Set Theory  SET

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.
Df, Rf are sets. Classes, operations.   Symmetric difference.
Equivalence, cardinals. a ≤ b, a<b for cardinals. Properties, a≤ b, b≤a imply a=b.
a+b, ab for cardinals, properties. ∑i ai
General Cartesian product. ∏ i a    ab, 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)|,2a>a  for every cardinal  a.
The monotonicity of ab.
Every nontrivial real interval is of cardinality c.
c=20. 1 · 2 · 3  · · · = c. c=c2=c3= · · ·=c20.
There are c continuous real functions. The cardinals 2c, 22c, ....
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 an+ · · · + 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<x1 then some value is obtained ℵ1 times.
Automaton that returns ℵ 1 Forint coins if a coin of 1 Forint is inserted.
Throwing darts on the plane.