Slavitch, Noah2025-08-262025-08-262025-08-262025-08-19https://hdl.handle.net/10012/22275In this thesis, we describe the state of the field of generic absoluteness, that is, the study of which statements retain their truth value under any forced generic extension or series of generic extensions of $V$, along with what large cardinal assumptions are necessary or sufficient to cause formulas to become generically absolute. We detail some of the tools used in generic absoluteness, such as homogeneously Suslin trees, direct limits of models, extenders, and the incredibly powerful Stationary Tower used for stationary tower forcing, along with the effect specific large cardinal properties have when combined with the stationary tower. We describe various landmark generic absoluteness results including the following: Shoenfield's $\Sigma_2^1$-Absoluteness theorem, which establishes that all $\Sigma_2^1$ statements retain their truth across all transitive models of $ZF$ containing $\omega_1$. We also examine more advanced generic absoluteness results in the projective hierarchy, such as the equiconsistency projective absoluteness with the existence of $\om$-many strong cardinals, and that projective absoluteness follows from $\om$-many Woodin cardinals. Further up the hierarchy, we describe $\Sigma_1^2$-absoluteness conditioned on $CH$, which states that assuming a proper class of measurable Woodin cardinals and $CH$ holding in $V$, if $\phi$ is a $\Sigma_1^2$ statement and $G$ is a generic filter for $\p$ some partial order such that and $V[G] \models CH$, then $V \models \phi$ if and only if $V[G] \models \phi$. We also cover $\Sigma_2^2$-absoluteness with generic $\lozenge$, a generic form of Jensen's $\lozenge$ principle, and its relation to Neeman Games in $\Omega$-logic. Furthermore, we cover the generic absoluteness properties of universally Baire sets, $uB$, such as a proper class of Woodin cardinals implying for $A \in uB$ and and any generic extension $V[G]$ an elementary embedding $j:L(A, \R) \to L(A_{G}, \R_{G})$ such that $j(A) = A$ and $j\restriction_\R = id$, along with the more recent axiom of Sealing, which seeks to 'seal' the truth of the universally Baire sets, along with examinations of the $uB$-powerset of $uB$, $uBp$, such as Weak Sealing for $uBp$, a strengthening of Sealing. We examine particular inner models of interest, such as Chang-like models and their unique generic absoluteness properties. Finally, we give proofs of the Shoenfield $\bSigma_2^1$ absoluteness theorem, and the fact that $\bSigma^1_3$ absoluteness follows from a class of measurable cardinals.enMaster Thesis