• Iterated Forcing

This thesis belongs to the research area of iterated forcing, which is a powerful tool used in proofs of independence within set theory.
Forcing has been introduced by Cohen in 1963 in his proof of the independence of the continuum hypothesis from the ZF axiomatic system.

Soon after this, several mathematicians started using and expanding this technique. One of the main generalization has been iterated forcing.

While forcing consists of a transformation of a model into another one once and for all, iterated forcing makes infinite transformations which give additional power to the technique.

In this thesis we analyze two different approaches to iterated forcing and we prove that one of the two can simulate the other one.

The first approach is the poset iterated forcing and the second one is the complete Boolean forcing.

Chapter 1 of this thesis contains all the necessary background from combinatorics (posets, complete Boolean algebras, complete and regular embeddings, and the related theorems.)

Chapter 2 presents the two approaches to forcing. After giving the background definitions, it provides the basic theorems, without proofs, and shows the equivalence between the two approaches, in the basic, non iterated, version.

Chapter 3 is devoted to present Boolean iterated forcing. In the first part, we show with some detail the two-step iteration. In the second part, we present the general version.

Chapter 4 is organized as the previous one and is dedicated to poset forcing.

Chapter 5 contains the full proof of a result which has been folklore in the community, but it lack a rigorous proof in the current literature. This is the fact that Boolean iterated forcing can simulate poset iterated forcing.
We close the chapter with a wealth of conjectures, notably the possibility of the other direction of the equivalence. After a closer look at this, it becomes clear that such equivalence if true would not be trivial to prove.