to topological spaces. The Cantor space 2 and Baire space 
Chapter 8 introduces the notion of a model of set theory. Conditions are given under which a given set A can satisfy certain of the axioms, such as the Union Axiom, the Power Set Axiom, and so on. It is shown that the hereditarily finite sets satisfy all axioms except for the Axiom of Infinity. The topic of the possible models of fragments of the axioms is examined. In particular, we consider the axioms that are satisfied by Vα when α is for example a limit cardinal, or an inaccessible cardinal. The hereditarily finite and hereditarily countable, and more generally hereditarily < κ, sets are also studied in this regard. The hereditarily finite sets are shown to satisfy all axioms except regularity. This culminates in the proof that Vκ is a model of ZF if and only if κ is a strongly inaccessible cardinal.
Chapter 9 is a brief introduction to Ramsey theory, which studies partitions. This begins with some finite versions of Ramsey’s theorem and related results. There is a proof of Ramsey’s theorem for the natural numbers as well as the Erdős–Rado theorem for pairs. Uncountable partitions are also studied.
This additional material gives the instructor options for creating a course which provides the basic elements of set theory and logic, as well as making a solid connection with many other areas of mathematics.
Chapter 2
Review of Sets and Logic
In this chapter, we review some of the basic notions of set theory and logic needed for the rest of the book. There is a very close connection between the Boolean algebra of sets and the formulas of predicate logic. We will present some aspects of so-called naive set theory and indicate the methods of proof used there as a foundation for more advanced notions and theorems. Topics here will include functions and relations, in particular, orderings and equivalence relations, presented at an informal level. We will return to these topics in a more formal way once we begin to study the axiomatic foundation of set theory. Students who have had a transition course to higher mathematics, such as a course in sets and logic, should be able to go right to the next chapter.
2.1. The Algebra of Sets
In naive set theory, there is a universe U of all elements. For example, this may be the set
Definition 2.1.1. For any element a of U and any subsets A and B of U,
(1) a ∈ A ∪ B if and only if a ∈ A ∨ a ∈ B;
(2) a ∈ A ∩ B if and only if a ∈ A ∧ a ∈ B;
(3) a ∈ A∁ if and only if ¬ a ∈ A.
Here we use the symbols ∨, ∧, and ¬ to denote the logical connectives or, and, and not. We will frequently write x ∉ A as an abbreviation for ¬ x ∈ A.
The convention is that two sets A and B are equal if they contain the same elements, that is
This is codified in the Axiom of Extensionality, one of the axioms of Zermelo–Fraenkel set theory which will be presented in detail in Chapter 3. The family of subsets of U composes a Boolean algebra, that is, it satisfies certain properties such as the associative, commutative, and distributive laws. We will consider some of these now, and leave others to the exercises. We will put in all of the details at first, and later on leave some of them to the reader.
Proposition 2.1.2 (Commutative Laws). For any sets A and B,
(1) A ∪ B = B ∪ A;
(2) A ∩ B = B ∩ A.
Proof. (1) Let x be an arbitrary element of U. We want to show that, for any x ∈ U, x ∈ A ∪ B
Part (2) is left to the exercises.
The notion of subset, or inclusion, is fundamental.
Definition 2.1.3. For any sets A and B, we have the following conditions:
(1) A ⊆ B
(2) A ⊊ B
