Problems and Theorems in Classical Set Theory
A problem book that appeared at
This book contains over 1000 problems in classical set theory.
The book starts with introductory topics as: set operations, cardinal
operations, countable sets, sets of cardinality continuum,
ordered and well ordered sets, ordinals.
Next, important classical
results are covered as the well ordering theorem,
the definition and properties of alephs,
Zorn's lemma, cofinalities, stationary sets.
Along the way, we apply these techniques in proving various
reasults in analysis, graph theory, algebra by
using transfinite methods, the continuum hypothesis,
Hamel bases, etc.
Special attention is given to such tradionally Hungarian topics as
infinite graphs and combinatorial set theory.
Some of the highligts:
- scattered order types,
- Goodstein's theorem
- the existence of Hausdorff gap,
- equivalents of CH,
- the Banach-Tarski paradox,
- Solovay's decomposition theorem
- Ramsey's theorem
- the Erdős-Dushnik-Miller theorem,
- the Erdős-Rado theorem,
- the Todorcevic partition of the pairs of ω1,
- the Δ-system lemma,
- Hajnal's set mapping theorem,
- Galvin's tree game,
- the existence of finitely additive, isometry invariant measure
on all subsets of R and R2,
- Aronszajn trees, Specker types, Countryman types.
Full, detailed solution is provided to every problem.
"The book is well-written and self-contained, a choice
collection of hundreds of tastefully selected problems related to
calssical set theory, a wealth of naturally arising, simply
formulated problems not only in pure set theory but also in
real analysis, geometry, graph theory, discrete mathematics, algebra,
topology, etc., connected to set theory." (Tamas Erdelyi)
"This is a welcome addition to the literature, which should be useful to
students and researchers alike." (Mathematical Reviews)
"This is not a book from which to learn set theory, but rather it is a book
that allows one to savor set theory." (Zentralblatt)