INSTYTUT MATEMATYKI
Uniwersytet Śląski
40-007 Katowice, ul. Bankowa 14
Introduction
The targeted group of readers for this book consists primarily of undergraduate and graduate
students of mathematics. It can also be useful for mathematicians conducting research in other
fields of mathematics or even mathematics enthusiasts. This work is more suited to be considered
as a textbook than a scientific monograph. Therefore the authors did not feel obligated to give detailed
sources of all the cited definitions and theorems. However each section ends with Comments that provide
additional pertinent information along with necessary citations and literature references. Moreover,
the Comments can also serve as Problem Sections since the theorems presented there contain merely
hints instead of complete proofs. This should allow an attentive reader to carry out proofs on his own.
The textbook consists of four parts. The first part contains a concise discussion on what at
Polish universities is referred to as the introduction to mathematics. The subject itself is
treated far more thoroughly in Wojciech Guzicki’s and Piotr Zakrzewski’s book entitled
“Wykłady ze wstępu do matematyki” (Lectures on the introduction to mathematics) published by the
Scientific Publishing House PWN. In our textbook, introduction to mathematics is presented somewhat
differently. The authors’ aim was to ascertain that the content of the remaining three parts is independent
of other publications devoted to set theory. The reader who after perusal of this part finds it somewhat
lacking in depth, particularly as far as examples are concerned, is advised to refer to the aforementioned book.
The second part of the textbook deals with the material that constitutes the basis of any contemporary set theory.
We included there: the Zermelo-Fraenkel axioms, the structure of ordinary numbers, and the arithmetic of ordinal
and cardinal numbers. We also presented some of the most important combinatorial theorems such as the Ramsey
theorem or the Erdös–Rado Theorem. Moreover, we discuss the arithmetic of ordinal numbers that is used to prove
the Goodstein theorem. The theorem itself, which is independent of the Peano axiomatic system of arithmetic,
exemplifies the enormous diversity of set theory. For it not only encompasses cardinal and ordinal arithmetic
but also goes far beyond this scope. We end this part with a discussion about Martin’s axiom, whose importance
is due to powerful applications in real analysis and measure theory.
The third part of the textbook is the most extensive one. In authors’ opinion, this part contains the most
important topics and results from the lattice theory, trees, topology, measure theory, Boolean algebras, as
well as Ramsey’s theory. The authors’ guiding principle, while writing this section, was to put emphasis on
the existing connections between set theory and other fields of mathematics. Hence the particular scope of
subjects chosen for considerations. It was also the authors’ ambition to write a book which would be to a great
extent self-contained. They devoted substantial considerations to topological structures which are indispensable
in the development of some parts of set theory like, for instance, the theory of Boolean algebras. The ties
between topology and set theory are ubiquitous and profound. They exist at a much deeper level than we managed
to focus and present here.
The fourth part focuses on the axiom of choice, the one among the axioms of set theory which is the most
difficult to acquiesce. We present various equivalent versions of this axiom. We also discuss weaker versions
of the axiom of choice with the purpose of deriving from them a number of important theorems that belong to
different areas of mathematics. We make an attempt to support the claim that contemporary mathematics would be
incomplete without the axiom of choice. Towards this goal, we show that the axiom of choice is equivalent to
the existence of a base in every linear space. We also present the Banach-Tarski paradox. It has been well
known among mathematicians and thoroughly discussed in several publications worldwide. Yet, in the Polish
mathematical literature, the Banach-Tarski paradox can only be found in the monograph by Kuratowski and Mostowski
from 1978.
The textbook under discussion is by no means an in-depth and complete
treatise on set theory. In particular, the authors have completely
passed over the issues associated with forcing which is used to prove
relative consistency of some statements of the Zermelo-Fraenkel set
theory. The chapter devoted to Boolean algebras, which are an
indispensable tool in the theory of forcing may serve as an
introduction to forcing theory. Another topic omitted from
considerations here is the descriptive set theory. There exists a vast
literature on this subject. In Polish publications, descriptive set
theory is discussed in Kazimierz Kuratowski’s and Andrzej
Mostowski’s monograph “The theory of sets” [27] ,
whose last edition was published by PWN in 1978 as well as in Wojciech
Guzicki’s and Paweł Zbierski’s book “ Fundamentals of
the theory of sets”[15] also published by PWN in the same year.
We had the opportunity of presenting a number of chapters of the textbook during our lectures to many
mathematics majors at the University of Silesia. Their questions and inquires enabled us to improve the
text and we are grateful to them for their involvement. We would like to thank professors Karol Baron
and Piotr Wojtylak for their invaluable comments regarding the manuscript. We received much help from
our colleagues and co-workers from the Faculty of Set Theory of the University of Silesia: doctors
Wiesław Kubiś, Andrzej Kucharski, and Michał Machura. We are also indebted to Wojciech Bielas for the
extremely thorough reading of the text and for pointing out several mistakes and errors.
We are particularly grateful to professors Piotr Zakrzewski and Zbigniew Lipecki for the very
careful perusal of the manuscript and for the valuable comments and advice they gave us.
We would like to express our gratitude to the Rector of the Silesian University for the help and
understanding in publishing our book. We would like also to acknowledge the support we received
from the publisher’s office of PWN.
Aleksander Błaszczyk
Sławomir Turek
Katowice, November 2005
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