Generic Souslin sets

Homogeneously Souslin sets in small inner models

We prove that every homogeneously Souslin set is coanalytic provided that either (a) 0long does not exist, or else (b) V = K, where K is the core model below a μ-measurable cardinal. 1 Homogeneously Souslin sets. In this paper we shall deal with homogeneously Souslin sets of reals, or rather with sets of reals which admit an ω-closed embedding normal form. Definition 1.1 (Cf. [4, p. 92].) Let A...

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.

On Mathias Generic Sets

We present some results about generics for computable Mathias forcing. The n-generics and weak n-generics in this setting form a strict hierarchy as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n-generic with n ≥ 3 then it satisfies the jump property G(n−1) = G′ ⊕ ∅(n). We prove that every such G has generalized high degree, ...

Examples for Souslin Forcing

On full Souslin trees

In the present note we answer a question of Kunen (15.13 in [Mi91]) showing (in 1.7) that it is consistent that there are full Souslin trees. ∗ We thank the NSF for partially supporting this research under grant #144-EF67. Publication No 624.