*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*