Instructor: Dr. Lajos SOUKUP
Website of the course: http://www.renyi.hu/~soukup/set_16f.php
Text: The course is based on printed handouts
Prerequisite: Some familiarity with "higher" mathematics. No specific knowledge is expected.
Course description
This course is designed as an introduction to basic set theoretic notions and methods.
 we learn how to use set theory as a powerful tool in algebra, analysis, combinatorics, and even in geometry;
 we study how to build up a rich mathematical theory from simple axioms;
 we get an insight how set theory can serve as the foundation of mathematics.
Topics:
 Classical set theory: "By a set we are to understand any collection onto a whole of definite and separate objects of out intuition or our thought." (Cantor)
Basic principles: Extensionality : Two sets are equal if and only if they have the same elements.
 General principle of comprehension of Frege : If \(P(x)\) is a property, then there is a set \(Y=\{x:P(x)\}\) of all elements having property P.
 Countable and uncountable sets. A sample problem: any family of disjoint letters T on the plain is countable.
 Inductive constructions. A sample problem: "A flea is moving on the integer points of the real line by making identical jumps every seconds. You can check one integer every seconds. Catch the flea!"
 Ramsey Theory. How to prove the finite Ramsey theorem from the infinite one? Konig lemma: an infinite, locally finite tree should contain infinite paths. Applications: a countable graph is ncolorable if and only if its every finite subgraph is ncolorable.
 Cardinalities. Comparing the sizes of infinite sets. Cardinalities. Basic operation on cardinalities. Elementary properties of cardinal numbers. CantorBernstein 'Sandwich' Theorem and its consequences, \(A < P(A)\).
 More on cardinal numbers: Calculations with cardinals, \(2^{\aleph_0}= \mathfrak c\) (the cardinality of the real line), there are \(\mathfrak c\) many continuous functions. Infinite operations on cardinals: \(1\cdot 2 \cdot 3 \cdots = \mathfrak c\). Konig's Inequality.
 The crucial notion of "wellordering", ordinal numbers: Definition, properties, calculations with ordinals.
 The heart of the matter: Zorn lemma and the Well Ordering Theorem of Zermelo: we can enumerate everything, the Theorem of Transfinite Induction and Recursion, the Fundamental Theorem of Cardinal Arithmetic: \(\kappa^2=\kappa\) for every cardinal \(\kappa\).
 Applications (as many as time permits):
 Every vector space has a basis; Hamel basis; the additive groups of the reals and of the complex numbers are isomorphic.
 Mazurkiewicz theorem: there is a subset of the plain which intersects every line in exactly two points
 Cauchy's Functional Equation: find nontrivial solutions of the function equations f(x)+f(y)=f(x+y),
 Dehn's Theorem about decompositions of geometric bodies
 the Long Line
 the function f(x)=x is the sum of two periodic functions,
 Sierpinski's Theorem and the Continuum Hypothesis,
 decomposition of \(\mathbb R^3\) into congruent circles,
 Goodstein's Theorem.
 Contradictions in mathematics? The comprehension principle of Frege leads to contradictions.
 Russel's Paradox: Does the set of all those sets that do not contain themselves contain itself?

The solution:
 keep the Extensionality, and
 replace the faulty General Comprehension with some weaker hypotheses, axioms , which are necessary for the proofs of the fundamental results and seemingly free of contradiction. (The majorities of axioms will be special cases of General Comprehension.)
 prove all results from the axioms.
 Axioms of the ZermeloFraenkel set theory
 Basic Set Theory from the Axioms: Ordered pairs. Basic operations on sets. Relations and functions. Cartesian product. Partial and linearorder relations. Natural numbers. Ordinals
 A glimpse of independence proofs: How can you prove that you can not prove something?
 Contemporary Axiomatic Set Theory: extensions of the classical ZermeloFraenkel Set Theory with new axioms
BOOKS:
A. Shen, and N. K. Vereshchagin, Basic Set Theory, AMS Student Mathematical Library 17,
P. Halmos: Naive Set Theory
P. Hamburger, A. Hajnal: Set Theory
K. Ciesielski: Set Theory for the Working Mathematician
GRADING:
Homework assignments: 40%, midterm exam: 20%, final exam: 40%.
12 homework assigments, the best 10 count.
Extra hw problems for extra credits.
A: 80100%, B: 6079%, C: 5059%, D: 4049%