Amoeba-absoluteness and projective measurability

Amoeba-Absoluteness and Projective Measurability

We show that Σ4–Amoeba–absoluteness implies that ∀a ∈ R (ω L[a] 1 < ω V 1 ), and hence Σ3–measurability. This answers a question of Haim Judah (private communication).

Projective Absoluteness under Sacks Forcing

We show that Σ3-absoluteness under Sacks forcing is equivalent to the Sacks measurability of every ∆2 set of reals. We also show that Sacks forcing is the weakest forcing notion among all of the preorders which always add a new real with respect to Σ3 forcing absoluteness.

Generic absoluteness under projective forcing∗

We study the preservation of the property of L(R) being a Solovay model under projective ccc forcing extensions. We compute the exact consistency strength of the generic absoluteness of L(R) under forcing with projective ccc partial orderings and, as an application, we build models in which Martin’s Axiom holds for Σ ∼ 1 n partial orderings, but it fails for the Σ ∼ 1 n+1.

Souslin Absoluteness, Uniformization and Regularity Properties of Projective Sets

We show that Souslin Absoluteness and Projective Regularity holds ii Souslin Uniformization does. As a result, Souslin Absoluteness plus 1 n Projective Regularity implies 1 n+1 Projective Regularity. Another result is that 1 5 Souslin Absoluteness implies 1 4 Projective Regularity, and 1 6 Souslin Absoluteness implies 1 5 Projective Regularity.

Forcing absoluteness and regularity properties