The Consistency of the ${\mathbf{\Sigma }}_3^1$-Separation Property

Abstract

We generically construct a model in which the ${\mathbf{\Sigma }}_3^1$-separation property is true, i.e. every pair of disjoint ${\mathbf{\Sigma }}_3^1$-sets can be separated by a ${\mathbf{\Delta }}_3^1$-definable set. This answers an old question from the problem list $"$Surrealist landscape with figures$"$ by A. Mathias from 1968. We also construct a model in which the (lightface) ${\Sigma }_3^1$-separation property is true.

42 pages. Slightly changed coding method which makes several definitions more transparent, while leaving the general flow of ideas and proofs almost unchanged

Cited by in corpus (4)

• Forcing Axioms, the Uniformization and the Basis Property
• Forcing the ${\Pi }_3^1$-Reduction Property and a Failure of ${\Pi }_3^1$-Uniformization
• $\mathsf{MA}(\mathcal{I}$) and a Failure of Separation on the third Level
• The global ${\Sigma }_{n+1}^1$-Uniformization Property and $\mathsf{BPFA}$