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

Stefan Hoffelner

doi:10.1142/S0219061322500088

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

Logic 2025-06-17 v6

Authors: Stefan Hoffelner

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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.

Keywords

sigma-1-3 separation property delta-1-3 definable set forcing model construction descriptive set theory mathias problem

Cite

@article{arxiv.1912.11811,
  title  = {The Consistency of the $\bf{\Sigma^1_3}$-Separation Property},
  author = {Stefan Hoffelner},
  journal= {arXiv preprint arXiv:1912.11811},
  year   = {2025}
}

Comments

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

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

Abstract

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