BSM Set Theory — SET 2014S.
Instructor: Dr. Lajos SOUKUP
Website of the course:
http://www.renyi.hu/~soukup/set_14f.html
Text: The course is based on printed handouts
Books:P. Halmos:
Naive Set Theory
P. Hamburger, A. Hajnal:
Set Theory
K. Kunen:
Set Theory, Chapter 1.
T. Jech:
Set Theory, Chapters 16.
K. Ciesielski:
Set Theory for the Working Mathematician
Prerequisite: Some familiarity with "higher" mathematics.
No specific knowledge is expected.
Course description
The goal of the course is threefold:
 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 learn how to use set theory as a powerful tool in algebra,
analysis, and even geometry,
 Since set theory is also an independent branch of mathematics, like algebra or geometry, with its own subject matter, basic results, open problems, the course tries to catch a glimpse of some results and
problems from contemporary set theory, especially from infinite combinatorics.
Grading:Grading: Homework assignments: 50%, midterm exam: 20%, final exam: 30%.
A: 80100%, B: 7079%, C: 6069%, D: 5059%
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:
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. An application: there are uncountably many transcendental real numbers.
 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?
König 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 size 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^{.} = c (the
cardinality of the real line), there are c many continuous functions, 1· 2
· 3 ··· = c, the cardinal numbers c, 2^{c}, etc., K.nig's Inequality.
 The crucial notion of "wellordering", ordinal numbers: Definition, properties, calculations with
ordinals.
 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: x^{2}=
x for every cardinal x.
 Applications (as many as time permits):
 Contradictions in mathematics? The fall of naive set theory.
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?
 Berry's Paradox:
'The least integer not nameable in
fewer than nineteen syllables'
 The solution: Axiomatic approach (without tears):
Mathematical logic in a nutshell. Variables, terms and formulas.
The language of settheory. ZermeloFraenkel Axioms.
 Basic Set Theory from the Axioms: Ordered pairs. Basic operations on
sets. Relations and functions. Cartesian product. Partial and
linearorder relations.
 A glimpse of independence proofs: How can you prove that you can not prove something?