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.