Souslin Absoluteness, Uniformization and Regularity Properties of Projective Sets

The Extent of Definable Scales

The notion of a scale on a pointset is implicit in the classical proof of the Kondo uniformization theorem for E^ sets and in the uniformization theorem of Martin and Solovay [2] for X3 sets, from the assumption that measurable cardinals exist. It was formally isolated in Moschovakis [3], where it was shown (granting projective determinacy) that all projective sets admit projective scales — and...

Forcing absoluteness and regularity properties

The Continuum Hypothesis, the generic-multiverse of sets, and the Ω Conjecture

Is this really evidence (as is often cited) that the Continuum Hypothesis has no answer? Another prominent problem from the early 20th century concerns the projective sets, [8]; these are the subsets of Rn which are generated from the closed sets in finitely many steps taking images by continuous functions, f : Rn → Rn, and complements. A function, f : R→ R, is projective if the graph of f is a...

Cichoń's diagram, regularity properties and Δ13 sets of reals

We study regularity properties related to Cohen, random, Laver, Miller and Sacks forcing, for sets of real numbers on the ∆13 level of the projective hieararchy. For ∆12 and Σ 1 2 sets, the relationships between these properties follows the pattern of the well-known Cichoń diagram for cardinal characteristics of the continuum. It is known that assuming suitable large cardinals, the same relatio...

Bounds for the Regularity of Edge Ideal of Vertex Decomposable and Shellable Graphs