Combining Forcing Notions

Abstract

In this chapter we shall investigate how one can combine various forcing notions. For this we first consider just two (not necessarily distinct) forcing notions, say ℙ=(P,≤ _P ) and ℚ=(Q,≤ _Q ).

The simplest way to combine ℙ and ℚ is to form the disjoint union of ℙ and ℚ (where conditions of ℙ are incomparable with those of ℚ). Obviously, a generic filter of the disjoint union is either ℙ-generic or ℚ-generic, and therefore, this construction is useless for independence proofs.

Another way to combine ℙ and ℚ is to build the product ℙ×ℚ=(P×Q,≤ _P×Q ). Since the forcing notion ℙ×ℚ belongs to V, forcing with ℙ×ℚ is in fact just a one-step extension of V. Products of forcing notions will be investigated in the first part of this chapter, where the focus will be on products of Cohen-forcing notions.

A more sophisticated way to combine ℙ and ℚ is to iterate ℙ and ℚ, i.e., we first force with ℙ and then—in the ℙ-generic extension—by ℚ. In this case, the forcing notion ℚ does not necessarily belong to V. To see this, let G be ℙ-generic over V and let ℚ=(Fn (G,2),⊆). Obviously, the forcing notion ℚ does not belong to V. However, since ℚ belongs to V[G], there is a ℙ-name $\mathop {\rm \mathbb{Q}}\limits_ \sim$ in V such $\mathop {\rm \mathbb{Q}}\limits_ \sim[G] = {\rm \mathbb{ Q}}$. Two-step iterations of this type are denoted by ${\rm \mathbb{P}}*\mathop {\rm \mathbb{Q}}\limits_ \sim$. In the second part of this chapter we shall see how to transform a two-step iteration into a one-step forcing extension. Furthermore, we shall see different ways to define general iterations of forcing notions.