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. ∑_i a_i
General Cartesian product. ∏ _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 a≥ ℵ₀ and a+ℵ₀=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^ℵ₀. 1 · 2 · 3  · · · = c. c=c²=c³= · · ·=c2^ℵ₀.
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 a∈ 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)≥ x holds for every x∈ 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 α is an ordinal, then ^~α is a well ordered set of ordinal α.
α+1, successor, limit ordinals.   The minimality principle of ordinals.
Theorems on transfinite induction/recursion.        Addition and multiplication of ordinals, rules.
Ordinals of the form ωⁿ a_n+ · · · + ωa₁+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.
ω₁, ℵ₁. The continuum hypothesis.   Sierpinski decomposition of the plane.
In ^~ω₁ every countable set is bounded.
If f(x)<x for every 0<x<ω₁ then some value is obtained ℵ₁ times.
Automaton that returns ℵ₀  1 Forint coins if a coin of 1 Forint is inserted.
Throwing darts on the plane.