Projective Well-orderings of the Reals

The bounded proper forcing axiom and well orderings of the reals

“…The well ordering obtained in Corollary 1 is in general not optimal. Friedman has shown the consistency of BPFA together with a Σ 1 3 well ordering of the reals by class forcing techniques, see [6].…”

Universally Baire Sets and Generic Absoluteness

“…A bit higher in the complexity hierarchy we reach an obstacle: the continuum hypothesis is a Σ 2 1 statement that cannot be generically absolute, so generic absoluteness principles at this level of complexity must be limited in some way in order to be consistent. One approach is to consider only the generic extensions that satisfy the continuum hypothesis.…”

“…One approach is to consider only the generic extensions that satisfy the continuum hypothesis. Woodin showed that the absoluteness of Σ 2 1 statements to such generic extensions follows from large cardinals (see Larson [8,Theorem 3.2.1]).…”

“…The approach we take in this article is to replace Σ 2 1 with (Σ 2 1 ) Γ where Γ is a pointclass and a statement is called (Σ 2 1 ) Γ if it has the form ∃A ∈ Γ (HC; ∈, A) |= ϕ for some statement ϕ in the language of set theory expanded by a unary predicate symbol. We will consider pointclasses Γ that are "well behaved" and in particular do not contain well orderings of R. The pointclass uB of universally Baire sets of reals and its local version uB are both examples of such pointclasses, and indeed Woodin has shown that if is a limit of Woodin cardinals then generic absoluteness holds for (Σ 2 1 ) uB statements with respect to generic extensions by posets of cardinality less than (see Steel [17,Theorem 6.1]).…”

“…In this article, we investigate generic absoluteness principles for pointclasses beyond (Σ 2 1 ) uB in relation to strong axioms of infinity and determinacy, and also in relation to the extent of the universally Baire sets. First, we consider generic absoluteness for ∃ R (Π 2 1 ) uB statements.…”