Discovering Modern Set Theory

Winfried Just and Martin Weese

Topics covered in Volume II

From the Preface:

"Our aim is to present the most important set-theoretic techniques that have found applications outside of set theory.

We think of Volume II as a natural continuation of Volume I of the same text, but it is sufficiently self-contained to be studied separately. The main prerequisite is a knowledge of basic naive and axiomatic set theory. Moreover, some knowledge of mathematical logic and general topology is indispensible for reading this volume. A minicourse in mathematical logic was given in Chapters 5 and 6 of Volume I, and we include an appendix on general topology at the end of this volume. Our terminology is fairly standard. For the benefit of those readers who learned their basic set theory from a different source than our Volume I we include a short section on somewhat idiosyncratic notations introduced in Volume I. In particular, some of the material on mathematical logic covered in Chapter 5 is briefly reviewed. The book can be used as a text in the classroom as well as for self-study."

Chapter 13. Filters and Ideals in Partial Orders

Filters and ideals in arbitrary partial orders. An application: the proof of Tychonoff's Theorem. Chain conditions and closedness conditions in partial orders.

A second application: ultraproducts of models of first-order theories. Wellfoundedness of ultrapowers. Measurable cardinals make a first appearance.

Fields of sets. Definition and elementary properties of lattices and Boolean algebras. Representation of Boolean algebras as fields of sets. Stone spaces. Stone Duality Theorem. Separative partial orders vs. Boolean algebras. Complete Boolean algebras. Algebra of regular open sets of a topological space.

Chapter 14. Trees

Basic concepts: nodes, branches, levels. An application: the size of uncountable closed subsets of the real line. König's Lemma. Compactness in logic, topology, and combinatorics: What do these notions have in common?

Kurepa, Aronszajn, Suslin trees. Weakly compact cardinals make a first appearance. Constructions of Aronszajn trees. Special Aronszajn trees. The Suslin Problem and the Suslin Hypothesis. Nonproductivity of the c.c.c. is consistent. Lexicographical ordering of a tree. Construction of Aronszajn and Suslin lines.

Chapter 15. A Little Ramsey Theory

Pigeonhole Principle. Arrow notation. Ramsey's Theorem: infinite and finite versions. Paris-Harrington Theorem. Some bounds for Ramsey numbers. Ketonen-Solovay Theorem. Erdös-Rado Theorem. Some negative partition relations and the Negative Stepping-Up Lemma. Characterizations of weakly compact cardinals. Arrow relations with infinite superscripts. Erdös and Ramsey cardinals. Erdös-Dushnik-Miller Theorem. Partition theorems for ordinals. Miller-Rado Theorem. Specker's Theorem. Open Coloring Axiom.

Chapter 16. The Delta-System Lemma

The Delta-System Lemma is proved. An application: If the c.c.c. is finitely productive, then it is productive.

Chapter 17. Applications of the Continuum Hypothesis

Luzin sets and Sierpinski sets. Erdös-Sierpinski Duality Theorem. Cardinal invariants. Additivity of the ideals of null sets and meager sets. Strong measure zero sets. Borel Conjecture. Universal measure zero sets. Perfectly meager sets. Strongly meager sets. Galvin-Mycielski-Solovay Theorem. Dual Borel Conjecture.

Dominating number. Bounding number. Cofinality and covering number of an ideal. Splitting number. Construction of the Kunen line. Almost disjoint families. Fichtenholz-Kantorovitch-Hausdorff Theorem.

Chapter 18. From the Rasiowa-Sikorski Lemma to Martin's Axiom

Rasiowa-Sikorski Lemma. A sample proof using recursion over $\omega$ is step by step transformed into one using the Rasiowa-Sikorski Lemma. Martin's Axiom (MA) formulated. Some consistent and some inconsistent modifications of MA. $\sigma$-centered and $\sigma$-linked partial orders. The Knaster Property. Amoeba forcing and the partial order $Fn(I,J)$.

Chapter 19. Martin's Axiom

Consistency of the Suslin Hypothesis. Solovay's Lemma. Influence of MA on cardinal arithmetic and regularity of $2^{\aleph_0}$. Topological and Boolean algebra versions of MA. Consequences of MA: Every Aronszajn tree is special. Every c.c.c. partial order has precaliber $\aleph_1$. The c.c.c. is productive.

Influence of MA on cardinal invariants: p, t, a, s, b, d, and the additivities of the ideals of meager and null sets are all equal to $2^{\aleph_0}$. Inequalities between p, t, a, s, b, d that are provable in ZFC. Rothberger's Theorem: If $\kappa$ is less than t, then $2^\kappa \leq 2^{\aleph_0}$. Bell's Theorem: MA for $\sigma$-centered partial orders is equivalent to $p = 2^{\aleph_0}$. Wage's theorem on almost disjoint families.

Ultrafilters on $\omega$: P-points, Q-points, selective ultrafilters. Constructions of ultrafilters using MA. Ketonen's construction of a P-point. Combinatorial characterizations of selective ultrafilters.

Chapter 20. Hausdorff Gaps

Gaps and pregaps in $P(\omega)$. Rothberger's theorem on $< \kappa, \omega^*>$-gaps and the bounding number. Constructions of Hausdorff and Luzin gaps. Gaps and the Open Coloring Axiom.

Chapter 21. Closed Unbounded Sets and Stationary Sets

Closed unbounded subsets of ordinals. Normal functions and their sets of fixed points. Closure under families of functions. Diagonal intersections. Closure properties of the CLUB-filter. Stationary subsets of ordinals. Regressive functions and Fodor's Pressing Down Lemma. Applications: Maximal almost disjoint sets of transversals. The topological space $\omega_1 \times (\omega_1 + 1)$ is not normal. Silver's Theorem.

Closure under functions revisited: universes of submodels of a given model. An application: MA(\kappa) restricted to partial orders of size at most $\kappa$ is equivalent to MA(\kappa). Closed unbounded and stationary subsets of $[X]^{< \kappa}$. Closure properties of the CLUB-filter on $[X]^{< \kappa}$. The Pressing Down Lemma for $[X]^{< \kappa}$. Fixed points of normal functions on $[X]^{< \kappa}$.

Chapter 22. The $\diamondsuit$-Pinciple

Towards a formulation of the $\diamondsuit$-principle: What does it take to construct a Suslin tree? Variants of the $\diamondsuit$-principle. Construction of a Kurepa tree from $\diamondsuit^+$. The $\clubsuit$-principle. Under CH, $\diamondsuit$ and $\clubsuit$ are equivalent. The stick-principle. The stick-principle is strictly weaker than the $\cluibsuit$-principle.

Chapter 23. Measurable Cardinals

Probability measures. Ulam matrices. Applications: Families of pairwise disjoint stationary sets. Under CH, Lebesgue measure cannot be extended onto all subsets of the reals. If $\diamondsuit(E)$ holds, then there are disjoint subsets $F,G$ of $E$ such that $\diamondsuit(F)$ and $\diamondsuit(G)$ hold.

Real-valued measurable cardinals and measurable cardinals. These cardinals are inaccessible. Measurable cardinals are weakly compact. Normal ultrafilters. Measurable cardinals are Ramsey. Atomic vs. atomless measures: Ulam's Dychotomy. Extendability of Lebesgue measure. If $2^{\aleph_0}$ is real-valued measurable, then the bounding number is small. Solovay's equiconsistency theorem for measurable and real-valued measurable cardinals.

Chapter 24. Elementary Submodels

Submodels vs. elementary submodels. The Tarski-Vaught Criterion. An application: Elementary chains. Skolemized models. The universes of elementary submodels form a closed unbounded set. An application: Proof of the Löwenheim-Skolem Theorem.

"Sufficiently large" fragments of ZFC and of its models. ZFC is not finitely axiomatizable. What do elementary submodels of "sufficiently large" fragments of the universe $V$ look like? Applications: Alternative proof of the Delta-System Lemma. Short proofs of two famous topological theorems of Arhangel'skii. Characterizations of strongly inaccessible and weakly compact cardinals in terms of elementary submodels. Characterization of measurability of $\kappa$ in terms of elementary embeddings of $V_{\kappa + 1}$. Construction of a normal ultrafilter. A measurable cardinal is not the first where CH fails.

Chapter 25. Boolean algebras

Boolean rings. Characterization of the algebra of regular open subsets of the Cantor set. The cardinalities of complete Booelan algebras. Interval algebras and linear generating systems. Vaught relations and Vaught's Theorem. Superatomic Boolean algebras and their Cantor-Bendixson characteristics. Erdös-Tarski theorem on attainment of cellularity. Balcar-Vojtas theorem on disjoint refinements. Characterization of $P(\omega)/Fin$ under CH. McKenzie's theorem on maximal irredundant sets. Baumgartner-Komjath theorem on embeddability in $P(\omega)$.

Chapter 26. Appendix: Some General Topology

Review of concepts and results from general topology that are used in the text.

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