Text: A. Hajnal, P. Hamburger: Set Theory + handouts
The goal of the course is twofold. On the one hand, we get an insight how
set theory can serve as the foundation of mathematics, and on the
other hand, we learn how to use set theory as a powerful tool in algebra,
analysis, geometry and even number theory.
Notation, empty set, union, intersection, complement, subset,
power set, equality of sets, N, Z, Q, R, countable and uncountable sets.
Elementary properties of cardinal numbers: Equivalence of sets, cardinals, the
Cantor-Bernstein 'Sandwich' Theorem and its consequences, |A| < |P(A)|.
'Oops': Russel's Paradox.
The axiomatic approach: Zermelo-Fraenkel Axioms.
More on cardinal numbers: Calculations with cardinals, 2ω = c (the
cardinality of the continuum), there are c many continuous functions, 1· 2
· 3 ··· = c, the cardinal numbers ω,
2ω, 22ω, etc., König's Inequality.
Ordered sets: Definition, isomorphism, initial segment, initial segment
determined by an element, order type, well ordered set.
The crucial notion, ordinal numbers: Definition, properties, calculations with
The heart of the matter: The Well Ordering Theorem*: we can enumerate
everything, the Theorem of Transfinite
Induction and Recursion, the Fundamental Theorem of Cardinal Arithmetic:
&kappa2=&kappa for every infinite cardinal κ, Zorn's lemma.
Applications (as many as time permits): Every vector space has a basis, Hamel
basis*, Cauchy's Equation, Dehn's Theorem about decompositions of geometric
bodies, the Long Line, f(x)=x is the sum of two periodic functions, 2-point
Sierpinski's Theorem* and the Continuum Hypothesis, throwing darts at the
plane, decomposition of R3 into circles, Goodstein's Theorem*, the Problem of
13 Numbers, nonstandard analysis (infinitesimal numbers and how to make dy/dx precise), how to define the
limit of nonconvergent sequences (Banach limits and ultrafilters).
* The theorems marked by asterisk are, in the instructor's opinion, among
the ten most beautiful results of mathematics.