• absoluteness
• forcing
• stationary
• stationary tower
• ultrapower
• well-founded
• Woodin cardinal

We discuss the stationary tower, a forcing notion introduced by H. Woodin and based on a general notion of stationarity, and apply it to prove an absoluteness result for \(\undertilde{\Sigma}^2_1\) formulas. The thesis is divided into three chapters.
In the first chapter, we present the preliminaries, focusing in particular on Woodin cardinals.
In the second chapter, we describe the countable and the full stationary tower forcings (up to a strongly inaccessible cardinal \(\kappa\)). We will see that in both cases a \(V\)-generic \(G\) induces a tower of ultrapowers, and then we consider the direct limit of this directed system. Moreover, we will emphasize the case in which \(\kappa\) is a Woodin cardinal, since it guarantees that the direct limit is well-founded and closed under \(<\kappa\)-sequences in \(V[G]\).
In the third chapter, we apply the stationary tower forcing to prove that (under some large cardinal hypotheses) the Continuum Hypothesis fixes the truth values of \(\undertilde{\Sigma}^2_1\)-formulas with respect to forcing.