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

TL;DR

This paper constructs models demonstrating the $f{ heta^1_3}$-separation property, showing that disjoint $f{ heta^1_3}$-sets can be separated by $f{ heta^1_3}$-definable sets, resolving a question from 1968.

Contribution

It provides the first known models where the $f{ heta^1_3}$-separation property holds, answering a longstanding open problem.

Findings

Constructed a model with $f{ heta^1_3}$-separation property.

Constructed a model with lightface $f{ heta^1_3}$-separation property.

Resolved an open question from 1968.

Abstract

We generically construct a model in which the $\bf{\Sigma^1_3}$-separation property is true, i.e. every pair of disjoint $\bf{\Sigma^1_3}$-sets can be separated by a $\bf{\Delta^1_3}$-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) $\Sigma^1_3$-separation property is true.