Instructor: Dr. Lajos SOUKUP
Website of the course: http://www.renyi.hu/~soukup/set_15s.html
Text: The course is based on printed handouts, which are distributed after classes
Prerequisite: ---
Course description
The goal of the course is threefold:
- we learn how to use set theory as a powerful tool in algebra, analysis, combinatorics, and even in geometry,
- we get an insight how set theory can serve as the foundation of mathematics: all mathematical concepts, methods, and results can be represented within set theory,
- we study how to build up a rich mathematical theory from simple axioms.
- Introduction: Notation, empty set, union, intersection, complement, subset, power set, equality of sets, \(\mathbb N, \mathbb Z,\mathbb Q,\mathbb R\), countable and uncountable sets.
- Cardinalities of sets: Comparing the sizes of infinite sets \( \mathbb N, \mathbb Z, \mathbb Q \) and \( \mathbb R\). Cantor-Bernstein Theorem: \(|A|\le |B|\) and \(|B|\le |A|\) implies \( |A|=|B| \). Cantor's Theorem: \( |A| < |\mathcal P(A)|\).
- The fall of naive/classical set theory: Russel's Paradox: Does the set of all those sets that do not contain themselves contain itself?
- The axiomatic approach: Zermelo-Fraenkel Axioms. The Axiom of Choice.
- Basic notions revisited: operations on sets, ordered pairs, relations and functions, Cartesian products, partial- and linear-order relations.
- Zorn's Lemma and its applications: (i) every vector space has a basis; (ii) the additive groups of the reals and of the complex numbers are isomorphic; (iii) every connected graph has a spanning tree; (iv) Cauchy's Functional Equation: find the non-trivial solutions of the function equations f(x)+f(y)=f(x+y); (v) a rectangle can be decomposed into finitely many non-overlapping squares if and only if the ratio of the slides of the rectangle is rational.
- Well-ordered sets: ordinals and ordinal arithmetic with applications: (i) Hercules and the Hydra game;, (ii) Goodstein's theorem.
- Cardinalities of infinite sets: addition, multiplication and exponentiation of cardinals; Fundamental Theorem of Cardinal Arithmetic: \( |X|^2=|X| \) for all infinite set \(X \) ; König's Inequality.
- The heart of the matter: the Well Ordering Theorem: every set has a well-ordering.
- Theorem of Transfinite Induction and Recursion with applications: (i) Mazurkiewicz's theorem: there is a subset of the plain which intersects every line in exactly two points, (ii) \( \mathbb R^3 \) can be decomposed into disjoint congruent circles.
- Infinite combinatorics: infinite Ramsey theorem, König's lemma
- A glimpse of independence proofs: How can you prove that you can not prove something?
Books:
P. Hamburger, A. Hajnal: Set Theory
Ernest Schimmerling: A Course on Set Theory