.Segments, segments determined by elements.
Ordinals, ordinal comparison, it is irreflexive, transitive, trichotomic. If a is an ordinal, then ^~a is a well ordered set
of ordinal a.
a+1, successor, limit ordinals.   The minimality principle of ordinals.
Theorems on transfinite induction/recursion.        Addition and multiplication of ordinals, rules.
Ordinals of the form wⁿ a_n+ · · · + w¹a₁+a₀. Comparison and addition of them.
The well ordering theorem. Trichotomy of cardinal comparison.
Every vector space has a basis. Hamel basis, Cauchy-functions.
f(x)=x is the sum of 2 periodic functions.
w₁, À₁. The continuum hypothesis.   Sierpinski decomposition of the plane.
In ^~w₁ every countable set is bounded.
If f(x)<x for every 0<x<w₁ then some value is obtained À₁ times.
Automaton that returns À₀  1 Forint coins if a coin of 1 Forint is inserted.
Throwing darts on the plane.