By Jindrich Zapletal

This booklet unites descriptive set thought and definable right forcing and explores the family members among them. either forcing and descriptive set conception are defined independently, their sub-areas defined, following their dedication to one another. Containing unique examine, this article highlights the connections that forcing makes with different components of arithmetic, and is vital interpreting for educational researchers and graduate scholars in set idea, summary research, and degree idea.

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This is a nontrivial -ideal, the forcing PI is not proper though. Let M be a countable elementary submodel of a large structure and consider the Borel set B = x ∈ 2 x is Mgeneric . I will show that the set A ∩ B is Borel, so B ∈ I and PI is not proper. Suppose x ∈ B is a point. We have x ∈ A iff M x = x ∈ A (by analytic absoluteness) ˙ (by the forcing theorem) iff x ∈ C ∈ PI ∩M iff ∃C ∈ PI ∩M x ∈ C ∧C • x˙ gen ∈ A ˙ . Thus A ∩ B is Borel as desired. 4. Let X denote the Hilbert cube 0 1 . 5. Let I be the -ideal on X generated by the zerodimensional sets.

Suppose that I is a -ideal on a Polish space X such that the forcing PI is proper. Suppose that B ∈ PI is a Borel set, Y is a Polish space, f B → Y is a Borel function, and An n ∈ are analytic or coanalytic subsets of the space X. Then there is a Borel set C ⊂ B such that for every number n ∈ , the image f C ∩ An ⊂ Y is a Borel set. Proof. The expression y˙ = f˙ x˙ gen is a PI -name for a point in the space Y . Write ˙ where P adds the point y˙ and Q ˙ is the remainder forcing, adding the PI = P ∗ Q point x˙ gen .

Proof. On one hand, a review of the definitions shows that if B C D witness the ˙ x˙ and the latter set is in the ideal J by an statement I ⊥ J then B • C ∩ V ⊂ D gen absoluteness argument. On the other hand, if B • C˙ ∩ V ∈ J then there is a name ˙ ∈ J such that B • C˙ ∩ V ∈ I. 2, thinning out for a Borel set A ˙ =D ˙ x˙ . the set B if necessary, I can find a Borel set D ⊂ B × C such that B • A gen Using an absoluteness argument, thinning out the set B if necessary again, I can find the set D in such a way that its vertical sections are in the ideal J .

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