## BSM Set Theory — SET, Fall 2011.

*
Homepage of the course*:

http://www.renyi.hu/~soukup/set_11f.html
*Instructor*: Dr. Lajos SOUKUP

*
Homepage*:

http://www.renyi.hu/~soukup
*Text*: The course is based on handouts

*Books*:
P. Hamburger, A. Hajnal:

*Set Theory*
K. Kunen:

*Set Theory, * Chapter 1.

T. Jech:

*Set Theory,* Chapters 1--6.

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,
- we learn how to use set theory as a powerful tool in algebra,
analysis, and even geometry,
- we study how to build up a rich mathematical theory from simple axioms.

*
Grading*: Homework assignments: 40%, midterm exam: 20%, final exam: 40%.

A: 80-100%, B: 60-79%, C: 40-59%, D: 30-39%

*
Homeworks* are distributed and collected on ....

*Topics:*
*Naive set theory*: general principle of comprehension, due to
Frege (1893):
*If P is a property then there is a set Y={X:P(X)} of all elements
having property P.*
*Contradictions in mathematics?*

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 set-theory. Zermelo-Fraenkel Axioms.
*Basic Set Theory from the Axioms*: Ordered pairs. Basic operations on
sets. Relations and functions. Cartesian product. Partial- and
linear-order relations.
*The crucial notion of "well-ordering", ordinal numbers*: Definition, properties, calculations with
ordinals.
*Elementary properties of cardinal numbers*:
Equivalence of sets, cardinals, the
Cantor-Bernstein 'Sandwich' Theorem and its consequences, |A| < |P(A)|.
*More on cardinal numbers*: Calculations with cardinals, 2^{ω} = c (the
cardinality of the continuum), there are c many continuous functions, 1· 2
· 3 ··· = c, the cardinal numbers c, 2^{c}, etc., KÅ‘nig's Inequality.
*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)*: Every vector space has a basis, Hamel
basis, Cauchy's Functional Equation, Dehn's Theorem about decompositions of geometric
bodies, the Long Line, f(x)=x is the sum of two periodic functions,
Sierpinski's Theorem and the Continuum Hypothesis, decomposition of R^{3} into circles, Goodstein's Theorem^{*}.