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.
Topics:
• 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