ABSTRACT
The goal of this chapter is to prove several preservation theorems for both countable and finite support iteration. Let us mention the following two kinds of preservation theorems. Let Φ be a property of forcing notions, and let https://www.w3.org/1998/Math/MathML"> P α , Q α : α < δ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064630/6df7636a-a9ba-4aa3-a87d-b69f569c8e4c/content/eq_9083.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be an iterated forcing (either finite or countable support iteration), and let https://www.w3.org/1998/Math/MathML"> P δ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064630/6df7636a-a9ba-4aa3-a87d-b69f569c8e4c/content/eq_9084.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be the limit of this system.
Assume that for each https://www.w3.org/1998/Math/MathML"> α < δ , ⊩ α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064630/6df7636a-a9ba-4aa3-a87d-b69f569c8e4c/content/eq_9085.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> " https://www.w3.org/1998/Math/MathML"> Q ˙ α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064630/6df7636a-a9ba-4aa3-a87d-b69f569c8e4c/content/eq_9086.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> has the property https://www.w3.org/1998/Math/MathML"> Φ . " https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064630/6df7636a-a9ba-4aa3-a87d-b69f569c8e4c/content/eq_9087.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Does https://www.w3.org/1998/Math/MathML"> P δ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064630/6df7636a-a9ba-4aa3-a87d-b69f569c8e4c/content/eq_9088.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> have the property Φ ?
Assume that for each https://www.w3.org/1998/Math/MathML"> α < δ , P α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064630/6df7636a-a9ba-4aa3-a87d-b69f569c8e4c/content/eq_9089.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> has the property Φ. Does https://www.w3.org/1998/Math/MathML"> P δ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064630/6df7636a-a9ba-4aa3-a87d-b69f569c8e4c/content/eq_9090.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> have the property Φ ?
