2 - Basics

Summary

Forcing with ideals

The key definition

Definition 2.1.1.Suppose that X is a Polish space and I is a σ-ideal on the space X. The symbol P_Idenotes the partial order of I-positive Borel sets ordered by inclusion.

I will always tacitly assume that the Polish space X is uncountable and the ideal I contains all singletons. There are several cases in which this will not hold, and they will be pointed out explicitly. Note that the poset P_I depends only on the membership of Borel sets in the ideal I, but it will frequently be of interest to look at the membership of non-Borel sets in I.

It is clear that the partial order P_I is not separative, and its separative quotient is the σ-algebra B(X) mod I. There is exactly one property all partial orders of this kind share.

Proposition 2.1.2.The poset P_Iadds an element ẋ_genof the Polish space X such that for every Borel set B ⊂ X coded in the ground model, B ∈ G iff ẋ_gen ∈ B.

Proof. It is easy to see that the closed sets contained in the generic filter form a collection closed under intersection which contains sets of arbitrarily small diameter.