Our approach to forcing will use “Boolean-valued universes”, a method which originates with Scott and Solovay (see the notes to Chapter 7). It is therefore necessary for us to present some basic facts about Boolean algebras; we shall prove only the results that we shall require later. In fact, our discussion will be carried out in the more general context of partially ordered sets. In Example 2.2(vi) we shall describe examples of partially ordered sets which will be very prominent in later chapters.

In this chapter, we shall introduce the completion and the Stone space of a Boolean algebra B. The latter is the set of ultrafilters on B, and we shall recognize βK as the Stone space of the Boolean algebra P(N). This allows us to reformulate our basic question as a question about where U is a free ultrafilter on N.

We shall conclude the chapter by discussing the P-points of the compact space βN/N.

DEFINITION

Let P be a non-empty set. A strict partial order on P is a binary relation < such that:

(i) if a < b and b < c in P, then a < c;

(ii) a a for each a ε P.

A total order on P is a strict partial order < such that:

(iii) for each a, b ε P, either a < b or a = b or b < a.

A partially ordered set is a pair P = (P, <), where P is a non-empty set and < is a strict partial order on P.

Throughout this book, we shall work in the naïve set theory familiar to analysts: the formalization of this theory is the axiom system ZFC, which will be discussed in Chapter 4.

We first summarize some elementary facts about Banach algebras that we shall use. The account in the standard text of Rudin [58, Chapters 10 and 11] covers essentially all that we shall require.

All the algebras that we shall consider are linear and associative, and their underlying field is either the complex field C or the real field ℝ. An algebra is unital if it has an identity; in this case, we often denote the identity by e. The algebra formed by adjoining an identity to a non-unital algebra A is denoted by A#, and we take A# = A if A is unital.

The set of invertible elements of a unital algebra A is denoted by Inv A.

A character on a complex algebra A is a non-zero homomorphism from A onto ℂ. The set of characters on A is the character space of A, written ϕA.

Let A be a commutative algebra. An ideal I of A is modular if the quotient algebra A/I is unital. The (Jacobson) radical, written rad A, of A is defined to be the intersection of the maximal modular ideals of A; if A has no such ideals, then rad A = A, and in this case A is a radical algebra. (So our convention is that a radical algebra is necessarily commutative.) The algebra A is semisimple if rad A = {0}.

The purpose of this book is to explain what it means for a proposition to be independent of set theory, and to describe how independence results can be proved by the technique of forcing. We do this by presenting an application of forcing to a deep and interesting problem in analysis. Our application is, by current standards in set theory, fairly non-technical, and so it offers an excellent setting in which to exhibit to analysts these new techniques from set theory.

Most analysts will have a certain acquaintance with logic and set theory. They will know naïve set theory up to the level of ordinals and cardinals. They will have heard that forcing is a powerful technique that enables one to prove that certain propositions of set theory are independent of specified axioms, and, in particular, that Cohen developed the method of forcing in his proof that the Continuum Hypothesis (CH) is independent of the basic axioms of set theory, ZFC. They may also know of more recently formulated axioms, such as Martin's Axiom (MA), which can be used to establish independence results without the necessity of knowing any of the technicalities of forcing.

However, it is possible that analysts harbour two negative feelings about these matters. First, they may feel that, although logic and set theory are of interest in their own right, they have little to contribute concerning questions which “really” arise in mathematical analysis, and so can be safely left to their disciples.

We shall now complete our proof of the relative consistency with ZFC of the assertion that each norm on each algebra C(X,ℂ) is equivalent to the uniform norm. To do this, we shall construct a complete Boolean algebra A with

We recall that MA is Martin's Axiom (Definition 5.14) and that NDH is the sentence “For each compact space X, each homomorphism from C(X,ℂ) into a Banach algebra is continuous” (Definition 4.18). The existence of discontinuous homomorphism from C(X,ℂ) is equivalent to the existence of norms on C(X,ℂ) which are not equivalent to the uniform norm. We have explained how the construction of such a Boolean algebra gives our consistency result.

In this chapter, we write [[…]] for a Boolean value in VB. Let us discuss how we might construct a Boolean algebra A with [[MA]]A = 1 in VA.

Suppose, for example, that B is a complete Boolean algebra and that t is a term in VB which is a counter-example to MA in VB, in the sense that

[[‘t is a counter-example to MA’]]B = 1.

Then we shall show that there exists a complete Boolean algebra C containing B as a complete subalgebra such that

In this way, we can eliminate the counter-example t in VC. The hope is that, by iterating this process, we can exterminate all possible counter-examples to MA. Of course, new counter-examples could well arise at each stage of the iteration, and these must also be eliminated.