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},^{ B}A.
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≥ ℵ_{0}
and a+ℵ_{0}=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=c^{2}=c^{3}=
· · ·=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 ω^{n} a_{n}+
· · · + ωa_{1}+a_{0}.
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.
ω_{1}, ℵ_{1}.
The continuum hypothesis. Sierpinski decomposition of the plane.
In^{ ~}ω_{1} every countable
set is bounded.
If f(x)<x for every 0<x<ω_{1}
then some value is obtained ℵ_{1}
times.
Automaton that returns ℵ_{0 }
1 Forint coins if a coin of 1 Forint is inserted.
Throwing darts on the plane.