Instructor: Dr. Péter KOMJÁTH
Text: P. Hamburger, A. Hajnal: Set Theory and handouts.
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|< |P(A)|, Russell's paradox.
a+b, ab for cardinals, properties. Si ai
General Cartesian product. Pi ai 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 a³À0 and a+À0=a. 2|A| =|P(A)|,2a>a for every cardinal a.
The monotonicity of ab.
Every nontrivial real interval is of cardinality c.
c=2À0. 1 · 2 · 3 · · · = c. c=c2=c3= · · ·=cÀ0.
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 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
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 an+ · · · + 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.